The Turing Valley: How AI Capabilities Shape Labor Income

Introduction

The rapid advancement of Artificial Intelligence (AI) is set to transform the future of work,² prompting widespread debate over its potential societal impact. While some contend that AI will boost productivity and drive economic growth, benefiting society as a whole, others worry that the technology could substitute human labor, reduce wages, and weaken the economic and political power of workers ([?]; [?]; [?]; [?]; [?]; [?]; [?]).

Motivated by this debate, this paper develops a model of a knowledge economy where production requires solving multidimensional problems, and firms organize production by matching humans with AI-powered machines in hierarchies designed to use knowledge efficiently. The novelty of our approach lies in placing the multidimensional nature of knowledge at the center of the analysis. This focus allows us to explore the possibility that AI may outperform humans in certain knowledge dimensions—such as pattern recognition or data analysis—while underperforming in others—such as reasoning and decision-making. This creates opportunities for human-AI collaboration to outperform both humans and AI alone.³

Using this model, we then seek to answer the following questions: Which types of AI improvements increase labor income, and which ones reduce it? Do AI advancements that allow humans to focus on more exceptional, less routine problems necessarily increase labor income? How does the AI that maximizes labor income compare to the AI that maximizes capital income (i.e., the income of machine owners)? How do these outcomes depend on the availability of computing power (or “compute”) and the user-friendliness of the AI technology?⁴

Formally, we consider a competitive economy where firms can employ humans or “machines” (i.e., units of compute) to produce. Production requires time and knowledge to solve problems, which vary in difficulty across two dimensions. Successful production occurs when the knowledge of the human or machine attempting a problem exceeds the problem’s difficulty in each dimension; in other words, both dimensions must be addressed for production to succeed. For example, in healthcare, patient care requires not only an accurate diagnosis but also clear and effective interaction with the patient.

Each human is endowed with one unit of time and an exogenous knowledge profile, which we assume to be identical for all humans. Machines, which are in fixed supply, also possess one unit of time each, and their knowledge is powered by an exogenous AI algorithm common to all machines. Hence, when the AI algorithm improves, the knowledge of all machines simultaneously improves. We impose no restrictions on the relationship between the knowledge profiles of humans and machines; machines may be inferior to humans in both dimensions of knowledge, superior in only one dimension, or superior in both.

Humans and machines can also communicate at a cost, enabling them to assist each other in solving problems. This interaction allows for the formation of two-layer organizations, where one factor specializes in production while the other specializes in problem-solving.⁵ Successful production in these organizations occurs when the combined human and machine knowledge in each dimension surpasses the problem’s difficulty. This interaction potentially creates knowledge synergies, meaning the set of problems that humans and machines can jointly solve may be larger than the union of the sets of problems each can solve alone.

As the baseline scenario, we examine the case where machines are sufficiently abundant so that the binding constraint in human-AI interactions is human time. In other words, not all machines can be matched with humans in two-layer organizations. This assumption is motivated by the exponential growth of computational capacity over the past two centuries [?]. In this case, the equilibrium rental rate of machines equals the fraction of problems they can solve independently, and labor captures the full value of human-machine collaboration. Furthermore, improvements in AI in any dimension of knowledge always increase total output, total capital income, and average labor productivity (defined as output per human). Our primary focus, however, is on understanding how advancements in AI impact equilibrium labor income, which is determined by the marginal product of labor.

Our first main result states that labor income decreases when AI improves in dimensions where its knowledge is currently inferior to humans but increases when AI improves in dimensions where it is already superior to humans. Viewed through the lens of [?]'s famous test of machine intelligence,⁶ improvements in AI in dimensions where machines cannot pass the Turing Test reduce labor income, while improvements in AI in dimensions where machines can pass the Turing Test increase it. For this reason—and given the topography that arises—we refer to the function that maps AI’s knowledge to labor income, illustrated in Figure 1, as the “Turing Valley.”

For intuition, let us focus on the case where machines are placed at the bottom layer, so they seek assistance from humans when they cannot solve the problem independently. AI improvements in dimensions where machines are stronger than humans increase labor income because they enhance the value of human-machine collaboration. This occurs for two distinct reasons. First, every new problem machines can now solve on their own is one that the organization could not solve before. Hence, machines now pose fewer questions that humans cannot answer, effectively improving the pool of problems humans have to deal with. Second, these improvements expand the set of problems that humans and machines can jointly solve more significantly than the set of problems machines can solve alone, i.e., they increase the extent of knowledge synergies between humans and machines.

In contrast, AI improvements in dimensions where machines are weaker than humans reduce labor income by reducing the value of human-machine collaboration. Such improvements only broaden the set of problems that machines can solve independently without increasing those that humans and machines can solve together. As a result, machines ask more difficult questions, worsening the pool of problems humans have to deal with.

Our second main result characterizes the AI that maximizes total labor income. We show that when humans are sufficiently weak in both knowledge dimensions, labor income is maximized when AI is as good as possible in both dimensions. Otherwise, labor income is maximized when AI simultaneously performs as poorly as possible in the dimension of knowledge where humans are relatively strong and as well as possible in the dimension where humans are relatively weak.

The reasoning is as follows: Machines that perform well in both dimensions are always useful for humans because they enable them to solve a large fraction of problems. However, these machines are also relatively expensive due to their ability to solve many problems independently. In contrast, machines that excel in one dimension but underperform in another are less costly—because they lack autonomy—and their usefulness depends on whether humans can complement the knowledge they lack. Therefore, unless humans are sufficiently strong in at least one dimension, the AI that maximizes labor income is the most advanced AI possible.

Our two main results have several important implications. First, AI advancements that further automate routine problems that could already be solved by combining human and machine knowledge necessarily reduce labor income. This occurs despite the fact that more knowledgeable machines allow humans to assist more machines with exceptional problems, as each machine requires less assistance, thereby increasing average labor productivity. The decrease in labor income happens because these improvements lead to a larger portion of output being captured by machines, outweighing the additional output generated by humans who assist a larger number of more advanced machines.

Second, if machines initially underperform humans across all dimensions of knowledge and AI improvements are gradual, these advancements will inevitably decrease labor income before potentially increasing it. Politically, pursuing or allowing technological advancements that initially reduce labor income could be contentious, especially since these initial reductions may only be reversed by substantial subsequent AI advancements—a scenario that may or may not materialize. Therefore, temporary labor income support policies might be necessary to address these political challenges. Alternatively, AI developers could be required to meet discrete improvement thresholds before deploying new AI models.

Third, the AI that maximizes labor income may differ significantly from the one that maximizes capital income (before considering the costs of developing this technology). This discrepancy occurs when humans are relatively strong in at least one dimension of knowledge. In this case, labor prefers relatively cheap, non-autonomous machines that can complement humans in the dimension where they are relatively weak. In contrast, capital prefers autonomous machines that can perform as well as possible in both dimensions of knowledge.

Our baseline setting assumes that (i) machines are abundant and (ii) all humans have the same knowledge profile. Our two main extensions relax each of these assumptions. In the first extension, we consider the general case where there is an arbitrary amount of machines. In this context, we identify a new force that can cause improvements in AI to reduce labor income: AI improvements may increase the amount of machines each human can assist to such an extent that the equilibrium shifts from a state where not all machines are matched with humans to one where they are. This shift induces a discontinuous drop in labor income, as labor is then forced to share with capital the value created by human-machine collaboration.

This extension also reveals that making machines more user-friendly—by reducing the communication costs between humans and machines—can be a double-edged sword from labor’s perspective. Specifically, while lower communication costs always increase labor income when machines are abundant (by increasing the value of human-machine collaboration), the resulting rise in machine demand from a more user-friendly technology can lead to a discontinuous drop in labor income akin to the one described above. More generally, this extension suggests that policymakers concerned about labor income might want to ensure an adequate supply of machines (i.e., sufficient compute) to keep machines abundant as AI advances.

In our second extension, we examine a model with two types of humans, each with distinct knowledge profiles. We show that our first main result—regarding which types of AI improvements increase total labor income—generalizes as follows: Total labor income decreases when AI improves in dimensions where it is inferior to all types of humans (i.e., machines fail the Turing Test against any human in that dimension) but increases when AI improves in dimensions where it surpasses all types of humans (i.e., machines pass the Turing Test against all humans in that dimension).

Regarding our second main result—the type of AI that maximizes labor income—we show that this result generalizes on a type-by-type basis. However, the same AI may not maximize the labor income of both types of humans. This result indicates that the direction of AI development could become politically divisive not only along capital-labor lines, as previously discussed, but also within the labor force itself.

The remainder of the paper is organized as follows: After reviewing the most relevant literature, we introduce our baseline model in Section 2. Section 3 characterizes the competitive equilibrium, and Section 4 presents our main results. Sections 5 and 6 discuss our two main extensions. Finally, Section 7 concludes.

Related Literature

The main contribution of this paper is the development of a new model of human-AI collaboration with multidimensional knowledge to analyze how various AI improvements impact labor income. In doing so, we contribute to two distinct literatures: the literature on knowledge hierarchies and the literature on the effects of technological progress on labor income.

In the context of the literature on knowledge hierarchies, the most closely related paper is our own recent work ([?]). That paper is the first to introduce AI into a canonical model of a knowledge economy: The knowledge hierarchies of [?], [?], and [?].⁷ The present paper differs from [?] in two key respects. First, the research question is different. [?] focus on characterizing the effects of AI on organizational and labor outcomes by comparing the equilibrium before and after AI’s introduction. In contrast, the present paper seeks to understand the effects of different types of AI improvements once AI is already in use. This question is important because this technology is improving at the same time as it is being deployed, and there is an active debate on whether the current direction of AI developments will benefit society as a whole or concentrate economic power among a select few ([?, ?, ?, ?, ?, ?]).

Second, while [?], in line with the previous literature on knowledge hierarchies, examines a setting where knowledge is unidimensional, the present paper introduces multidimensional knowledge. In this context, we show that organizations can be valuable for reasons beyond the traditional specialization motive found in one-dimensional models. For instance, when machines struggle with dimensions of knowledge in which humans excel—and vice versa—communication between humans and machines can allow them to solve problems that neither could solve independently.⁸

This paper is also related to the literature examining the effects of technological progress on labor income (e.g., [?, ?, ?, ?, ?, ?]). A key insight from this body of work is that the conventional wisdom—that technological progress increasing average labor productivity is ultimately beneficial for workers—is flawed. As explained by [?] and [?], this is because wages are determined by marginal productivity, and there is no guarantee that average and marginal productivity move together.⁹

We contribute to this line of work by studying the effects of a specific type of technological progress—AI advancements—on labor income. To do this, and motivated by the fact that AI primarily impacts knowledge work, we develop a framework inspired by the theory of knowledge hierarchies. Through this framework, we identify which types of AI improvements benefit or harm labor and identify the AI that serves labor’s best interests as a function of the knowledge of the existing population, the availability of compute, and the user-friendliness of the AI technology.

The Model

We begin by formally describing the model, followed by a discussion of its main elements and assumptions.

The Baseline Setting

The Basics.—There is a unit mass of identical humans, each endowed with one unit of time and exogenous knowledge $\boldsymbol{h}=(h_1,h_2)\in(0,1)^2$. Additionally, there are $\mu>0$ units of compute, normalized such that one unit of compute is equivalent to one unit of time. An exogenous AI algorithm provides these units of compute with knowledge $\boldsymbol{m}=(m_1,m_2)\in[0,1]^2$. For simplicity, we refer to one unit of compute as a “machine,” with each machine having one unit of “time.”

There are many identical competitive firms. Production occurs within these firms, which are the residual claimants of output. Time and knowledge are the only inputs. Firms have no fixed costs and a maximum of two layers.

Single-layer firms can be automated or non-automated. Single-layer automated firms hire a single machine to produce. This machine dedicates her full unit of time to a single production opportunity. Each production opportunity is associated with a problem with unknown difficulty $\boldsymbol{x}\equiv(x_1,x_2)$ drawn from a distribution with full support on $[0,1]^2$, cumulative distribution function $F$ and probability density function $f$. If the knowledge of the machine exceeds the problem’s difficulty in both dimensions, i.e., if $m_1\geq x_1$ and $m_2\geq x_2$, the machine solves the problem and produces one unit of output. Otherwise, no output is produced. Single-layer non-automated firms are identical to their automated counterparts, except that they use humans for production instead of machines.

Two-layer organizations combine humans and machines and can be bottom-automated or top-automated. Bottom-automated (“$b$”) firms rent machines for production and employ a single human to assist with problems the machines cannot solve. Production occurs when the combined knowledge of the human and machine exceeds the problem’s difficulty, i.e., if $\max\{m_1,h_1\}\geq x_1$ and $\max\{m_2,h_2\}\geq x_2$. Otherwise, no production occurs. Assistance consumes $c\in(0,1)$ units of the human’s time. Since each machine cannot solve the problem it encounters with probability $1-F(\boldsymbol{m})$, $b$ firms optimally employ $n(\boldsymbol{m})$ machines to fully utilize the human’s time, where:

Top-automated (“$t$”) firms are similar to $b$ firms, except the roles are reversed: Humans engage in production, while machines specialize in problem-solving. In this case, consulting with a machine consumes $c\in(0,1)$ units of the machine’s time. Consequently, $t$ firms always employ $n(\boldsymbol{h})$ humans to fully exploit the time of the machine that assists these humans.

Figure 2 depicts the four firm configurations that can potentially arise in equilibrium. Note that a two-layer organization never uses only humans (or only machines) in both layers of the organization, as the human (or machine) engaged in production would solve the same set of problems as the one specialized in problem-solving. A similar reasoning also implies that focusing on two-layer organizations is without loss of generality in this setting: organizations with more than two layers cannot solve any problems that two-layer organizations cannot solve.

For brevity, we refer to agents (humans or machines) in two-layer organizations as “workers” if they engage in production and as “solvers” if they specialize in problem-solving. We call agents in single-layer organizations “independent producers” as they engage in unassisted production work.

Wages, Prices and Profits.— Let $w$ be the wage of humans and denote by $r$ the rental rate machines. All agents in this economy are risk neutral and maximize their income. This implies, in particular, that labor supply is inelastic. We normalize the value of each unit of output to one.

