Optimal Taxation of Intermediate Goods in A Partially

Automated Society *

Hideto Koizumi

October 20, 2020

Abstract

The recent automating technologies have sparked discussions on “robot taxes,”aimed to dissuade the displacement of labor and to generate revenue to redistribute tothe displaced laborers. Implementing such taxes is challenging, however, in part be-cause of the difficulty in clearly separating which technologies substitute for labor fromthose which complement it. Modeling automating technologies as intermediate goods,I consider optimal tax policy in this environment. As in standard models, non-linearlabor taxes are assigned without knowledge of a laborer’s type. Additionally, due to taxavoidance concerns and arbitrage opportunities, intermediate goods are uniformly andlinearly taxed without knowledge of their complementarity or substitutability with la-bor. Despite the potential for automating technologies to be complementary to workers,I find that the optimal tax regime includes a strictly positive tax on these intermediategoods. I discuss the implications of these findings for the robustness of robot-tax policyproposals.

Keywords: Automation, Optimal Taxation of Intermediate Goods, Optimal Taxa-tion, Robot Tax

JEL Codes: H21, H25

“I’m sure I can come up with a robot that isn’t a robot, according to the [robot] tax code.”—

Shu-Yi Oei, a Boston College law professor1

*The author thanks Benjamin Lockwood, Juuso Toikka, Alexander Rees-Jones, and Kent Smet-ters for extensive guidance and instructions in this work. The author also thanks Eduardo Azevedofor comments and guidance, and thanks seminar and conference discussants and audiences for usefulfeedback. This project is funded by Mack Institute Research Grants for presentations abroad.

1This quote is excerpted from a Wall Street Journal article, “The ‘Robot Tax’ Debate HeatsUp” on January 8, 2020.

1

1 Introduction

Historically, routine labor has been displaced by automating technologies. This trend has

been accelerated during the last three decades (Acemoglu and Restrepo (2019)), leading

to increasing inequality (Acemoglu and Restrepo (2020)). Furthermore, using the monthly

Current Population Survey, Ding et al. (2020) document that the COVID-19 pandemic has

been encouraging even further displacement of routine labor by automation2. The rapidly

declining labor share and rising inequality have led some researchers including, Costinot and

Werning (2018), Thuemmel (2018), and Guerreiro et al. (2020) to consider taxing robots, to

redistribute income to these displaced workers3.

Implementing taxes such as a “robot tax” is challenging, however, in part because of

the difficulty in clearly separating which intermediate goods perfectly substitute for labor

from those which complement it. There are numerous examples of these difficult cases

such as self-check-out cash registers vs. conventional cash registers, self-driving trucks vs.

conventional trucks, and industrial robots that displace assembly line workers vs. robots

designed to augment the productivity of assembly line workers. Screening intermediate goods

that perfectly substitute for routine labor from those which complement it is administratively

costly and may not be even feasible. Moreover, even if the planner decides to pay these

administrate costs and imposes a robot tax4, tax avoidance will be a concern as encapsulated

by the epigraph above and as documented in Slemrod and Kopczuk (2002)5.

To address these practical issues, this paper studies the welfare consequences of imposing

a tax on intermediate goods (such as robots) when their type cannot be determined by

the planner. In particular, my model considers a two-by-two scenario: two types of labor

and two types of intermediate goods. Workers are categorized into (low-skill) routine and

(high-skill) non-routine labor. On the other hand, intermediate goods are dichotomized into

2Some news articles after the inception of the COVID-19 pandemic, such as the ones from Time(https://time.com/5876604/machines-jobs-coronavirus/) and the New York Times (https://www.nytimes.com/2020/04/10/business/coronavirus-workplace-automation.html) raise aserious concern that the pandemic accelerates the already concerning automation movement.

3Bill Gates suggested a robot tax to compensate the tax revenue lost from the displacement, adifferent reason from Costinot and Werning (2018), Thuemmel (2018), and Guerreiro et al. (2020)

4Or ‘automation tax’ suggested by Acemoglu et al. (2020). They suggest to impose a higher taxon the use of capital in tasks where labor has a comparative advantage. However, this tax policyalso has the same issues of task screening and tax avoidance.

5Also, see Slemrod and Yitzhaki (2002).

2

(i) displacing intermediate goods (e.g., self-check-out cash registers and self-driving trucks),

which are complements to non-routine labor but perfect substitutes for routine labor and

(ii) complementing intermediate goods (e.g., conventional cash registers and conventional

trucks), which are more complementary to routine labor than to non-routine labor. The two

types of intermediate goods are perfect substitutes. I believe that this model assumption is

natural in modeling automation; for instance, if a firm buys a self-driving truck for a certain

distribution task, there is no point of additionally purchasing a conventional truck for this

task.