The profit of a single-layer firm as a function of its type is given by:

In other words, the profit of a single-layer non-automated firm is the expected fraction of problems that a human independent producer can solve, $F(h)$, minus her wage, $w$. The profit of a single-layer automated firm is obtained similarly, with the human replaced by a machine.

The profit of a two-layer organization as a function of its type (i.e., $b$ or $t$) is given as follows:

where $m \mathbin{\vee} h$ denotes the maximum dimension per dimension between $m$ and $h$.¹⁰ In both cases, the profit of a firm is its expected output minus the cost of the resources it uses. For instance, in the case of a $b$ firm, its total expected output is $n(m)F(m \vee h)$, as such organization attempts $n(m)$ problems and successfully solves a fraction $F(m \vee h) = F(\max\{m_1,h_1\},\max\{m_2,h_2\})$ of them. In turn, the cost of the resources is $w+n(m)r$, as the firm employs one human solver and $n(m)$ machine workers.

Competitive Equilibrium.—Let $\alpha_s$, $\alpha_b$, and $\alpha_t$ be the mass of humans hired by single-layer non-automated, bottom-automated, and top-automated firms, respectively. Similarly, denote by $\mu_s$, $\mu_b$, and $\mu_t$, the mass of machines rented by single-layer automated, bottom-automated, and top-automated firms.

Definition 1 (Competitive Equilibrium). An equilibrium consists of a set of non-negative allocations $(\alpha_s,\alpha_b,\alpha_t,\mu_s,\mu_b,\mu_t)$ and a set of non-negative prices $(w,r)$, such that:

1. Firms optimally choose their structure (while earning zero profits).

2. Single-layer non-automated firms hire a mass $\alpha_s$ of humans.

3. Single-layer automated firms hire a mass $\mu_s$ of machines.

4. $b$ firms hire a mass $\alpha_b$ of humans and rent a mass $\mu_b=\alpha_b n(m)$ of machines.

5. $t$ firms hire a mass $\alpha_t$ of humans and rent a mass $\mu_t=\alpha_t n(h)^{-1}$ of machines.

6. Markets clear: $\alpha_s+\alpha_b+\alpha_t=1$ and $\mu_s+\mu_b+\mu_t=\mu$.

Machines are “Abundant”.—As our baseline, we focus on the case where machines are sufficiently abundant so that the binding constraint in human-AI interactions is human time, not machine time. More precisely, we assume that:

This condition ensures that, regardless of machine knowledge $m \in[0,1]^2$, there are always more machines available than those demanded in equilibrium by two-layer organizations, i.e., $\mu> \alpha_b^*n(m)+\alpha_t^*/n(h)$.¹¹ In Section 5, we consider the general case where condition (1) does not necessarily hold.

Discussion of the Model

Before proceeding with the analysis, we briefly comment on the main assumptions underlying our model.

First, our framework emphasizes the multidimensional nature of knowledge. This focus enables us to explore the possibility that AI may outperform humans in certain knowledge dimensions while underperforming in others. This creates opportunities for human-AI collaboration to outperform both humans and AI alone. Moreover, it allows for the possibility that machines may excel in certain dimensions of knowledge but still lack “autonomy” because they struggle in other dimensions.

Second, we assume that communication between humans and machines allows them to combine their knowledge, potentially creating knowledge synergies—meaning the set of problems that humans and machines can jointly solve may be larger than the union of the sets of problems each can solve alone. In the Online Appendix, we show the sense in which our results hold when humans and machines cannot combine their knowledge (so solving a problem requires that one of them knows how to solve it independently).

Third, we assume that humans and machines cannot identify the specific dimensions in which they fail to solve a problem. Consequently, they cannot selectively choose which problems to seek assistance with among those they cannot solve. If, however, humans and machines could identify exactly which dimensions they know how to solve and which they do not, firms would optimally develop complex protocols—which could, in principle, be contingent on the knowledge of humans and machines, as well as the difficulty of the problem at hand—dictating when production workers should escalate unresolved issues. While this extension is intriguing, the complexity it introduces is beyond the scope of this paper and is left for future research.

Fourth, to focus on the matching between humans and machines rather than between humans, our baseline model assumes that all humans share the same knowledge profile. However, in Section 6, we show how our main insights generalize to the case in which there are two different types of humans. The analysis of this multidimensional model with more general distributions of human knowledge is beyond the scope of this paper and is left for future research.

Fifth, we assume that all machines are equipped with the same AI and that when AI improves, the knowledge of all machines simultaneously improves. This assumption is motivated by the fact that algorithms, as digital information, are non-rival and have nearly zero marginal reproduction costs [?]. Consequently, even if multiple AI algorithms exist, all firms will choose to equip their “machines” (i.e., the units of compute they rent) with the best available AI.

Finally, in the baseline model, we assume that machines are sufficiently abundant, making human time the limiting factor in human-AI interactions.¹² This assumption is motivated by the exponential growth of computational capacity over the past two centuries ([?]). However, some commentators are beginning to caution that constraints on energy and chip supply could cause the availability of computational resources to lag behind rising demand.¹³ For this reason, in Section 5 we consider the general case in which machines may or may not be abundant.

Equilibrium Characterization

Our main goal is to characterize how equilibrium labor income depends on machine knowledge. Towards this end, this section characterizes the competitive equilibrium of the economy. In Section 4, we present our main results.

The abundance of machines relative to humans implies that a positive mass of machines must be rented by single-layer automated firms. Hence, by the zero-profit condition of these firms, $r^*=F(\mathbf{m})$. In other words, the equilibrium rental rate of machines is equal to the fraction of problems they can solve on their own.

Define $\bar{w}_i$ as the highest wage an organization of type $i \in\{s,b,t\}$ can offer, given that $r^*=F(\mathbf{m})$. The wage $\bar{w}_i$ is determined by the zero-profit condition of a firm of type $i$:

Finally, let $R_s=\{\mathbf{m}\in[0,1]^2:\bar{w}_s\ge\max\{\bar{w}_b,\bar{w}_t\}\}$, $R_b=\{\mathbf{m}\in[0,1]^2:\bar{w}_b>\max\{\bar{w}_s,\bar{w}_t\}\}$, and $R_t=[0,1]^2\setminus(R_s\cup R_b)$. We then have the following proposition:

Proposition 1. The competitive equilibrium maximizes total output. The equilibrium rental rate of machines is $r^*=F(\mathbf{m})$ and the equilibrium wage is $w^*=\max\{\bar{w}_s,\bar{w}_b,\bar{w}_t\}$. Moreover:¹⁴

• If $\mathbf{m}\in R_s$, only single-layer organizations form: $\alpha_s^*=1$ and $\mu_s^*=\mu$.

• If $\mathbf{m}\in R_b$, only $b$ and single-layer automated organizations form: $\alpha_b^*=1$, $\mu_b^*=n(\mathbf{m})$, and $\mu_s^*=\mu-\mu_b^*$.

• If $\mathbf{m}\in R_t$, only $t$ and single-layer automated organizations form: $\alpha_t^*=1$, $\mu_t^*=n(\mathbf{h})^{-1}$, and $\mu_s^*=\mu-\mu_t^*.$

Proof. See Appendix A.1.

future-f0c256b1 (accessed August 21, 2024).

The equilibrium maximizes total output because the First Welfare Theorem holds in this economy (there is perfect competition, complete information, and all externalities are pecuniary). Furthermore, the equilibrium price of each input is equal to its marginal product, defined as the increase in output from introducing one additional unit of the input into this economy.

In particular, given that machines are sufficiently abundant relative to humans, the increase in output from introducing an additional machine is its expected output as an independent producer. This is because the new machine will necessarily be rented by a single-layer firm. Thus, the (equilibrium) marginal product of machines, MPK, is $F(m)$, which is equal to $r^*$.

Similarly, the marginal product of labor in a firm type $i$—the increase in output from introducing one additional human and employing her at firm $i$—is exactly $\bar{w}_i$, as defined by (1). For instance, the marginal product of labor in top-automated firms is $\bar{w}_t = F(m \vee h) - F(m)/n(h)$. This is because introducing an additional human in the economy and employing her at such a firm produces $F(m \vee h)$ more units of output, but requires taking $1/n(h)$ machines out of independent production, which has an associated output cost of $F(m)/n(h)$.

Consequently, the competitive equilibrium allocates all available humans to the firms with the highest marginal product of labor, implying that the (equilibrium) marginal product of labor (MPL) is:

Moreover, since firms earn zero profits, the equilibrium total output, which we denote by $Y^*$, is equal to the sum of machine (i.e., capital) income and labor income:

Figure 3 shows the regions $R_s$, $R_b$, and $R_t$ as a function of $m$ when problem difficulty is uniformly distributed, i.e., $F(x) = x_1x_2$. Three fundamental properties are apparent from this figure. First, $R_s$ defines a region around $h$, indicating that two-layer organizations form only when machines differ significantly from humans. Second, when machines are significantly worse than humans in either dimension, only $b$ firms emerge. Finally, $t$ firms only arise when machines are sufficiently good in both dimensions.

The intuition behind these properties relates to the value of forming two-layer organizations. Such organizations are beneficial when they enable the top layer—whether human or machine—to generate more output than under independent production. This benefit depends on two key factors. First, in two-layer organizations, solvers address more problems than in independent production because providing assistance requires less than a full unit of time (i.e., $c < 1$). Second, the likelihood of successfully solving each problem differs from that of an independent producer due to two reasons: (i) the top layer receives a different set of problems because the bottom layer has already attempted them, and (ii) the combination of human and machine knowledge allows the top layer to solve a higher proportion of problems.

When machines and humans are similar to each other, two-layer organizations do not emerge because the probability that the top layer can help solve a problem already attempted by the first layer is small. Hence, in this case, every agent’s time (human or machine) is better spent in production than in providing assistance.

When two-layer organizations arise, the key question is whether machines will be placed at the bottom or top layer. When humans outperform machines in both dimensions, machines are necessarily placed at the bottom because if a human cannot solve a problem, neither can the machine. For the same reason, when machines surpass humans in both dimensions, humans must necessarily be at the bottom layer.

In contrast, when humans outperform machines in one dimension but underperform in the other (e.g., $m_1 > h_1$ and $m_2 < h_2$), then both $b$ and $t$ firms are viable because both humans and machines can solve problems that the other cannot. In this case, top-automated firms arise if machines significantly surpass humans in one dimension and perform similarly to humans in the other dimension. The reason is that the price of machines is relatively high in this case, and $t$ firms use fewer machines per human than $b$ firms.

We finish this section by showing that the three fundamental properties of Figure 3 discussed above hold regardless of the distribution $F$:

Proposition 2. $R_s$, $R_b$, and $R_t$ are non-empty. Moreover:

• $R_s$ contains a neighborhood of $h=(h_1,h_2)$.

• $R_b \cap([h_1,1] \times[h_2,1]) = \varnothing$. Furthermore,

  • For every $m_1 \in[0,1]$, there exists $\delta> 0$ such that $(m_1,\epsilon) \in R_b$ for all $\epsilon\in[0,\delta)$.

  • For every $m_2 \in[0,1]$, there exists $\delta> 0$ such that $(\epsilon,m_2) \in R_b$ for all $\epsilon\in[0,\delta)$.

• $R_t \cap([0,h_1) \times[0,h_2)) = \varnothing$. Furthermore, there exists $\delta> 0$ such that $(1-\epsilon,1-\epsilon) \in R_t$ for all $\epsilon\in[0,\delta)$.

Proof. See Appendix A.2. $\square$

The Turing Valley

A superior AI algorithm—resulting in higher machine knowledge—always increases equilibrium total output $Y^*$, capital income $\mu r^* = \mu F(\mathbf{m})$, and average labor productivity (defined as output per human).¹⁵ This section, which presents our main results, characterizes the equilibrium labor income as a function of machine knowledge, i.e., $w^* : [0,1]^2 \to\mathbb{R}_{\geq0}$. For reasons that will become clear shortly, we refer to this map as the “Turing Valley.”

How Labor Income Changes with Machine Knowledge

Let $\operatorname{int} A$ denote the interior of an arbitrary set $A$. Our first main result is the following:

Proposition 3. $w^*$ is continuous and almost everywhere differentiable in $\mathbf{m}$. Moreover, for $i = 1,2$:

1. If $\mathbf{m} \in\operatorname{int} R_s$, then $\partial w^*/\partial m_i = 0$.

2. If $\mathbf{m} \in(\operatorname{int} R_b \cup\operatorname{int} R_t)$ and $m_i < h_i$, then $\partial w^*/\partial m_i \leq0$ (with strict inequality if $m_{j \ne i} \ne0$).

3. If $\mathbf{m} \in(\operatorname{int} R_b \cup\operatorname{int} R_t)$ and $m_i > h_i$, then $\partial w^*/\partial m_i > 0$.

Proof. See Appendix B.1. $\square$

According to Proposition 3, when humans and machines match in equilibrium, improvements in AI in dimensions in which machines are worse than humans decrease labor income. In contrast, improvements in AI in dimensions in which machines are better than humans increase labor income. Viewed through the lens of [?]’s famous test of machine intelligence, advancements in AI in dimensions where machines fail the Turing Test reduce labor income, while advancements in areas where machines pass the Turing Test increase it. For this reason—and considering the resulting topography—we refer to the map $w^* : [0,1]^2 \to\mathbb{R}_{\geq0}$ as the “Turing Valley.” Figure 4 illustrates the Turing Valley.

The high-level intuition behind this result is as follows. First, because machines are abundant, labor captures the entire value of forming organizations (i.e., the value of matching humans and machines in two-layer firms). Second, conditional on matching, improvements in AI make machines more useful for humans (because they allow them to solve more problems) but also more expensive (as machines can solve more problems on their own).

Proposition 3 follows because improvements in AI in dimensions where machines are worse than humans make machines more expensive than useful, while improvements in AI in dimensions where machines are better than humans make machines more useful than expensive. The exact reason why this happens differs when $\mathbf{m} \in R_b$ compared to $\mathbf{m} \in R_t$, so we consider each case individually.

When $\mathbf{m} \in R_b$, humans are solvers assisting the production work of machines. The equilibrium wage is given by:

This means that the equilibrium wage, or marginal product of labor, is equal to the number of times a human can provide assistance, $1/c$, multiplied by the probability that such assistance is useful. Hence, in bottom-automated firms, AI improvements make machines more useful than expensive when they increase the likelihood that human assistance is useful.

In this scenario, increases in $m_i$ when $m_i < h_i$ reduce the likelihood that human assistance is useful because such an improvement only allows machines to solve problems that the organization was already solving. As a result, machines ask more difficult questions, worsening the pool of problems humans have to deal with. This effect is illustrated in Figure 5, panel (a) in the case of an increase in $m_2$ when $m_1 > h_1$ and $m_2 < h_2$.

In contrast, an increase in $m_i$ when $m_i > h_i$ increases the likelihood that human assistance is useful. This occurs for two reasons. First, every new problem machines can now solve on their own is one that the organization could not solve before. Hence, machines now pose fewer questions that humans are unable to answer, effectively improving the pool of problems humans have to deal with. Second, an increase in $m_i$ when $m_i > h_i$ also increases knowledge synergies: there is a new set of problems that humans and machines can now jointly solve, but neither can solve alone. Figure 5, panel (b) illustrates these two effects in the case of an increase in $m_1$ when $m_1 > h_1$ and $m_2 < h_2$.

Consider next $m \in R_t$. Humans now engage directly in production and receive the assistance of machines when they cannot solve a problem on their own. The equilibrium wage is given by:

That is, the equilibrium wage is equal to the output per human $F(m \vee h)$ (note that each human here tackles a single problem and solves its with probability $F(m \vee h)$ with the help of machines) minus the cost of the machines each human uses, $F(m)/n(h)$.

In this scenario, increases in $m_i$ when $m_i < h_i$ reduce labor income because they raise the price of machines $F(m)$ without increasing the set of problems that a human can solve with the help of machines $F(m \vee h)$. Hence, this improvement only allows machines to appropriate a larger share of the organization’s output. In contrast, increases in $m_i$ when $m_i > h_i$ increases $F(m \vee h) - F(m)$ (and, therefore, $F(m \vee h) - F(m)/n(h)$) because there is a new set of problems that humans and machines can now jointly solve, but neither can solve alone.

Proposition 3 has three noteworthy implications. First, improvements in AI that further automate routine problems that could already be solved by combining human and machine knowledge necessarily reduce labor income. This happens even though more knowledgeable machines enable humans to assist more machines with exceptional problems, as each machine requires less assistance, thereby increasing average labor productivity. Put differently, AI improvements that further automate routine problems increase the average labor productivity while simultaneously reducing the marginal product of labor.

Second, since total output is the sum of capital and labor income, Proposition 3 implies that when AI is inferior to humans, improvements in AI result in capital income gains that exceed the additional output created. In contrast, once AI surpasses human capabilities, AI improvements result in larger output gains than capital income gains. Thus, in this case, both capital and labor benefit from AI improvements.

Third, if AI initially performs worse than humans in all dimensions and its improvements are gradual, these improvements will necessarily decrease labor income before potentially increasing it. This phenomenon is illustrated in Figure 6. Politically, pursuing or allowing technological advancements that initially reduce labor income may be infeasible. This is especially true if the initial reductions in labor income can only potentially be reversed by sufficiently large subsequent advancements—a scenario that may or may not materialize.

The fact that gradual AI progress necessarily decreases labor income before potentially increasing it—and the political backlash this may entail—poses a serious challenge for society. Prohibiting AI improvements could be highly inefficient in terms of output and detrimental to future labor income. Temporary labor income support policies might help alleviate these political concerns. Another approach could be requiring AI developers to meet discrete thresholds of improvements before deploying new AI models.

The AI that Maximizes Labor Income

The previous section analyzed how marginal changes in machine knowledge affect labor income. This section describes the AI that maximizes labor income globally, i.e., the peak of the Turing Valley. The key insight is that the AI maximizing labor income may differ significantly from the AI that maximizes capital income and total output.

We start with the following definition:

Definition 2 (Stronger Dimension). Humans are stronger in dimension 2 than in dimension 1 if:

Otherwise, humans are weaker in dimension 2 than in dimension 1.

Intuitively, humans are stronger in dimension 2 than in dimension 1 if the fraction of problems that humans cannot solve on their own because they lack knowledge in dimension 2, i.e., $F(h_1,1)-F(h_1,h_2)$, is smaller than the fraction of problems that they cannot solve on their own because they lack knowledge in dimension 1, i.e., $F(1,h_2)-F(h_1,h_2)$.

For the remainder of this section, we assume, without loss of generality, that humans are stronger in dimension 2 than in dimension 1. Hence, we restrict attention to $(h_1,h_2) \in(0,1) \times[h_2(h_1),1)$, where $h_2:(0,1) \to(0,1)$ is strictly increasing in $h_1$ and implicitly defined by $F(h_1,1)=F(1,h_2(h_1))$.

The following result, which is our second main result, shows that if humans are sufficiently weak in both dimensions (i.e., both $h_1$ and $h_2$ are sufficiently small), then labor income is maximized when AI is as good as possible in both dimensions, i.e., at $m=(1,1)$. Otherwise, labor income is maximized when AI simultaneously performs as poorly as possible in the dimension where humans are strongest and as well as possible in dimensions where humans are weakest, i.e., at $m=(1,0)$.

Proposition 4. There exist $\bar{h}_1 \in(0,1)$ such that the following holds:

• If $h_1 \ge\bar{h}_1$, then $m=(1,0)$ maximizes labor income.

• If $h_1 < \bar{h}_1$, then there exists $\bar{h}_2(h_1) \in[h_2(h_1),1)$ such that:

  • If $h_2 \in[\bar{h}_2(h_1),1)$, then $m=(1,0)$ maximizes labor income.

  • If $h_2 \in[h_2(h_1),\bar{h}_2(h_1))$, then $m=(1,1)$ maximizes labor income.

Proof. See Appendix B.2.

Figure 7 provides an illustration of Proposition 4. To understand this result, note that Proposition 3 implies that equilibrium labor income has four local maxima, one in each vertex of $[0,1]^2$. Moreover, by Proposition 2, the three local maxima $(0,0)$, $(1,0)$ and $(0,1)$ are in $R_b$, while the local maximum $(1,1)$ is in $R_t$. As a result, total labor income at each of these local maxima is:

Clearly, $w^*(0,0)$ is strictly dominated by either $w^*(1,0)$ or $w^*(0,1)$. Moreover, because humans are stronger in dimension 2 than in dimension 1—that is, $F(1,h_2) \ge F(h_1,1)$—$w^*(0,1)$ is dominated by $w^*(1,0)$.¹⁶ Hence, to determine the point at which labor income achieves its global maximum, we just need to compare $w^*(1,0)$ with $w^*(1,1)$.

When humans are sufficiently strong in their strongest dimension, $w^*(1,0)$ dominates $w^*(1,1)$. This occurs because humans add more value by assisting machines with $m=(1,0)$ than by engaging directly in production with the assistance of machines with $m=(1,1)$. Specifically, humans can assist $1/c$ machines of type $m=(1,0)$, and each machine, with assistance, can solve a relatively large fraction of problems. This approach generates more output than successfully addressing a single problem with the help of machines of type $m=(1,1)$.

Conversely, when humans are weak in both dimensions, $w^*(1,1)$ dominates $w^*(1,0)$ because at-

Notes. $F(x)=x_1x_2$. Parameter values: Both panels have $c=1/2$. For panel (a), $h_1=1/6$ and $h_2=1/5$, while for panel (b), $h_1=1/6$ and $h_2=4/5$. tempting the $1/c$ problems from machines with $m=(1,0)$ would result in a large fraction of failures, as humans would struggle to provide useful assistance. Therefore, focusing on a single problem with an assured success is more beneficial in this scenario.

An important implication of Proposition 4 is that whether the same AI simultaneously maximizes output, capital income, and labor income (before considering the costs of developing this technology) depends on whether the world resembles panel (a) or panel (b) of Figure 7. In panel (a), where AI can significantly surpass human performance in all dimensions, output, capital, and labor income are jointly maximized with the best possible AI.

In contrast, in panel (b), where AI cannot significantly outperform humans in certain dimensions of knowledge, the AI that maximizes output and capital income ($m=(1,1)$) differs substantially from the AI that maximizes labor income ($m=(1,0)$). Moreover, in this scenario, the AI that maximizes labor income yields no capital income, indicating a potentially significant conflict between the interests of labor and capital.

Extension I: Non-Abundant Machines

In the baseline model, we assume that machines are sufficiently abundant, i.e., condition (1) holds. This assumption implies that regardless of $m\in[0,1]^2$, the supply of machines always exceeds the equilibrium demand for machines from two-layer organizations. Hence, $r^*$ is determined by the zero-profit condition of single-layer automated firms.

In this section, we consider the general case with an arbitrary amount of machines. We start by noting that if $\mu<1/n(h)$, then humans are always abundant, in the sense that not all humans can match with machines in equilibrium. As a result, labor income in this case is $F(h)$ regardless of $m$, and the equilibrium can be characterized in a manner analogous to the characterization in Section 3. Hence, in this section we focus on the case $\mu\ge1/n(h)$.

In this scenario, we identify a new force that can cause improvements in AI to reduce labor income: AI improvements may increase the number of machines each human can assist to such an extent that the equilibrium shifts from a state where not all machines are matched with humans to one where they are. This shift induces a discontinuous drop in labor income, as labor is then forced to share with capital the value created by human-machine collaborations.

Equilibrium Characterization

Let $K\equiv\{m\in[0,1]^2:\mu\ge n(m)\}$ and $R_m\equiv\{m\in[0,1]^2:\overline{w}_b-\overline{w}_s\le n(m)n(h)(\overline{w}_t-\overline{w}_s)\}$. The following result generalizes Proposition 1 of Section 3.

Proposition 5. The competitive equilibrium maximizes total output. Moreover, if $m\in R_s\cup R_t$ or $m\in R_b\cap K$, the equilibrium is as described in Proposition 1. Otherwise,

• If $m \in R_b \cap K^C \cap R_m$, then only $b$ and $t$ organizations form: $\alpha_b^* + \alpha_t^* = 1$ and $\mu_b^* + \mu_t^* = \mu$ (where $\mu_b^* = \alpha_b^* n(m)$ and $\mu_t^* = \alpha_t^* n(h)^{-1}$). Equilibrium prices are:

• If $m \in R_b \cap K^C \cap R_m^C$, then only $b$ and single-layer non-automated organizations form: $\alpha_b^* = \mu/n(m)$, $\alpha_s^* = 1-\alpha_b^*$, and $\mu_b^* = \mu$. Equilibrium prices are $r^* = F(m \vee h)-F(h)/n(m)$ and $w^* = F(h)$.

Proof. See Appendix C.1.

According to the proposition, if $m$ is in $R_s$, $R_t$, or in $R_b \cap K$, then the equilibrium with abundant machines, as described in Proposition 1, remains the equilibrium in this broader context where $\mu\ge1/n(h)$. Intuitively, this occurs because the number of available machines exceeds the demand by

two-layer organizations when $m \in R_s \cup R_t \cup(R_b \cap K)$ and prices are $w^* = \max\{\bar{w}_s, \bar{w}_b, \bar{w}_t\}$ and $r^* = F(m)$.

The difference between Propositions 1 and 5 arises when $m \in R_b \cap K^C$. In this case, there are insufficient machines to employ all humans in bottom-automated firms (which would be the equilibrium outcome if machines were sufficiently abundant). Consequently, the market clears by reallocating some humans to either top-automated firms (when $m \in R_b \cap K^C \cap R_m$) or single-layer non-automated firms (when $m \in R_b \cap K^C \cap R_m^C$). In the former case, the equilibrium wage is strictly between $\bar{w}_s$ and $\bar{w}_b$, while in the latter case it is equal to $\bar{w}_s$.

Figure 8 shows the evolution of the equilibrium as $\mu$ increases when problems are uniformly distributed. Panel (a) represents the case where $1/n(h) \leq\mu< n(0)$. Here, regardless of $m$, there are insufficient machines for all humans to be employed in bottom-automated firms (so $R_b \cap K = \varnothing$). Panels (b) and (c) demonstrate how, as $\mu$ increases, the set $K$ progressively covers a larger portion of $R_b$, eventually covering the entire set $R_b$, as shown in panel (d). In the latter case, the equilibrium is the same as in the baseline.

How Labor Income Changes with Machine Knowledge

We now analyze how equilibrium labor income is influenced by machine knowledge. For the result that follows, let $\partial K$ be the set of boundary points of $K$.

Proposition 6.

1. $w^*$ and $r^*$ are continuous in $m$ for all $m \notin\partial K \cap\operatorname{int} R_b$. However, if $m \in\partial K \cap\operatorname{int} R_b$, then marginal increases in $m_1$ and $m_2$ lead to a discontinuous drop in $w^*$ and a discontinuous increase in $r^*$.

2. When $m$ is in the interior of $R_s$, $R_t$ or $R_b \cap K$, the effects of marginal changes of $m_i$ on $w^*$ are as described in Proposition 3. Otherwise,

(a) If $m \in(\operatorname{int} R_b \cap K^C \cap R_m^C)$, then $\partial w^*/\partial m_i = 0$.

(b) If $m \in\operatorname{int}(R_b \cap K^C \cap R_m)$ and $m_i < h_i$, then $\partial w^*/\partial m_i \leq0$ (with strict inequality if $m_{j \ne i} \ne0$).

(c) If $m \in\operatorname{int}(R_b \cap K^C \cap R_m)$ and $m_i > h_i$, then $\partial w^*/\partial m_i > 0$ provided that $c \leq\sqrt{2}/2$.¹⁷

Proof. See Appendix C.2.

Figure 9 illustrates Proposition 6. Panels (a) and (b) compare labor income under two scenarios: when $K \supset R_b$ (so the supply of machines exceeds equilibrium demand from two-layer organizations for all $m \in[0,1]^2$, as analyzed in Sections 3 and 4) and when $K \subset R_b$ (so the equilibrium demand for machines from two-layer organizations may or may not exceed supply depending on $m$). As shown in panel (b), labor income drops discontinuously at the boundary of $K \cap R_b$. Panels (c) and (d) illustrate the equilibrium capital income for the same situations of panels (a) and (b), respectively. As can be seen in panel (d), capital income increases discontinuously whenever labor income decreases discontinuously.