As in standard models, labor taxes must be assigned without knowledge of a laborer’s

type and inputs, while I go beyond standard models and additionally impose that interme-

diate goods must be taxed without knowledge of their complementarity or substitutability

with routine labor. The government observes labor income levels and taxes them with a

nonlinear schedule. An intermediate good tax is imposed with a uniform, proportional rate

over different types of intermediate goods. The non-discriminatory tax rate over different

intermediate good types addresses the aforementioned screening and tax avoidance concerns.

Meanwhile, following Guesnerie (1998), I focus on a proportional intermediate good tax:

non-linear taxes on intermediate goods could generate arbitrage opportunities at the resale

market.

Due to asymmetric information on intermediate good types, there are two opposing forces

of non-discriminatory intermediate good taxation in welfare. On one hand, by taxing just

displacing intermediate goods, the planner can reduce the wage gap between the two labor

types. Taxing a complement of non-routine workers will decrease non-routine-worker wage

rates while taxing a substitute for routine workers will increase routine-worker wage rates.

The reduction of the wage gap will relax the incentive compatibility constraint of non-routine

workers to mimic routine workers and reduce their hours of work to that of routine workers—

that is, a reduction in the information rent of non-routine types—in the planner’s welfare

maximization program. On the other hand, taxing just complementing intermediate goods

will decrease the wage rates of not only non-routine workers but also routine workers since

complementing intermediate goods are complements to both types of labor. A decrease

in the routine-workers’ wage rates will decrease welfare and possibly dominate the positive

3

redistribution effects from taxing displacing intermediate goods.

Despite these competing forces and complex settings, I find that there is a simple so-

lution in which on top of optimal income taxation, the planner imposes a strictly positive

intermediate-good tax that is non-discriminatory over types of intermediate goods, for redis-

tributive purposes. Note that this result holds regardless of the degree of complementarity

between routine labor and complementing intermediate goods. Therefore, the result is not

driven solely by the relative degree of substitutability and complementarity among the two

intermediate good types and routine labor.

A novel key force driving the positive result exploits the price difference between the

two types of intermediate goods. As a unique feature of an automation model, perfectly

substituting intermediate goods such as robots need zero routine labor by definition. This

implies that the price of complementing intermediate goods has to be lower than that of

displacing intermediate goods in a partially automated economy since complementing inter-

mediate goods require routine labor for potentially automatable tasks6. Thus, if we impose

a uniform proportional tax on both types of intermediate goods, the tax burden is placed

disproportionally on displacing intermediate goods than complementing intermediate goods

since displacing intermediate goods is more expensive. This differential tax burden reduces

the wage gap between two worker types. The reduced wage gap relaxes the incentive com-

patibility constraints of non-routine workers, resulting in the first-order informational gain.

Utilizing parameter estimates from other papers, I conduct a simulation and demon-

strate a non-negligible welfare improvement measured by an increase in the routine laborer’s

consumption amount. While the degree of improvement depends largely on the degree of

complementarity between complementing intermediate goods and routine labor, intermedi-

6Note that there are two corner solutions: one without any use of routine labor and thus com-plementing intermediate goods, and the other without any use of displacing intermediate goodsand with the composite of routine labor and complementing intermediate goods and non-routinelabor. At the former corner solution, only displacing intermediate goods and non-routine labor willbe used in production. Then, the setting becomes equivalent to one of the corner solutions fromGuerreiro et al. (2020), and the optimal intermediate good tax rate is zero. This is because there isno redistributive gain from intermediate good taxation in the absence of routine labor. In contrast,at the latter corner solution, only the composite of routine labor and complementing intermediategoods together with non-routine labor will be used in production. In this case, the optimal interme-diate good tax rate is generally negative since this setting in the absence of displacing intermediategoods tends to be the opposite case of the interior solution of Guerreiro et al. (2020). These aretrivial cases and thus will not be explored in the main text.

4

ate good taxation can increase the consumption amount by 0.6% that is a lower bound (a

mean estimate is 4.8%), even after the planner has already imposed the optimal non-linear

labor income tax. This is when the price of displacing intermediate goods is sufficiently

higher than that of complementing intermediate goods; the welfare improvement dies out as

the price gap becomes reduced.