The intuition for the discontinuity of $w^{*}$ and $r^{*}$ was outlined above: When $m \in R_b$, an increase in machine knowledge raises the demand for machines by bottom-automated firms. This surge in demand can shift the equilibrium from a state where machine supply exceeds demand from bottom-automated organizations to one where demand from these organizations fully consumes the available supply. Consequently, the equilibrium wage and price of machines experience a discontinuous

change, as labor is compelled to share with capital the value created by forming two-layer organizations.

Regarding the effects of marginal changes of $m_i$ on $w^*$ when $\mathbf{m}$ is in the interior of $R_b \cap K^C \cap R_m$, note that $w^*$ can be expressed as:¹⁸

This indicates that the equilibrium wage is a fraction $(n(\mathbf{h})-1)/(n(\mathbf{h})-1/n(\mathbf{m}))$ of the expected output of a human worker. Consequently, when $m_i < h_i$, an increase in $m_i$ reduces labor income because it does not alter the expected output per worker $F(\mathbf{m}\vee\mathbf{h})$, but decreases the share of this output that accrues to labor $(n(\mathbf{h})-1)/(n(\mathbf{h})-1/n(\mathbf{m}))$, as machines become more scarce.

In contrast, when $m_i > h_i$, increases in $m_i$ have two opposing effects on $w^*$: they raise the expected output per worker, $F(\mathbf{m}\vee\mathbf{h})$, but simultaneously decreases the share of output that accrues to labor. While either effect could dominate in general, we show in the proof of Proposition 6 that the first effect necessarily dominates if $c \le\sqrt{2}/2 \approx0.7071$.

We conclude this section by emphasizing four implications of Proposition 6. First, as noted in Section 4.1, gradual AI progress initially reduces labor income before eventually increasing it, which may call for temporary labor income support policies. Proposition 6 further suggests that policymakers concerned about labor income might want to ensure an adequate supply of machines (i.e., sufficient compute) to keep machines abundant as AI advances.

Second, improvements in AI that reduce the communication cost $c \in(0,1)$ between humans and machines—making machines more user-friendly—can be a double-edged sword. While a lower $c$ increases labor income when $\mathbf{m} \in R_t$ or $\mathbf{m} \in R_b \cap K$ by enhancing the value of two-layer organizations (see (8) and (9)), it also raises the demand for machines from bottom-automated organizations, shrinking the set $K$, and increasing the likelihood that the economy falls on the lower side of the labor income discontinuity, i.e., $\mathbf{m} \in R_b \cap K^C$.

Third, AI advancements that improve compute efficiency—interpreted in our model as an increase in the mass $\mu$ of machines—make it more likely that the demand for machines from two-layer organizations does not exceed supply. Consequently, such technological improvements, if they expand the set $K$ sufficiently to include $\mathbf{m}$, may actually reduce capital income while boosting labor income.

Finally, Proposition 6 offers another argument, from labor’s perspective, for maintaining machine autonomy at a relatively low level (i.e., limiting AI’s knowledge in at least one dimension). When machine knowledge is constrained in at least one dimension, the demand for machines remains approximately $n(0) = 1/c$, regardless of how advanced machines become in other dimensions, since human assistance is almost always required. Thus, low autonomy not only helps keep the price of machines/compute low by limiting their expected output in independent production but also ensures that the demand for machines/compute does not escalate as AI improves.

Extension II: Human Heterogeneity

For simplicity, the baseline model assumes that all humans are homogenous. In this section, we show that our main insights remain valid when there is human heterogeneity in terms of knowledge.

The Model

The model is exactly as described in Section 2, except that there are two types of humans. There is a measure $\phi^{A} \in(0,1)$ of type-$A$ humans with knowledge $h^{A} \in(0,1)^{2}$, and a measure $\phi^{B} = 1-\phi^{A}$ of type-$B$ humans with knowledge $h^{B} \in(0,1)^{2}$. We place no restrictions on these knowledge profiles; one type might be more knowledgeable than the other in both dimensions, or one type may excel in one dimension while the other excels in the other dimension.

As in the baseline model, humans can be hired as independent producers, as solvers in bottom-automated firms, or as workers in top-automated firms. However, human heterogeneity introduces a new potential type of hierarchical firm consisting of humans in both the bottom and top layers (a “two-layer non-automated firm”). For simplicity, we assume that the cost of communication between humans is the same as the cost $c \in(0,1)$ of communication between humans and machines, but our results do not depend on this assumption. Figure 10 depicts all possible types of two-layer organizations in this scenario.

We denote by $w^{A}$ the equilibrium wage of type $A$, and by $w^{B}$ the equilibrium wage of type $B$. Total labor income is $w^{*} \equiv\phi^{A}w^{A} + \phi^{B}w^{B}$. Finally, we focus on the case in which machines are sufficiently abundant relative to time. This means that regardless of $m \in[0,1]^{2}$, the supply of machines always exceeds the equilibrium demand for machines from two-layer organizations.

The Turing Valley Under Human Heterogeneity

As mentioned above, human heterogeneity opens up the possibility of creating two-layer non-automated firms. The following example demonstrates that this implies Proposition 3 cannot hold type-by-type.

Example 1. Suppose problem difficulty is uniformly distributed, i.e., $F(x)=x_1x_2$. Additionally, assume that $c=1/2$, $\phi_A=0.8$, $h^A=(0.375,0.475)$, $h^B=(0.625,0.725)$, and $m=(0.29,0.7)$. As demonstrated in Section 2.3 of the Online Appendix—and illustrated in Figure 11—three distinct firm configurations simultaneously emerge in equilibrium under these conditions: (i) top-automated firms that hire type-$A$ humans, (ii) two-layer non-automated firms that hire type-$B$ humans as solvers and type-$A$ humans as workers, and (iii) single-layer automated firms.

In this scenario, $m_1<h^A_1<h^B_1$. Consider a marginal increase in $m_1$. Since some type-$A$ humans are employed in top-automated firms and $m_1<h^A_1$ a marginal increase in $m_1$ reduces $w^A$ (by the same argument as in Proposition 3). The reduction in $w^A$ leads to an increase in $w^B$, as it allows type-$B$ humans to claim a larger share of the output from two-layer non-automated firms. Therefore, a marginal increase in $m_1$ results in an increase in $w^B$ even though $m_1<h^B_1$.

Nevertheless, the following result shows that despite not holding type-by-type, the essence of

Proposition 3 still holds for total labor income. To present this result, let $R_h \equiv\{m \in[0,1]^2 : \text{no human matches with machines in equilibrium}\}$. In Section 2 of the Online Appendix, we characterize the competitive equilibrium of this economy with two-types in full detail.

Proposition 7. The equilibrium total labor income $w^*$ is continuous and almost everywhere differentiable in $m$. Moreover, if $m \in\operatorname{int} R_h$, then $\partial w^*/\partial m_i = 0$ for $i = 1, 2$. Otherwise, at every differentiable point of $w^*$:

• If $m_i < \min\{h_i^A,h_i^B\}$, then $\partial w^*/\partial m_i \leq0$ (with strict inequality if $m_{j\ne i} \ne0$).

• If $m_i > \max\{h_i^A,h_i^B\}$, then $\partial w^*/\partial m_i > 0$.

Proof. See Section 2.4 of the Online Appendix.

Figure 12 illustrates Proposition 7 for the case where problem difficulty is uniformly distributed. According to this proposition, total labor income decreases in any dimension of machine knowledge where AI cannot pass the Turing Test for any type of human but increases in any dimension where AI can pass the test for all types of human.

For intuition, note first that if no humans are matched with machines, wages remain independent of machine knowledge. Otherwise, there are two possibilities: humans either match with each other, or they do not.

If humans do not match with each other, then Proposition 3 holds type-by-type. Consequently, Proposition 7 follows because wages for all types decrease with $m_i$ if $m_i < \min\{h_i^A,h_i^B\}$ and increase with $m_i$ if $m_i \geq\max\{h_i^A,h_i^B\}$. When $\min\{h_i^A,h_i^B\} < m_i < \max\{h_i^A,h_i^B\}$, the wage of one type increases while that of the other decreases with $m_i$. Thus, the net effect on $w^* \equiv\phi^A w^A + \phi^B w^B$ is, in general, ambiguous.

The more interesting case is when humans match with each other in equilibrium. In this scenario, even when $m_i < \min\{h_i^A,h_i^B\}$ or $m_i > \max\{h_i^A,h_i^B\}$, an increase in $m_i$ can sometimes reduce the wage of one type while increasing the wage of the other type, as illustrated in Example 1. Nevertheless, the net effect on $w^* \equiv\phi^A w^A + \phi^B w^B$ is unambiguous in this case. This is because changes in $m$ only change the distribution of output among the different types of humans in non-automated two-layer firms; they do not affect the total output produced by these firms. Hence, the net effect of an increase in $m_i$ on $w^*$ is equal to the labor income gain or loss of those humans who are matched with machines.

We conclude this section by showing that the process of deciding the direction of AI development may be politically divisive not only along capital-labor lines—as discussed in Section 4.2—but also within labor lines.

Echoing Definition 2, we say that type $k$ is stronger in dimension 2 than in dimension 1 if $F(1,h_2^k) \ge F(h_1^k,1)$ and is weaker in dimension 2 than in dimension 1 otherwise. The following result shows that Proposition 4 holds type-by-type but not for total labor income.

Proposition 8.

1. For any type $k = A,B$, if both $h_1^k$ and $h_2^k$ are sufficiently small, then $w^k$ is maximized when AI is as good as possible in both dimensions. Otherwise, $w^k$ is maximized when AI simultaneously performs as poorly as possible in the dimension where $k$ is strongest and as well as possible in the dimension where $k$ is weakest.

2. The AI that maximizes total labor income $w^*$ need not be a vertex of $[0,1]^2$.

Proof. See Section 2.5 of the Online Appendix.

The intuition for Proposition 8 is as follows. When choosing $m$ to maximize $w^k$, it suffices to consider the case where type $k$ matches with machines. This is because there always exists a machine that can perfectly replicate a type-$k'$ human and is (weakly) cheaper than that human (as machines are abundant). Hence, the problem of choosing an $m$ to maximize $w^k$ is the same regardless of the existence of other types of humans, so the first part of Proposition 8 directly follows from Proposition 4. The second part of Proposition 8, in turn, follows because maximizing total labor income may involve balancing the gains of some types against the losses of others, leading to an interior solution.

Proposition 8 implies that different types of labor may have different preferences regarding machine knowledge—and none of them may be aligned with maximizing either output or capital income. This discrepancy occurs, for example, when one type is only strong in one dimension and another type is only strong in the other dimension. This suggests that choosing the direction of AI progress may be politically divisive not only along capital-labor lines but also within labor lines.

Final Remarks

The impressive advances in AI over the past decade have sparked intense debate on whether the current trajectory of AI development will benefit society as a whole or concentrate economic power among a select few. Motivated by this debate—and recognizing that AI primarily impacts knowledge work—we develop a model of a knowledge economy in which production involves solving multidimensional problems, and humans collaborate with AI-powered machines to optimize the use of their knowledge.

We characterize how labor income depends on AI’s capabilities and demonstrate that this relationship is closely connected to [?]’s famous test of machine intelligence: improvements in AI in dimensions of knowledge where machines fail the Turing Test reduce labor income, while improvements in AI in dimensions where machines pass the Turing Test increase it. Hence, if AI initially underperforms humans in all dimensions and improves gradually, total labor income initially declines before rising. Furthermore, we show that the AI that maximizes labor income may differ significantly from the one that maximizes capital income, indicating that the direction of AI development can create significant divisions between the interests of labor and capital.

Our findings suggest a role for policymakers in guiding the development and deployment of AI technologies. For instance, income support policies during transitional phases in which labor income falls could help alleviate political pressures to halt AI progress. Additionally, policymakers must be prepared to balance the potentially conflicting interests of labor and capital when trying to shape the direction of AI development. This includes incentivizing the production of sufficient computational resources to ensure that humans fully benefit from human-machine collaboration.

We believe the framework developed in this paper can be fruitfully applied to study other important questions. For example, a version of our model with heterogeneous humans, like the one analyzed in Section 6, could be used to explore the effects of AI improvements on labor income inequality. We leave these potential applications for future research.

Proofs Omitted From Section 3

Proof of Proposition 1

We first characterize the equilibrium and then show that it maximizes total output.

As explained in the main text, the abundance of machines relative to humans implies that $r^{*}=F(m)$. Moreover, given the definitions of $\bar{w}_{i}$ and $R_{i}$ for $i \in\{s,b,t\}$, it is evident that if $m \in R_{i}$, all humans are employed by firms of type $i$ in equilibrium, as these firms can offer the highest possible wages. Consequently, when $m \in R_{i}$, only type $i$ firms and single-layer automated firms arise, i.e., the equilibrium allocations are $\alpha_{i}^{*}=1$, $\alpha_{j}^{*}=0$ for all $j \ne i$, $\mu_{b}^{*}=\alpha_{b}^{*}n(m)$, $\mu_{t}^{*}=\alpha_{t}^{*}n(h)^{-1}$, and $\mu_{s}^{*}=\mu-\mu_{b}^{*}-\mu_{t}^{*}$. This also implies that the equilibrium wage is $w^{*}=\bar{w}_{i}$ (since type $i$ firms must obtain zero profits), so $w^{*}=\max\{\bar{w}_{s},\bar{w}_{b},\bar{w}_{t}\}$.

We now show that the equilibrium allocations maximize total output. Consider an arbitrary allocation $(\alpha,\mu)$. Since it is always efficient to fully utilize the time of both human and machine solvers (i.e., $\mu_{b}=\alpha_{b}n(m)$ and $\mu_{t}=\alpha_{t}n(h)^{-1}$), the total output in this economy is:

That is, total output is equal to the sum of the output produced by each of the four possible types of organizations. Specifically, there are $\alpha_{b}$ bottom-automated firms (since each $b$ firm hires one human), each producing $n(m)F(m\vee h)$ units of output. Similarly, there are $\mu_{t}=\alpha_{t}n(h)^{-1}$ top-automated firms (since each $t$ firm rents one machine), each producing $n(h)F(m\vee h)$ units of output. The number of single-layer non-automated firms is $\alpha_{s}=1-\alpha_{b}-\alpha_{t}$, and each of them produces $F(h)$ units of output. The number of single-layer automated firms is $\mu_{s}=\mu-\alpha_{b}n(m)-\alpha_{t}n(h)^{-1}$, and each of them produces $F(m)$ units of output.

Rearranging terms, total output can be written as $Y=\mu F(m)+\alpha_{b}\bar{w}_{b}+\alpha_{t}\bar{w}_{t}+(1-\alpha_{b}-\alpha_{t})\bar{w}_{s}$. Maximizing this last expression subject to $\alpha\in[0,1]^{3}$ and the non-negativity constraint $\mu_{s}=\mu-\alpha_{b}n(m)-\alpha_{t}n(h)^{-1}\ge0$ yields:

where we use the fact that the non-negativity constraint $\mu_{s}=\mu-\alpha_{b}n(m)-\alpha_{t}n(h)^{-1}\ge0$ never binds, as machines are abundant. Except for the ties that arise when $\bar{w}_{i}=\max\{\bar{w}_{j},\bar{w}_{k}\}$ (which are irrelevant in terms of output), this is exactly the allocation emerging from the competitive equilibrium. Thus, the equilibrium maximizes total output.

Proof of Proposition 2

Properties of $R_s$.— To show that $R_s$ contains a neighborhood of $h=(h_1,h_2)$, note that when $m=h$, then $\bar{w}_s=F(h)$, $\bar{w}_b=0$, and $\bar{w}_t=F(h)(1-1/n(h))$. Hence, $\bar{w}_s>\max\{\bar{w}_b,\bar{w}_t\}$, so $h\in R_s$. The result then follows by continuity.

Properties of $R_b$.— We first show that $R_b\cap[h_1,1]\times[h_2,1]=\varnothing$. If $m\in[h_1,1]\times[h_2,1]$, then $\bar{w}_b=0<\bar{w}_s$, so $m\notin R_b$. Next, we show that for every $m_1\in[0,1]$, there exists $\delta>0$ such that $(m_1,\epsilon)\in R_b$ for all $\epsilon\in[0,\delta)$. To do this, we show that $m=(m_1,0)\in R_b$. The result then follows by continuity. When $m=(m_1,0)$, then $\bar{w}_b=F(\max\{m_1,h_1\},h_2)/c$, $\bar{w}_s=F(h_1,h_2)$, and $\bar{w}_t=F(\max\{m_1,h_1\},h_2)$. Thus, $\bar{w}_b>\max\{\bar{w}_s,\bar{w}_t\}$, so $m=(m_1,0)\in R_b$.

Finally, we show that for every $m_2\in[0,1]$, there exists $\delta>0$ such that $(\epsilon,m_2)\in R_b$ for all $\epsilon\in[0,\delta)$. To do this, we show that $m=(0,m_2)\in R_b$. The result then follows by continuity. Indeed, if $m=(0,m_2)$, then $\bar{w}_b=F(h_1,\max\{m_2,h_2\})/c$, $\bar{w}_s=F(h_1,h_2)$, and $\bar{w}_t=F(h_1,\max\{m_2,h_2\})$. Thus, $\bar{w}_b>\max\{\bar{w}_s,\bar{w}_t\}$, so $m=(0,m_2)\in R_b$.

Properties of $R_t$.— We first show that $R_t\cap([0,h_1)\times[0,h_2))=\varnothing$. If $m\in[0,h_1)\times[0,h_2)$, then $\bar{w}_t=F(m)(1-1/n(h))<F(h)=\bar{w}_s$, so $m\notin R_t$. Now we show that there exists $\delta>0$ such that $(1-\epsilon,1-\epsilon)\in R_t$ for all $\epsilon\in[0,\delta)$. To do this, we show that $m=(1,1)\in R_t$. The result then follows by continuity. When $m=(1,1)$, then $m\notin R_b$ (as $R_b\cap[h_1,1]\times[h_2,1]=\varnothing$). Moreover, $\bar{w}_t=1-1/n(h)$ and $\bar{w}_s=F(h)$, so $\bar{w}_t>\bar{w}_s$ if and only if $(1-F(h))(1-c)>0$, which always holds. Hence, $m=(1,1)\in R_t$.

Proofs Omitted From Section 4

Proof of Proposition 3

The continuity of $w^*$ follows from (i) $w^*$ being continuous in $m$ whenever $m\in\operatorname{int}R_i$ for $i\in\{s,b,t\}$, and (ii) $w^*\to\bar{w}_i=\bar{w}_j$ as $m\to\tilde{m}\in\operatorname{cl}R_i\cap\operatorname{cl}R_j$ for $i\ne j$ (where $\operatorname{cl}R_i$ denotes the closure of the set $R_i$). That $w^*$ is almost everywhere differentiable follows because $w^*$ is differentiable in $m$ whenever $m\in\operatorname{int}R_i$ for $i\in\{s,b,t\}$, and the boundaries of $(R_s,R_b,R_t)$ have zero Lebesgue measure.

To prove the remaining properties of the proposition, we first consider $m\in\operatorname{int}R_s$, then $m\in\operatorname{int}R_b$, and finally $m\in\operatorname{int}R_t$. For brevity, we focus on $\partial w^*/\partial m_1$, since the results regarding $\partial w^*/\partial m_2$ are analogous.

• $m\in\operatorname{int}R_s$.—In this case, $w^*=F(h)$. Hence, $\partial w^*/\partial m_i=0$, for $i=1,2$.

• $m\in\operatorname{int}R_b$.—In this case, $w^*=n(m)(F(m\vee h)-F(m))$. Consequently, if $m_1<h_1$, then:

This is weakly negative for all $m_2 \in[0,1]$ and strictly negative for all $m_2 \in(0,1]$.

If $m_1 > h_1$, then:

This last expression is strictly positive for all $m_2 \in[0,h_2)$ (note that $m_2 < h_2$ necessarily, since $R_b \cap([h_1,1] \times[h_2,1]) = \varnothing$ according to Proposition 2).

• $\mathbf{m} \in\operatorname{int} R_t$.— In this case, $w^* = F(\mathbf{m} \vee\mathbf{h}) - F(\mathbf{m})/n(\mathbf{h})$. When $m_1 < h_1$, marginal increase in $m_1$ does not change $F(\mathbf{m} \vee\mathbf{h})$, but increases $F(\mathbf{m})/n(\mathbf{h})$ (strictly so if $m_2 > 0$). Hence, $\partial w^*/\partial m_1 < 0$ when $m_1 < h_1$.

In contrast, when $m_1 > h_1$, then:

This expression is strictly positive for all $m_2 \in[h_2,1]$ (note that $m_2 \in[h_2,1]$ necessarily since $R_t \cap([0,h_1) \times[0,h_2)) = \varnothing$ according to Proposition 2). □

Proof of Proposition 4

We begin by constructing $\bar{h}_1$ and $\bar{h}_2(h_1)$ and then show that the statement is true. Let $\Delta(h_1,h_2) \equiv w^*(1,0)-w^*(1,1)=F(1,h_2)/c-1+c[1-F(h_1,h_2)]$, define $\bar{h}_1 \equiv\sup\{h_1 \in(0,1): \Delta(h_1,\underline{h}_2(h_1))=0\}$.

Note that $\lim_{h_1\to0}\Delta(h_1,\underline{h}_2(h_1))=-(1-c)<0$ and $\lim_{h_1\to1}\Delta(h_1,\underline{h}_2(h_1))=(1/c)-1>0$, so the set $\{h_1 \in(0,1):\Delta(h_1,\underline{h}_2(h_1))=0\}$ is non-empty. Thus, $\bar{h}_1 \in(0,1)$.

We now show that the statement of the Proposition is true. First, given that humans are stronger in dimension 2 than in dimension 1, the maximum labor income is given by either $w^*(1,0)$ or $w^*(1,1)$. Second, note that $\Delta(h_1,h_2)$ is strictly increasing in $h_2$:

Consequently, for any $h_1 \ge\bar{h}_1$, we have $\Delta(h_1,h_2) \ge\Delta(h_1,\underline{h}_2(h_1)) \ge0$ for all $h_2 \in[\underline{h}_2(h_1),1)$. Thus, if $h_1 \ge\bar{h}_1$, then $\mathbf{m}=(1,0)$ maximizes labor income, yielding the first part of the theorem (“If $h_1 \ge\bar{h}_1$, then...”).

To prove the second part of the theorem (“If $h_1 < \bar{h}_1$, then...”), fix $h_1 < \bar{h}_1$ and define $\bar{h}_2(h_1)$ as:

where $h_2^*(h_1)$ is the (unique) $h_2 \in[\underline{h}_2(h_1),1]$ that solves $\Delta(h_1,h_2)=0$. Note that $h_2^*(h_1)$ exists (as $\Delta(h_1,\underline{h}_2(h_1))<0$ and $\lim_{h_2\to1}\Delta(h_1,h_2)=1/c-1+c[1-F(h_1,1)]>0$), is unique (as $\Delta(h_1,h_2)$ is strictly increasing in $h_2$), and strictly between $\underline{h}_2(h_1)$ and $1$.

Existence and Characterization of a Competitive Equilibrium

Clearly, if $h_2 \in[\bar{h}_2(h_1),1)$, then $\mathbf{m}=(1,0)$ maximizes labor income. Indeed, if $\Delta(h_1,\bar{h}_2(h_1)) \ge0$, then $\Delta(h_1,h_2) \ge\Delta(h_1,\bar{h}_2(h_1)) \ge0$ for all $h_2 \in[\bar{h}_2(h_1),1)=[\bar{h}_2(h_1),1)$. In contrast, if $\Delta(h_1,\bar{h}_2(h_1))<0$, then $\Delta(h_1,h_2) \ge\Delta(h_1,h_2^*(h_1))=0$, for all $h_2 \in[h_2^*(h_1),1)=[\bar{h}_2(h_1),1)$.

Moreover, if $h_2 \in[\underline{h}_2,\bar{h}_2(h_1))$, then $\mathbf{m}=(1,1)$ maximizes labor income. Indeed, if $\Delta(h_1,\bar{h}_2(h_1)) \ge0$, then $[\underline{h}_2,\bar{h}_2(h_1))=[\bar{h}_2(h_1),\bar{h}_2(h_1))=\varnothing$, so the statement holds trivially. In contrast, if $\Delta(h_1,\bar{h}_2(h_1))<0$, then $\Delta(h_1,h_2)<\Delta(h_1,h_2^*(h_1))=0$, for all $h_2 \in[\underline{h}_2,h_2^*(h_1))=[\underline{h}_2,\bar{h}_2(h_1))$. Thus, we have proven the second part of the theorem. □

Proofs Omitted From Section 5

Proof of Proposition 5

We first verify that the candidate described in the proposition constitutes an equilibrium. Then, we show that the equilibrium is generically unique and that it maximizes total output.

The proof that the equilibrium is as described in Proposition 1 when $\mathbf{m} \in R_s \cup R_t \cup(R_b \cap K)$ is straightforward: The supply of available machines exceeds the demand by two-layer organizations when prices are $w^*=\max\{\bar{w}_s,\bar{w}_b,\bar{w}_t\}$ and $r^*=F(m)$.

So consider $\mathbf{m} \in R_b \cap K^C \cap R_m$. That markets clear at the candidate allocation follows directly from the fact that $\alpha_b^*+\alpha_t^*=1$ and $\mu_b^*+\mu_t^*=\mu$ (where $\mu_b^*=\alpha_b^*n(m)$ and $\mu_t^*=\alpha_t^*n(h)^{-1}$). Hence, we just need to verify that $b$ and $t$ firms obtain zero profits in equilibrium, and that single-layer automated and non-automated firms obtain at most zero profits in equilibrium. That $b$ and $t$ firms obtain zero profits follows because $w^*$ and $r^*$ are such that:

That single-layer automated and non-automated firms obtain at most zero profits follows because:

Finally, consider $\mathbf{m} \in R_b \cap K^C \cap R_m^C$. That markets clear at the candidate allocation follows directly from the fact that $\alpha_b^*=\mu/n(m)$, $\alpha_s^*=1-\alpha_b^*$, and $\mu_b^*=\mu$. Hence, we just need to verify that $b$ and single-layer non-automated firms obtain zero profits in equilibrium, and that single-layer automated and top-automated firms obtain at most zero profits in equilibrium. That $b$ and single-layer non-automated firms obtain zero profits follows because directly from the fact that $r^*=F(m\vee h)-F(h)/n(m)$ and $w^*=F(h)$. That single-layer automated firms obtain at most zero profits follows because $r^{*} \ge F(\mathbf{m})$ if and only if $\bar{w}_{b} \ge\bar{w}_{s}$, and this last condition holds since $\mathbf{m} \in R_{b}$. Finally, that top-automated firms obtain at most zero profits follows because:

which is satisfied since $\mathbf{m}\in R_{m}^{C}$.

Next, we show that the equilibrium is generically unique and that it maximizes total output. Consider the problem of choosing allocations to maximize total output subject to the economy’s resource constraints, i.e.,:

subject to $\alpha\in[0,1]^{3}$ and the non-negativity constraint $\mu_{s}=\mu-\alpha_{b}n(\mathbf{m})-\alpha_{t}n(h)^{-1}\ge0$ (see the proof of Proposition 1 in Appendix A).