The contribution of this paper is two-fold. One is a modeling contribution. By studying

a model of heterogeneous labor and intermediate good types with asymmetric information

over both labor and intermediate good types, this paper is able to deal with the significant

policy issues surrounding robot tax policy proposals. As far as I know, this is the first study

to analyze the novel setting with asymmetric information over both labor and intermediate

good types in the optimal taxation literature.

The other point of contribution is the discovery of a new force to mitigate the afore-

mentioned asymmetric information problems by a non-discriminatory (linear) tax rate. The

planner wants to tax the use of automating technologies but wants to subsidize the use of

complementing technologies. In automation models, since automating technologies are per-

fect substitutes to routine labor while complementing technologies require routine labor, the

factor price of complementing technologies has to be lower than that of automating coun-

terparts. Then, even if the planner imposes a positive, uniform proportional tax over these

different technologies, it results in differential burden on different technology types and thus

reduces the asymmetric information problems. Despite the level of complexity my model

permits, this novel channel leads to a simple, practical solution for the robustness of robot

tax policy proposals.

1.1 Literature Review

As the seminal paper of optimal taxation of intermediate goods, Diamond and Mirrlees

(1971a) finds no production distortion to be optimal. They demonstrate that if the planner

can tax net trades of different goods at different linear rates, then taxing intermediate goods

is suboptimal. This condition in Diamond and Mirrlees (1971a) implies that the planner can

distinguish between different labor types and can impose different tax rates over different

labor types even when these workers earn the same income, violating the premise of my

5

model. In fact, if the government can implement different tax schedules between different

labor types, this will achieve perfect redistribution, leaving no room for intermediate good

taxation in my model.

And yet, the restriction to the same income tax schedule is insufficient to justify pro-

duction distortion for redistribution. Atkinson and Stiglitz (1976) find that if workers with

different productivities are perfect substitutes, then production efficiency is still optimal.

My model assumes complementarity between routine and non-routine labor, and thus their

result is not applicable.

In fact, building on the work of Stiglitz (1982), Naito (1999) is the first study to show

that sacrificing production efficiency for redistribution may be optimal when heterogeneous

labor types are imperfect substitutes. The important channel of redistribution in Naito

(1999) is to indirectly increase the wage of low-skill workers and decrease that of high-skill

workers by taxing skilled labor-intensive goods that are complements to high-skill workers

but substitutes for low-skill workers.

This channel is used by the three recent papers mentioned in the introduction section

that find a positive robot tax rate to be optimal for income redistribution. One is Guerreiro

et al. (2020), the closest paper to mine78. They question if it is welfare-improving to tax

robots, using the task-based framework emphasized by Autor et al. (2003) and Acemoglu

and Restrepo (2018). In their settings, there are two types of labor, routine and non-routine,

between which the policymaker cannot distinguish or cannot impose different tax rates based

on labor types, and one type of intermediate goods—robots. Robots are complements to

non-routine workers, but substitutes for routine workers. Guerreiro et al. (2020) find an

optimal robot tax to be, in general, strictly positive, using the channel.

Note that they analyze a dynamic extension, focusing on their calibration-based quan-

7Thuemmel (2018), a concurrent work to Guerreiro et al. (2020), is similar to Guerreiro et al.(2020) but has richer elements in that he allows for additional labor type and heterogeneity withinoccupations. However, he also does not incorporate the additional type of intermediate goods.

8Among many related papers such as the direct taxation literature exemplified by Diamondand Mirrlees (1971a,b), Atkinson and Stiglitz (1976), Deaton (1979), Saez (2001, 2004), Rothschildand Scheuer (2013), Scheuer (2014), Stantcheva (2014), and Gomes et al. (2017), (2) the indirecttaxation literature exemplified by Naito (1999), Saez (2002), Slavık and Yazici (2014), and (3) thenew dynamic public finance literature exemplified by Golosov et al. (2003), Kocherlakota (2005),Golosov et al. (2006), Golosov et al. (2011), Farhi and Werning (2013), Golosov et al. (2016),Stantcheva (2017), I select and discuss on the most closely related ones in this section.

6

titative analysis of a dynamic version of their static model. In their dynamic model, they

incorporate an endogenous skill choice process with heterogeneous skill acquisition costs.