It is not difficult to prove that (i) this problem has a unique solution except when $\bar{w}_{i}=\max\{\bar{w}_{j},\bar{w}_{k}\}$ for $i,j,k\in\{s,b,t\}$ and $i\ne j\ne k$, and (ii) the equilibrium allocations are a solution to this problem. Hence, the equilibrium allocations are generically unique (as the set $\{\mathbf{m}\in[0,1]^{2}:\bar{w}_{i}=\max\{\bar{w}_{j},\bar{w}_{k}\}$ for $i,j,k\in\{s,b,t\}$ and $i\ne j\ne k\}$ has Lebesgue measure zero) and the equilibrium maximizes total output. Moreover, it is easy to prove that the equilibrium prices are the unique set of prices that support the output-maximizing allocations when the latter are unique. This is implies that the equilibrium is also generically unique. □

Proof of Proposition 6

Part 1.— First, we show that marginal increases in $m_{i}$ leads to a discontinuous drop in $w^{*}$ and a discontinuous increase in $r^{*}$ when $\mathbf{m}\in\partial K\cap\operatorname{int}R_{b}$. Note that if $\mathbf{m}\in\partial K\cap\operatorname{int}R_{b}$, then $\bar{w}_{b}>\max\{\bar{w}_{s},\bar{w}_{t}\}$ (because $\mathbf{m}\in\operatorname{int}R_{b}$). Moreover, if $\mathbf{m}\in\partial K\cap\operatorname{int}R_{b}$, then $\mathbf{m}$ is either in (i) the boundary between $R_{b}\cap K$ and $R_{b}\cap K^{C}\cap R_{m}$, or (ii) the boundary between $R_{b}\cap K$ and $R_{b}\cap K^{C}\cap R_{m}^{C}$.

In case (i), a marginal increase in $m_{i}$ induces the equilibrium wage to drop from $\bar{w}_{b}$ to $(n(\mathbf{m})n(h)\bar{w}_{t}-\bar{w}_{b})/(n(\mathbf{m})n(h)-1)<\bar{w}_{b}$, where the last inequality follows because $\bar{w}_{b}>\bar{w}_{t}$. Moreover, the equilibrium price of machines discontinuously raises from $r^{*}=F(\mathbf{m})$ to

In case (ii), a marginal increase in $m_{i}$ induces the wage to drop from $\bar{w}_{b}$ to $\bar{w}_{s}$ which is strictly smaller than $\bar{w}_{b}$. Moreover, the equilibrium price of machines raises discontinuously raises from $r^{*}=F(\mathbf{m})$ to:

Next, we show that equilibrium prices are continuous in $\mathbf{m}$ for all other $\mathbf{m}\in[0,1]^{2}$. The fact that $w^{*}$ and $r^{*}$ are continuous in the interior of the sets $R_{s}\cup R_{t}$, $R_{b}\cap K$, $R_{b}\cap K^{C}\cap R_{m}$ and $R_{b}\cap K^{C}\cap R_{m}^{C}$ is straightforward. Hence, we only need to check that prices are continuous in the boundary of any two of these sets, except for the boundary between $R_b \cap K$ and $R_b \cap K^C \cap R_m$, and the boundary $R_b \cap K$ and $R_b \cap K^C \cap R_m^C$.

It follows from Proposition 3 that prices are continuous in the boundary between $R_s$ and $R_t$, between $R_s$ and $R_b \cap K$, and between $R_t$ and $R_b \cap K$. Hence, we only need to check the following five additional boundaries:

• The boundary between $R_s$ and $R_b \cap K^C \cap R_m$. In this boundary, $\bar{w}_s = \bar{w}_b = \bar{w}_t$, so it is easy to check that prices do not jump discontinuously at this boundary.

• The boundary between $R_s$ and $R_b \cap K^C \cap R_m^C$. In this case, $\bar{w}_s = \bar{w}_b$, so it is easy to check that prices do not jump discontinuously at this boundary.

• The boundary between $R_t$ and $R_b \cap K^C \cap R_m$. In this case, $\bar{w}_s = \bar{w}_b$, so it is easy to check that prices do not jump discontinuously at this boundary.

• The boundary between $R_t$ and $R_b \cap K^C \cap R_m^C$. In this case, $\bar{w}_s = \bar{w}_b = \bar{w}_t$, so it is easy to check that prices do not jump discontinuously at this boundary.

• The boundary between $R_b \cap K^C \cap R_m$ and $R_b \cap K^C \cap R_m^C$. In this case, $\bar{w}_b - \bar{w}_s = n(h)n(m)(\bar{w}_t - \bar{w}_s)$, which implies that the wage is $\bar{w}_s$, and hence the price of machines does not jump discontinuously at this boundary.

Part 2.—Part 2(a) is straightforward, so we focus on parts (b) and (c). Note that if $m \in\operatorname{int}(R_b \cap K^C \cap R_m)$, then:

Part 2(b) then follows because when $m_i < h_i$, an increase in $m_i$ decreases $(n(h)-1)/(n(h)-1/n(m))$ and does not affect $F(m\vee h)$.

Finally, let us turn to part 2(c). By symmetry, it is enough to consider $\partial w^*/\partial m_1$ when $m_1 > h_1$. Note that since $m_1 > h_1$, then $m_2 < h_2$ (since $m \in R_b$). Thus,

where the last inequality follows because $\partial F(m_1,h_2)/\partial m_1 > \partial F(m_1,m_2)/\partial m_1$ since $h_2 > m_2$. Note then that:

The feasible values of $h$ and $m$ that minimize the right-hand side of the last inequality are $h_1 = 0$, $h_{2}=1$, $m_{1}=1$, and $m_{2}=0$, in which case:

Thus, if $c^{2}\leq1/2$, that is if $c\leq\sqrt{2}/2$, then $\partial w^{*}/\partial m_{1}>0$. $\square$

The Turing Valley: How AI’s Capabilities Shape Labor Income (Online Appendix - Not for Publication)

Ide; Talamàs

A Model without Knowledge Synergies

In our baseline model, we assume that communication between humans and machines allows them to combine their knowledge. In this appendix, we show the sense in which our results hold when this is not the case.

The Model

The model is exactly as described in Section 2 of the main text, except that humans and machines can no longer combine their knowledge. This implies that the profit of a two-layer organization as a function of its type (i.e., $b$ or $t$) is given as follows:

where $F(m \oplus h) \equiv F(m)+F(h)-F(m \wedge h)$ and $m \wedge h$ is the meet of $m$ and $h$ on $[0,1]^2$, i.e., $m \wedge h \equiv\inf\{z \in[0,1]^2:z \le m,z \le h\}$. Intuitively, $F(m \oplus h)$ is the probability that either humans or machines can solve any given problem on their own.

Equilibrium Characterization

As in the main text, the abundance of machines relative to humans implies that $r^* = F(m)$. Similarly, define $\widetilde{w}_i$ as the highest wage an organization of type $i \in\{s,b,t\}$ can offer, given that $r^* = F(m)$. The wage $\widetilde{w}_i$ is determined by the zero-profit condition of a firm of type $i$:

Finally, let $\widetilde{R}_s \equiv\{m \in[0,1]^2:\widetilde{w}_s \ge\max\{\widetilde{w}_b,\widetilde{w}_t\}\}$, $\widetilde{R}_b \equiv\{m \in[0,1]^2:\widetilde{w}_b > \max\{\widetilde{w}_s,\widetilde{w}_t\}\}$, and $\widetilde{R}_t = [0,1]^2 \setminus(\widetilde{R}_s \cup\widetilde{R}_b)$. We then have the following proposition:

Proposition 1.1. The competitive equilibrium maximizes total output. The equilibrium rental rate of machines is $r^* = F(m)$ and the equilibrium wage is $w^* = \max\{\widetilde{w}_s,\widetilde{w}_b,\widetilde{w}_t\}$. Moreover:

• If $m \in\widetilde{R}_s$, only single-layer organizations form: $\alpha_s^* = 1$ and $\mu_s^* = \mu$.

• If $\mathbf{m} \in\widetilde{R}_b$, only $b$ and single-layer automated organizations form: $\alpha_b^* = 1$, $\mu_b^* = n(\mathbf{m})$, and $\mu_s^* = \mu- \mu_b^*$.

• If $\mathbf{m} \in\widetilde{R}_t$, only $t$ and single-layer automated organizations form: $\alpha_t^* = 1$, $\mu_t^* = n(\mathbf{h})^{-1}$, and $\mu_s^* = \mu- \mu_t^*$.

Proof. The proof is analogous to that of Proposition 1 of the main text.

Figure 1(a) depicts the equilibrium regions with $\widetilde{R}_s$, $\widetilde{R}_b$, and $\widetilde{R}_t$ as a function of $\mathbf{m}$ when $F(\mathbf{x}) = x_1x_2$. Figure 1(b), depicts the equilibrium regions $R_s$, $R_b$, and $\widetilde{R}_t$ of the model with synergies of the main text. As the figure shows, the two panels are qualitatively similar.

The Turing Valley without Synergies

The following result shows how Proposition 3 generalizes when there are no knowledge synergies.

Proposition 1.2. $w^*$ is continuous and almost everywhere differentiable in $\mathbf{m}$. Moreover, for $i = 1, 2$:

1. If $\mathbf{m} \in\operatorname{int}\widetilde{R}_s$, then $\partial w^*/\partial m_i = 0$.

2. If $\mathbf{m} \in(\operatorname{int}\widetilde{R}_b \cup\operatorname{int}\widetilde{R}_t)$, $m_i < h_i$ and $m_j \le h_j$, then $\partial w^*/\partial m_i \le0$ (with strict inequality if $m_{j \ne i} \ne0$).

3. If $\mathbf{m} \in(\operatorname{int}\widetilde{R}_b \cup\operatorname{int}\widetilde{R}_t)$ and $m_i > h_i$, then $\partial w^*/\partial m_i \ge0$ (with strict inequality if $m_{j \ne i} \ne0$).

Proof. See Section 1.5 of this Online Appendix.

Figure 2 illustrates Proposition 1.2 and also depicts the case with synergies discussed in the main text for comparison (i.e., Proposition 3). As the figure shows, the effects of AI improvements on labor income are nearly identical, whether or not synergies are present. Specifically, without synergies, improvements in AI in areas where machines outperform humans still lead to increases in labor income. Similarly, when AI is worse than humans in both dimensions, any improvement in AI continues to reduce labor income.

The only difference between the two cases is that, without synergies, the effects of increases in $m_i$ on labor income are ambiguous when $\mathbf{m} \in(\operatorname{int}R_b \cup\operatorname{int}R_t)$, $m_i < h_i$ and $m_j > h_j$. To understand this, consider the case $\mathbf{m} \in\operatorname{int}\tilde{R}_b$. The equilibrium wage is given by:

This implies that—similar to the model with synergies—the equilibrium wage is determined by the number of times a human can provide assistance, $1/c$, multiplied by the probability that such assistance is useful. The difference is that without synergies, the probability that human assistance is useful conditional on a machine asking for assistance is given by:

In contrast, when synergies are present, that conditional probability is equal to:

Consider then the effects of increasing $m_2$ when $m_1 > h_1$ and $m_2 < h_2$ on these conditional probabilities. This situation is illustrated in Figure 3. As panel (a) shows, when no synergies exist, the effect of this increase is ambiguous. On the one hand, there is a new set of problems—those in area $\gamma$—that machines can now solve and that humans could already solve before the improvement. On the other hand, there is a new set of problems—those in area $\eta$—that machines can now solve on their own and that humans cannot solve.

The first effect reduces the probability that human assistance is useful conditional on a machine asking for assistance by leading machines to pose more difficult questions, thereby worsening the

pool of problems that humans must address. In contrast, the second effect increases that conditional probability, as machines now pose fewer questions that humans cannot answer, effectively improving the pool of problems humans need to address. The overall impact, therefore, depends on the probability measure over the sets of problems $\gamma$ and $\eta$.

For comparison, panel (b) of Figure 3 depicts the effect of the same increase in $m_2$ when synergies are present. As shown in the panel, in this case, the improvement always reduces the probability that human assistance is useful. This is because the new set of problems that machines can now solve—those in areas $\gamma+\eta$—are problems the organization could already solve before the improvement. Thus, the only effect of improving $m_2$ is that it causes machines to pose more difficult questions.

For comparison purposes, panel (b) of Figure 3 also depicts the effect of the same increase in $m_2$ when synergies are present. As the panel shows, in this case, the improvement always reduces the value of human assistance because the new set of problems that machines can now solve—those in areas $\gamma+\eta$—are problems that the organization could already solve before the improvement. Hence, the only effect of improving $m_2$ is inducing machines to ask more difficult questions.

The AI that Maximizes Labor Income

Even when there are no knowledge synergies, the AI that maximizes labor income can be very different from the AI that maximizes capital income. The only difference that arises when there are no synergies is that for labor income to not be maximum at the best possible AI, humans need to be sufficiently knowledgeable in both dimensions.

Indeed, absent knowledge synergies, an AI that is knowledgeable in only one dimension does not help humans solve any problems. As a result, in this case labor income is more likely to achieve its maximum at an AI that is as good as possible in all dimensions. However, as Figure 4 illustrates, when humans are sufficiently good in both dimensions, it is still the case that labor income is not maximized when AI is as good as possible in both dimensions.

Proof of Proposition 1.2

The proof that $w^*$ is continuous and almost everywhere differentiable in $m$ is analogous to that of Proposition 3 of the main text, so it is omitted.

To prove the remaining properties of the proposition, we first consider $m \in\operatorname{int}\widetilde{R}_s$, then $m \in\operatorname{int}\widetilde{R}_b$, and finally $m \in\operatorname{int}\widetilde{R}_t$. For brevity, we focus on $\partial w^*/\partial m_1$, since the results regarding $\partial w^*/\partial m_2$ are analogous.

• $m \in\operatorname{int}\widetilde{R}_s$.—In this case, $w^*=F(h)$. Hence, $\partial w^*/\partial m_i=0$, for $i=1,2$.

• $m \in\operatorname{int}\widetilde{R}_b$.—In this case, $w^*=n(m)(F(h)-F(m\wedge h))$. We first show that if $m_1<h_1$ and $m_2\le h_2$,

then $\partial w^*/\partial m_1\leq0$. Indeed,

This is weakly negative for all $m_2\in[0,h_2]$ and strictly negative for all $m_2\in(0,h_2]$.

Next, we show that if $m_1>h_1$, then $\partial w^*/\partial m_1\geq0$. First, note that if $m_1>h_1$, then $m_2<h_2$ (otherwise, bottom-automated firms would not arise in equilibrium). Thus:

This last expression is weakly positive for all $m_2 \in[0,h_2)$ and strictly positive for all $m_2 \in(0,h_2)$.

• $m \in\operatorname{int}\widetilde{R}_t$.— In this case, $w^* = F(h)-F(m \wedge h)+F(m)(1-1/n(h))$. Moreover, it cannot be that $m_1<h_1$ and $m_2 \le h_2$ (if that were the case, top-automated firms would not arise in equilibrium). Hence, we only focus on showing that if $m_1>h_1$, then $\partial w^*/\partial m_1 \ge0$. Notice then that:

This last expression is weakly positive for all $m_2 \in[0,1]$ and strictly positive for all $m_2 \in(0,1]$. □

Human Heterogeneity

Preliminaries

Before characterizing the equilibrium, we provide the profits of the different types of firms and formally define the competitive equilibrium of this economy.

Wages, Prices and Profits.— As mentioned in the main text, let $w^A$ and $w^B$ be the wage of type-$A$ and type-$B$ humans, respectively, and denote by $r$ the rental rate machines. All agents in this economy are risk neutral and maximize their income. We normalize the value of each unit of output to one.

There are now nine different firm configurations that can potentially arise in equilibrium; three different types of single-layer firms, and six different types of two-layer organizations. The profits of single-layer and two-layer automated firms are as in the baseline model of Section 2, except that we now must take into account whether they hire a type-$A$ or a type-$B$ human. In particular, the profit of a single-layer firm as a function of its type is given by:

Similarly, the profit of a two-layer non-automated firm as a function of its type (i.e., $b$ or $t$) and the human or humans that it hires (i.e., $k=A,B$) is:

The profit of the new type of firm configuration that can arise, i.e., of two-layer non-automated firms, depends on whether it hires type-$A$ humans as workers and type-$B$ humans as solvers (i.e., a “BA firm”) or it hires type-B humans as workers and type-A humans as solvers (i.e., an “AB firm”):²

The logic is the same as in two-layer automated firms: the profit of a firm is its expected output minus the cost of the resources it uses. For instance, in the case of a BA firm, its total expected output is $n(h^{A})F(h^{A} \vee h^{B})$, as such organization attempts $n(h^{A})$ problems and successfully solves a fraction $F(h^{A} \vee h^{B})$ of them. In turn, the cost of the resources is $w^{B} + n(h^{A})w^{A}$, as the firm employs one type-B human as a solver and $n(h^{A})$ type-A human as workers.

Competitive Equilibrium.— Let $\alpha_{s}^{k}$, $\alpha_{b}^{k}$, and $\alpha_{t}^{k}$ be the mass of humans of type-$k$ hired by single-layer non-automated, bottom-automated, and top-automated firms of type $k$, respectively. Similarly, denote by $\mu_{b}^{k}$ and $\mu_{t}^{k}$, the mass of machines rented by bottom-automated and top-automated firms of type-$k$, and by $\mu_{s}$ the mass of machines rented by single-layer automated firms. Finally, let $\alpha_{\mathrm{BA}}^{k}$ and $\alpha_{\mathrm{AB}}^{k}$ be the mass of humans of type-$k$ hired by BA and AB two-layer non-automated firms, respectively.

Definition 1 (Competitive Equilibrium). An equilibrium consists of a set of non-negative allocations $(\mu_{s}, \mu^{A}, \mu^{B}, \alpha^{A}, \alpha^{B})$, where $\mu^{k} \equiv(\mu_{b}^{k}, \mu_{t}^{k})$ and $\alpha^{k} \equiv(\alpha_{s}^{k}, \alpha_{b}^{k}, \alpha_{t}^{k}, \alpha_{\mathrm{BA}}^{k}, \alpha_{\mathrm{AB}}^{k})$ for $k = A, B$, and set of non-negative prices $(w^{A}, w^{B}, r)$, such that:

1. Firms optimally choose their structure (while earning zero profits).

2. Single-layer non-automated firms of type $k$ hire a mass $\alpha_{s}^{k}$ of humans.

3. Single-layer automated firms hire a mass $\mu_{s}$ of machines.

4. $b$ firms of type $k$ hire a mass $\alpha_{b}^{k}$ of humans and rent a mass $\mu_{b}^{k} = \alpha_{b}^{k}n(m)$ of machines.

5. $t$ firms of type $k$ hire a mass $\alpha_{t}^{k}$ of humans and rent a mass $\mu_{t}^{k} = \alpha_{t}^{k}n(h^{k})^{-1}$ of machines.

6. BA firms hire a mass $\alpha_{\mathrm{BA}}^{B}$ of type-B humans as solvers and a mass $\alpha_{\mathrm{BA}}^{A} = \alpha_{\mathrm{BA}}^{B}n(h^{A})$ of type-A humans as workers.

7. AB firms hire a mass $\alpha_{\mathrm{AB}}^{A}$ of type-A humans as solvers and a mass $\alpha_{\mathrm{AB}}^{B} = \alpha_{\mathrm{AB}}^{A}n(h^{B})$ of type-B humans as workers.

8. Markets clear: $\mu_{s} + \sum_{k=A,B}(\mu_{b}^{k} + \mu_{t}^{k}) = \mu$, and $\alpha_{s}^{k} + \alpha_{b}^{k} + \alpha_{t}^{k} + \alpha_{\mathrm{BA}}^{k} + \alpha_{\mathrm{AB}}^{k} = \phi^{k}$ for $k = A, B$.

Existence and Characterization of a Competitive Equilibrium

We begin with some preliminary results and by introducing some notation. As in the baseline model, the abundance of machines relative to humans implies that a positive mass of machines must be rented by single-layer automated firms. Hence, the zero-profit condition of these firms implies that the equilibrium rental rate of machines is equal to the fraction of problems they can solve on their own, $F(m)$.

Additionally, in a similar fashion as in the main text, define $\bar{w}^{k}_{i}$ as the highest wage an organization of type $i \in\{s,b,t\}$ that hires humans of type-$k$ can potentially offer, given the equilibrium price of machines is $F(m)$:

Define also $\bar{w}^{k} \equiv\max\{\bar{w}^{k}_{s},\bar{w}^{k}_{b},\bar{w}^{k}_{t}\}$. Note that if in equilibrium humans do not match with each other, then the equilibrium wage of type $k$ is equal to $\bar{w}^{k}$, i.e., $w^{k}=\bar{w}^{k}$. This is because the equilibrium is then “separable” in the two types of humans, so Proposition 1 of Section 3 of the main text holds type-by-type. Consequently, in this case, the equilibrium allocations are determined by the sets $R^{k}_{s}\equiv\{\mathbf{m}\in[0,1]^{2}:\bar{w}^{k}_{s}\geq\max\{\bar{w}^{k}_{b},\bar{w}^{k}_{t}\}\}$, $R^{k}_{b}\equiv\{\mathbf{m}\in[0,1]^{2}:\bar{w}^{k}_{b}>\max\{\bar{w}^{k}_{s},\bar{w}^{k}_{t}\}\}$, and $R^{k}_{t}\equiv[0,1]^{2}\setminus(R^{k}_{s}\cup R^{k}_{b})$, as indicated in that proposition.

We still need to characterize when humans match or do not match with each other in equilibrium, as well as the equilibrium prices and allocations when they do match. To do this, let:

Intuitively, $\bar{M}^{A}_{BA}$ is the marginal product of a human of type-A in a BA firm when $B$’s wage is $\bar{w}^{B}$. The interpretation of $\bar{M}^{B}_{BA}$, $\bar{M}^{A}_{AB}$ and $\bar{M}^{B}_{AB}$ is similar. Note then that:

Hence, $\operatorname{sign}(\bar{M}^{A}_{BA}-\bar{w}^{A})=\operatorname{sign}(\bar{M}^{B}_{BA}-\bar{w}^{B})$ and $\operatorname{sign}(\bar{M}^{B}_{AB}-\bar{w}^{B})=\operatorname{sign}(\bar{M}^{A}_{AB}-\bar{w}^{A})$.

We can now present the main result of this section. We first provide the formal statement and then explain it intuitively:

Proposition 2.1. The equilibrium is generically unique and maximizes total output. It is given as follows:

1. If $\max\{\bar{M}^{A}_{BA},\bar{M}^{A}_{AB}\}\leq\bar{w}^{A}$, then humans do not match with each other. Hence, $w^{k}=\bar{w}^{k}$ for $k=A,B$, and all type-$k$ humans are either doing independent production (if $\mathbf{m}\in R^{k}_{s}$), or matched with machines (if $\mathbf{m}\in(R^{k}_{b}\cup R^{k}_{t})$).

2. If $\max\{\bar{M}^{A}_{BA},\bar{M}^{A}_{AB}\}>\bar{w}^{A}$, then humans do match with each other. In this case:

(a) If $\phi^{A}\leq1/(1+n(h^{B}))$, then $w^{A}=\max\{\bar{M}^{A}_{BA},\bar{M}^{A}_{AB}\}$ and $w^{B}=\bar{w}^{B}$. Furthermore,

• If $\overline{M}^{A}_{BA} \ge\overline{M}^{A}_{AB}$, then all type-A humans are workers in BA firms, a mass $\phi^{A}/n(h^{A})$ of type-B humans are solvers in BA, and a mass $\phi^{B} - \phi^{A}/n(h^{A})$ of type-B humans are either doing independent production (if $m \in R^{B}_{s}$), or matched with machines (if $m \in(R^{B}_{b} \cup R^{B}_{t})$).

• If $\overline{M}^{A}_{BA} < \overline{M}^{A}_{AB}$, then all type-A humans are solvers in AB firms, a mass $\phi^{A}n(h^{B})$ of type-B humans are workers in AB, and a mass $\phi^{B} - \phi^{A}n(h^{B})$ of type-B humans are either doing independent production (if $m \in R^{B}_{s}$), or matched with machines (if $m \in(R^{B}_{b} \cup R^{B}_{t})$).

(b) If $1/(1+n(h^{B})) < \phi^{A} < n(h^{A})/(1+n(h^{A}))$, then:

• If $\overline{M}^{A}_{AB} \le\overline{M}^{A}_{BA}$ and $\overline{M}^{B}_{AB} \le\overline{M}^{B}_{BA}$, then $w^{A} = \overline{M}^{A}_{BA}$ and $w^{B} = \overline{w}^{B}$. Moreover, type-A humans are workers in BA firms, a mass $\phi^{A}/n(h^{A})$ of type-B humans are solvers in BA, and a mass $\phi^{B} - \phi^{A}/n(h^{A})$ of type-B humans are either doing independent production (if $m \in R^{B}_{s}$), or matched with machines (if $m \in(R^{B}_{b} \cup R^{B}_{t})$).

• If $\overline{M}^{A}_{BA} \le\overline{M}^{A}_{AB}$ and $\overline{M}^{B}_{BA} \le\overline{M}^{B}_{AB}$, then $w^{A} = \overline{w}^{A}$ and $w^{B} = \overline{M}^{A}_{AB}$. Moreover, all type-B humans are workers in AB firms, a mass $\phi^{B}/n(h^{B})$ of type-A humans are solvers in AB, and a mass $\phi^{A} - \phi^{B}/n(h^{B})$ of type-A humans are either doing independent production (if $m \in R^{A}_{s}$), or matched with machines (if $m \in(R^{A}_{b} \cup R^{A}_{t})$).

• Otherwise, humans only match with each other. A mass $\alpha^{A}_{BA}$ of type-A humans are workers in BA, a mass $\phi^{A} - \alpha^{A}_{BA}$ of type-A humans are solvers in AB, a mass $\alpha^{B}_{AB}$ of type-B humans are workers in AB, and a mass $\phi^{B} - \alpha^{B}_{AB}$ of type-B humans are solvers in BA, where:

Equilibrium wages, in turn, are:

(c) If $n(h^{A})/(1+n(h^{A})) \le\phi^{A}$, then $w^{A} = \overline{w}^{A}$ and $w^{B} = \max\{\overline{M}^{B}_{BA}, \overline{M}^{B}_{AB}\}$. Furthermore,

• If $\overline{M}^{B}_{BA} \ge\overline{M}^{B}_{AB}$, then all type-B humans are solvers in BA firms, a mass $\phi^{B}n(h^{A})$ of type-A humans are workers in BA, and a mass $\phi^{A} - \phi^{B}n(h^{A})$ of type-A humans are either doing independent production (if $m \in R^{A}_{s}$), or matched with machines (if $m \in(R^{A}_{b} \cup R^{A}_{t})$).

• If $\overline{M}^{B}_{BA} < \overline{M}^{B}_{AB}$, then all type-B humans are workers in AB firms, a mass $\phi^{B}/n(h^{B})$ of type-A humans are solvers in AB, and a mass $\phi^{A} - \phi^{B}/n(h^{B})$ of type-A humans are either doing independent production (if $m \in R^{A}_{s}$), or matched with machines (if $m \in(R^{A}_{b} \cup R^{A}_{t})$).

That the equilibrium maximizes total output follows because the First Welfare Theorem holds in this setting. The equilibrium is generically unique because it is uniquely determined whenever $m is such that the numbers $(\overline{w}_b^s,\overline{w}_b^k,\overline{w}_t^k,\overline{M}_{\mathrm{BA}}^k,\overline{M}_{\mathrm{AB}}^k)$ are all different from each other for both $k=A,B$. The subset of $[0,1]^2$ in which this occurs has Lebesgue measure one.

Regarding the nature of the equilibrium, point 1. is intuitive. Note that $\max\{\overline{M}_{\mathrm{BA}}^A,\overline{M}_{\mathrm{AB}}^A\}$ is an upper bound on the marginal product of a type-$A$ human in a two-layer non-automated organization, while $\overline{w}^A$ is the marginal product of a type-$A$ conditional on not matching with another human. Hence, whenever $\max\{\overline{M}_{\mathrm{BA}}^A,\overline{M}_{\mathrm{AB}}^A\}\leq\overline{w}^A$, it is never profitable to match a type-$A$ with another human, so no two-layer non-automated organization emerges in this case.