This addition introduces an additional negative effect of a positive robot tax since it reduces

incentives to acquire non-routine skill sets. Therefore, the optimality of a robot tax becomes

a quantitative question in the dynamic model. They calibrate the dynamic model with a

geometrically declining robot cost and find a positive robot tax in the beginning when the

cost of robots is still high. Given their result, the optimality of an intermediate good tax

in the dynamic model with the additional intermediate good types is also a quantitative

question. For this reason and partly because of technical and computational difficulties9, I

focus on a static case and add two additional components to their static model10. One is

complementing intermediate goods introduced in the introduction11. The other one is asym-

metric information on intermediate good types. As emphasized in the introduction, these

additions are not just theoretically interesting exercises, but also important elements for a

practical policy recommendation.

Another paper close to mine is Costinot and Werning (2018). They study the optimal

robot tax rate using a sufficient statistics approach. The goal of their study “is not to

sign the tax on robots, nor to explore a particular production structure, but instead to offer

tax formulas highlighting key sufficient statistics needed to determine the level of taxes, with

fewer structural assumptions” (p.4). Without imposing any structural assumption, they find

an optimal non-zero robot tax should converge to zero over time as the process of automation

deepens.

The important assumption in their setting is that the planner know which firms use “new

technology” such as robots and which firms use “old technology” involving conventional non-

robotic intermediate goods. One can translate displacing intermediate goods into the “new

technology firm” and complementing intermediate goods into “old technology firm.” In this

9As shown in the 2019 version of Guerreiro et al. (2020), a dynamic model can allow for greatergenerality exemplified in functional forms of production function. This is true, however, only whenthere is only one type of capital/intermediate goods. I have conducted similar exercises of their2019 version with my model of two intermediate goods and found that it quickly gets analyticallyintractable and computationally difficult.

10Displacing intermediate goods in my model corresponds to robots in their model.11Slavık and Yazici (2014) have two different types of intermediate goods in their dynamic model:

structure and equipment. Yet, they have no asymmetric information on intermediate good typesin their settings and thus do not allow for the situation of our interests.

7

sense, they study a scenario where there is no asymmetric information on intermediate good

types in my model without structural assumptions. In this paper, my main focus is on a

case that involves asymmetric information over intermediate good types.

2 Model Environment: Key Players in the Economy

And Equilibrium Conditions

To reiterate, I focus on a static model. In a static model, intermediate good taxation can

be interpreted in a way that the planner reduces the total amount (undiscounted sum)

of tax depreciation deductions, a widely practised tax policy in the world12. This paper

follows the notations from Guerreiro et al. (2020), so that the readers can closely compare

two papers. There are two types of households, whose utility is based upon consumption

of private goods and whose disutility comes from labor. Note that in the main body of

this paper for analytical results, to highlight the aforementioned novel key force, I drop

the term for household consumption of public goods, while all the analytical results carry

through with the addition of the term as depicted in the appendix. One type of household

provides routine labor, while the other type provides non-routine labor. As assumed in Naito

(1999), Slavık and Yazici (2014), and the static version of Guerreiro et al. (2020), I focus

on a short-term analysis with fixed occupations in which the current generation of routine

labor households cannot afford sufficient education to become non-routine households, while

they are allowed to change labor supply amounts13. Routine labor, non-routine labor, and

displacing and complementing intermediate goods all contribute to the production of the

single consumption good. There is a continuum of symmetric tasks in a unit range [0, 1]

12For example, the U.S. federal government allows firms to deduct tax for the purchase of de-preciable assets by the amount equal to the the asset values over time following the depreciationschedule set by the Internal Revenue Service. In the static version of my model, the federal plannercan impose a tax on intermediate goods by reducing the total deduction amounts of such pur-chase. For example, if a firm purchases $70,000 depreciable assets that follows 7-year depreciationschedule, currently, the firm is allowed to deduct the entire $70,000 in year one (written off as anexpense) or deduct $10,000 over seven years. The federal planner can impose 10% intermediategood tax by reducing the total deduction amount to $63,000.

13Saez (2004) allows workers of different types to switch their jobs/sector to work while prohibitthem to change labor supply amounts. He models that in the long run, the relative supply toeach occupation becomes infinitely elastic, even though workers of different types can be imperfectsubstitutes in production. This leads to the resurrection of the Diamond-Mirrlees and Atkinson-Stiglitz no production distortion results. In contrast, by allowing for both occupational choice andlabor supply adjustments, Gomes et al. (2017) found production inefficiency at optimum.

8

where routine workers together with complementing intermediate goods compete against

displacing intermediate goods14. I incorporate tasks in my model not only to closely follow

Guerreiro et al. (2020) for comparison, but also to precisely elicit what assumptions warrant

the aggregation of production. Intermediate good producers for both intermediate good

types come from perfectly competitive markets. The final good producer faces a production

function featuring constant returns to scale and thus faces zero profits.