Consider then point 2. In case 2a, type $A$ is so scarce that only a fraction of $B$ humans can match with $A$ humans, no matter whether $A$ or $B$ are placed in the top layer. This immediately implies that $w^B=\overline{w}^B$ and that $\overline{M}_{\mathrm{BA}}^A$ and $\overline{M}_{\mathrm{AB}}^A$ are the actual marginal product of a type-$A$ human in a BA-firm and AB-firm, respectively. Naturally, in equilibrium all $A$ humans are then placed in the organization where their marginal product is highest, explaining why $w^A=\max\{\overline{M}_{\mathrm{BA}}^A,\overline{M}_{\mathrm{AB}}^A\}$. The intuition for case 2c is similar, but in that case it is the type-$B$ humans who are scarce.

Case 2b is more subtle. This is because neither type of human is scarce, in the following sense: Only a fraction of $B$ humans can match with $A$ if $A$ humans are placed on the bottom layer, but only a fraction of $A$ humans can match with $B$ if $B$ humans are placed on the bottom layer. Here, there are three possibilities.

First, when the marginal product of both types is higher in BA firms than in AB firms, i.e., $\overline{M}_{\mathrm{BA}}^k\geq\overline{M}_{\mathrm{AB}}^k$ for $k=A,B$, then all the available type-$A$ humans are placed at the bottom and some $B$ humans are forced to match elsewhere. Second, if the marginal product of both types is higher in AB firms than in BA firms, all the available type-$B$ humans are placed at the bottom and some $A$ humans are forced to match elsewhere. Third, when $\overline{M}_{\mathrm{BA}}^A<\overline{M}_{\mathrm{AB}}^A$ but $\overline{M}_{\mathrm{BA}}^B>\overline{M}_{\mathrm{AB}}^B$ (note that $\overline{M}_{\mathrm{BA}}^A>\overline{M}_{\mathrm{AB}}^A$ and $\overline{M}_{\mathrm{BA}}^B<\overline{M}_{\mathrm{AB}}^B$ can never happen due to (8)), then all humans match with each other: some in BA firms, and some in AB firms.

Proof. The argument for why the equilibrium maximizes total output and is generically unique was given above. Hence, here we focus on showing that the allocations and prices described in the statement are indeed an equilibrium. Point 1. was already established in the text above, so we focus on point 2.

Case 2a.—First, we show that market clearing is satisfied. Then we verify that the operating firms are obtaining zero profits and that no firm has incentives to deviate.

According to the statement, if $\overline{M}_{\mathrm{BA}}^A\geq\overline{M}_{\mathrm{AB}}^A$, then all type $A$ humans are hired as workers in BA firms. Hence, for market clearing to be satisfied there must be enough $B$ humans to act as solvers of all type-$A$ workers. That is $\phi^A/n(h^A)\leq\phi^B$, which occurs if and only if $\phi^A\leq n(h^A)/(1+n(h^A))$. This last condition is satisfied since $\phi^A\leq1/(1+n(h^B))<n(h^A)/(1+n(h^A))$.

Similarly, if $\overline{M}_{\mathrm{BA}}^A<\overline{M}_{\mathrm{AB}}^A$, then all type $A$ humans are hired as solvers in AB firms. Hence, for market clearing to be satisfied there must be enough $B$ humans workers for all type-$A$ solvers, i.e., $\phi^{A}n(h^{B}) \le\phi^{B}$. This occurs if and only if $\phi^{A} \le1/(1+n(h^{B}))$.

Now we show that firms are obtaining zero profits and have no incentives to deviate. Given that $w^{B}=\bar{w}^{B}$, the profits of BA and AB firms can be written as $\Pi_{\mathrm{BA}}=n(h^{A})[\overline{M}_{\mathrm{BA}}^{A}-w^{A}]$ and $\Pi_{\mathrm{AB}}=\overline{M}_{\mathrm{AB}}^{A}-w^{A}$. Consequently, if $\overline{M}_{\mathrm{BA}}^{A}\ge\overline{M}_{\mathrm{AB}}^{A}$, then $w^{A}=\overline{M}_{\mathrm{BA}}^{A}$, so $\Pi_{\mathrm{BA}}=0$ and $\Pi_{\mathrm{AB}}<0$. Similarly, if $\overline{M}_{\mathrm{BA}}^{A}<\overline{M}_{\mathrm{AB}}^{A}$, then $w^{A}=\overline{M}_{\mathrm{AB}}^{A}$, so $\Pi_{\mathrm{BA}}<0$ and $\Pi_{\mathrm{AB}}=0$. Moreover, because $w^{A}>\bar{w}^{A}$, no single-layer non-automated firm or two-layer automated firm can profitably hire a type-$A$ human.

Additionally, because $w^{B}=\bar{w}^{B}$, the $B$-types not hired by BA firms (when $\overline{M}_{\mathrm{BA}}^{A}\ge\overline{M}_{\mathrm{AB}}^{A}$) or by AB firms (when $\overline{M}_{\mathrm{BA}}^{A}<\overline{M}_{\mathrm{AB}}^{A}$) match where their marginal product of labor is the highest. That is, they are hired by single-layer non-automated firms if $m\in R_{s}^{B}$, by bottom-automated firms if $m\in R_{b}^{B}$, and by top-automated firms if $m\in R_{t}^{B}$.

Case 2b.— Suppose first that $\overline{M}_{\mathrm{AB}}^{A}\le\overline{M}_{\mathrm{BA}}^{A}$ and $\overline{M}_{\mathrm{AB}}^{B}\le\overline{M}_{\mathrm{BA}}^{B}$. Verifying that this is an equilibrium is analogous to case 2a when $\overline{M}_{\mathrm{BA}}^{A}\ge\overline{M}_{\mathrm{AB}}^{A}$.

Suppose instead that $\overline{M}_{\mathrm{BA}}^{A}\le\overline{M}_{\mathrm{AB}}^{A}$ and $\overline{M}_{\mathrm{BA}}^{B}\le\overline{M}_{\mathrm{AB}}^{B}$. Verifying that this is an equilibrium is analogous to case 2c when $\overline{M}_{\mathrm{BA}}^{B}\le\overline{M}_{\mathrm{AB}}^{B}$.

Finally, consider $\overline{M}_{\mathrm{BA}}^{A}<\overline{M}_{\mathrm{AB}}^{A}$ but $\overline{M}_{\mathrm{BA}}^{B}>\overline{M}_{\mathrm{AB}}^{B}$ (note that $\overline{M}_{\mathrm{BA}}^{A}>\overline{M}_{\mathrm{AB}}^{A}$ and $\overline{M}_{\mathrm{BA}}^{B}<\overline{M}_{\mathrm{AB}}^{B}$ can never happen due to (8)). In this case,

Since $w^{A}>\bar{w}^{A}$ (as $\overline{M}_{\mathrm{BA}}^{B}>\overline{M}_{\mathrm{AB}}^{B}$) and $w^{B}>\bar{w}^{B}$ (as $\overline{M}_{\mathrm{BA}}^{A}<\overline{M}_{\mathrm{AB}}^{A}$), then all humans match with each other. Moreover, $w^{A}$ and $w^{B}$ are such that:

Hence, BA and AB firms are both obtaining zero profits. Market clearing is satisfied since:

Finally, since $1/(1+n(h^{B}))<\phi^{A}<n(h^{A})/(1+n(h^{A}))$ and $\phi^{B}=1-\phi^{A}$, we have that $\alpha_{\mathrm{BA}}^{A}\in(0,\phi^{A})$ and $\alpha_{\mathrm{AB}}^{B}\in(0,\phi^{B})$ (i.e., the non-negative constraint on allocations is satisfied).

Case 2c.— This case is analogous to case 2a.

We finish this section with the following result:

Lemma 2.1. The equilibrium total labor income $w^* \equiv\phi^A w^A + \phi^B w^B$ is uniquely determined for each $\mathbf{m} \in[0,1]^2$. Furthermore, $w^*$ is continuous and almost everywhere differentiable in $\mathbf{m}$.

Proof. The result that the equilibrium total labor income $w^* \equiv\phi^A w^A + \phi^B w^B$ is uniquely determined for each $\mathbf{m} \in[0,1]^2$ follows from the observations that (i) total output equals capital income plus labor income, and (ii) in every equilibrium capital income is $\mu F(\mathbf{m})$. Hence, if there were two equilibria with different labor income, this would imply that these two equilibria achieve different output levels, which is impossible, since every equilibrium maximizes output.

The proof that $w^*$ is continuous in $\mathbf{m}$ is as follows. Consider the problem of maximizing output subject to the economy’s resource constraints. Noting that the marginal product of a type-$k$ human who is not matched with another human is $\bar{w}^k$ (as machines are abundant), total output in this economy can be written as:

The resource constraints of this economy are:

By the First Welfare Theorem, the equilibrium maximizes (9) subject to (10) and the non-negativity constraints $\alpha^A_{\mathrm{BA}} \ge0$ and $\alpha^B_{\mathrm{AB}} \ge0$.

Since (10) coupled with the non-negativity constraints $\alpha^A_{\mathrm{BA}} \ge0$ and $\alpha^B_{\mathrm{AB}} \ge0$ implies that $\alpha^A_{\mathrm{BA}} \in[0,\phi^A]$ and $\alpha^B_{\mathrm{AB}} \in[0,\phi^B]$. Hence, the constraints of the problem are continuous in $\mathbf{m}$ and compact-valued. Furthermore, since $Y$ is continuous in $\mathbf{m}$, by the Continuous Maximum Theorem [1], the value function $Y^*(\mathbf{m})$ is continuous in $\mathbf{m}$. Thus, total labor income $w^* = Y^*(\mathbf{m})-\mu F(\mathbf{m})$ is continuous in $\mathbf{m}$.

Finally, let us argue that $w^*$ is almost everywhere differentiable. It follows from Proposition 2.1 that whenever $\mathbf{m}$ is such that the numbers $(\bar{w}^A_s,\bar{w}^A_b,\bar{w}^A_t,\bar{M}^A_{\mathrm{BA}},\bar{M}^A_{\mathrm{AB}})$ are all different from each other for both $k=A,B$, then $w^A$ and $w^B$ are both differentiable in $\mathbf{m}$, and hence so is $w^*$. The subset of $[0,1]^2$ in which this occurs has Lebesgue measure one. Thus $w^*$ is almost everywhere differentiable in $\mathbf{m}$.

Example 1 of Section 6

Recall that in this example, $F(\mathbf{x})=x_1,x_2$, $c=1/2$, $\phi_A=0.8$, $\mathbf{h}^A=(0.375,0.475)$, $\mathbf{h}^B=(0.625,0.725)$, and $\mathbf{m}=(0.29,0.7)$. Hence, we have that $n(\mathbf{h}^A)/(1+n(\mathbf{h}^A))=0.709$, $1/(1+n(\mathbf{h}^B))=0.215$. Moreover,

Thus, $\bar{w}^{A} = \bar{w}^{A}_{t} = 0.179$ and $\bar{w}^{B} = \bar{w}^{B}_{b} = 0.628$. This implies that:

Hence, $\max\{\overline{M}^{A}_{BA},\overline{M}^{A}_{AB}\} = 0.195 > \bar{w}^{A}$ and $\phi_{A} = 0.8 > n(h^{A})/(1+n(h^{A}))$, so this corresponds to Case 2c of Proposition 2.1. As a result, $w^{A} = \bar{w}^{A} = 0.179$, and $w^{B} = \max\{\overline{M}^{B}_{BA},\overline{M}^{B}_{AB}\} = \overline{M}^{B}_{BA} = 0.667$.

We conclude that all $\phi^{B} = 0.2$ type-$B$ humans are solvers in $BA$ firms, $\phi^{B}n(h^{A}) = 0.487$ of type-$A$ humans are workers in $BA$ firms, and the remaining $\phi^{A} - \phi^{B}n(h^{A}) = 0.313$ type-$A$ humans are workers in top-automated firms.

Proof of Proposition 7

The result that $w^{*}$ is continuous and almost everywhere differentiable in $m$ has been established by Lemma 2.1 above.

If $m \in\operatorname{int}R_{h}$, marginal changes in $m$ keep it at $R_{h}$, where humans do not match with machines. As a result, since the production function of non-automated firms is independent of $m$, total labor income is not affected by such changes.

Now suppose that $m$ is not in $R_{h}$. If $m$ is in the boundary of $R_{h}$, it is not differentiable, so assume otherwise (i.e., a strictly positive fraction of humans match with machines both before and after marginal changes in $m$). Since the production function of non-automated firms is independent of $m$, marginal changes in $m$ that strictly reduce the marginal product of every type of labor when matching with machines necessarily strictly reduce labor income. By Proposition 3, this is the case when $m_{i} < \min\{h^{A}_{i},h^{B}_{i}\}$ and $m_{j} \ne0$. Similarly, marginal changes in $m$ that do not change the marginal product of any type of labor when matching with machines necessarily keep labor income constant. By Proposition 3, this is the case when $m_{i} < \min\{h^{A}_{i},h^{B}_{i}\}$ and $m_{j} = 0$. Finally, marginal changes in $m$ that strictly increase the marginal product of every type of labor when matching with machines necessarily increase labor income. By Proposition 3, this is the case when $m_{i} > \min\{h^{A}_{i},h^{B}_{i}\}$.

Proof of Proposition 8

When choosing $m$ to maximize $w^{k}$, it suffices to consider the case where type $k$ matches with machines because there always exists a machine that can perfectly replicate a type-$k'$ human and is (weakly) cheaper than that human (as machines are abundant). Hence, the problem of choosing an $m$ to maximize $w^{k}$ is the same regardless of the existence of other types of humans, so the first part of Proposition 8 directly follows from Proposition Proposition 4.

Let us now show by example that the AI that maximizes total labor income $w^{*}$ need not be a vertex of $[0,1]^{2}$. Consider $h^{A} = (0.05, 0.05)$, $h^{B} = (0.05, 0.8)$ and $\phi^{A} = 2/3$. The equilibrium labor income $w^{k}(m)$ for each type and total labor income $w^{*}(m)$ as a function of $m$ in each of the vertices of $[0,1]^{2}$ is:

Consider then $w^{k}(m)$ and $w^{*}(m)$ when $m=(1,0.08)$. In this case:

Since $w^{*}(1,0.08)>\max\{w^{*}(0,0),w^{*}(0,1),w^{*}(1,0),w^{*}(1,1)\}$, we conclude that neither of the vertices of $[0,1]^{2}$ can be a global maximum.