2.1 Household

The economy has a continuum of households with a unit measure. The unit measure of

households are decomposed to πn non-routine worker households and πr routine worker

households, where subscript n and r denote the non-routine and routine labor types, respec-

tively. A household of type j derives utility from consumption of private goods, cj. Again,

note that in the main body of this paper for analytical results, to highlight the aforemen-

tioned novel key force, I drop the term for household consumption of public goods, while

all the analytical results carry through with the term as depicted in the appendix. Each

household draws disutility from the hours of labor it supplies, lj. Every household has a unit

of time per period, leading to lj ≤ 1. A household of type j’s optimization problem is

maximizecj ,lj

Uj = u(cj, lj)

subject to cj ≤ wjlj − T (wjlj),

where wj denotes the wage rate of type j and T (.) indicates the income tax schedule15.

For convenience, write ux = ∂u(c, l)/∂x where x = c, l and uxy = ∂2u(c, l)/∂x∂y. I make

the standard concavity and convexity assumptions that, uc > 0, ul < 0, ucc, ull < 0, and

that consumption and leisure are both normal goods. Furthermore, I assume that the utility

14See Autor et al. (2003) for the importance of tasks performed by routine workers.15In the appendix for general results, the maximization problem becomes

maximizecj ,lj

Uj = u(cj , lj) + v(G)

subject to cj ≤ wjlj − T (wjlj),

with assumptions that v′(G) > 0, v′′(G) < 0.

9

functions satisfy the single-crossing property and thus, we are given ulll/ul + 1 − ucll/uc >

016. Additionally, I assume that u(c, l) satisfies the standard Inada conditions for interior

solutions.

2.2 Final good producers

The representative producer of the final good employs non-routine workers (Nn) for the single

non-automatable task, routine labor (n(i)) for symmetric, automatable task i, and buys

intermediate goods, a displacing intermediate goods (xd(i)) and complementing intermediate

goods (xc(i)), for the same automatable task i as well. For routine task i, the final producer

faces the following task-production function:

y(i) = κixd(i) +[βo(i)xc(i)

q−1q + (1− β)z(i)n(i)

q−1q

] qq−1

, (1)

where q indicates an elasticity of substitution between routine workers and complementing

intermediate goods, while κ(i), o(i), z(i) represent the efficiency parameters of displacing

intermediate goods, complementing intermediate goods, and routine labor. Note that as

assumed in the static model of Guerreiro et al. (2020), for tractability and clarity, I also

assume constant efficiency across tasks—that is, κ(i) = o(i) = z(i) = 1. Furthermore, for

routine labor and complementing intermediate goods to be gross-complements, q is assumed

to be between 0 and 1. With these routine tasks, the producer has the following final

production function:

Y =A

(∫ 1

0

y(i)ρ−1ρ di

) ρρ−1

(1−α)

Nαn α ∈ (0, 1) , ρ ∈ [0,∞) , (2)

where ρ is an elasticity of substitution between tasks. Now, let m denote the share of task

production by displacing intermediate goods in automatable tasks —that is, m =∫ 10 xd(i)di∫ 10 y(i)di

.

For convenience, since tasks are symmetric and have no difference in needs for productivity,

I write that for the range [0,m], the final good producers only use displacing intermediate

goods, while for the range (m, 1], the final good producer only uses the composite of routine

16While the single-crossing condition is usually for reducing the dimension of continuously manylabor types, it is still useful to ease the analysis.

10

workers and complementing intermediate goods. Thus, m is a choice variable for the final

good producer.

Then, we can divide the integral into the two different ranges and rewrite the production

function as:

Y = A

[∫ m

0

xd(i)ρ−1ρ di+

(∫ 1

m

[βxc(i)

q−1q + (1− β)n(i)

q−1q

] ρ−1ρdi

) qq−1

] ρρ−1

(1−α)

Nαn ,

α ∈ (0, 1) , ρ ∈ [0,∞) . (3)

Note that this way of defining automated and not-yet automated tasks is different from

Acemoglu and Restrepo (2018). I abstract away from the quality of tasks a new technology

destroys and creates but rather focus on the quantity of tasks being destroyed and created17.

With the production function, the final good producer’s problem is to maximize profits,

Y − wnNn − wr∫ 1

m

n(i)di− (1 + τx)

∫ 1

0

(φdxd(i) + φrxc(i)) di, (4)

where τx is an ad-valorem uniform tax rate on intermediate goods. Notice that my production

function features constant returns to scale, leading to the zero profits property.

Proposition 1 below simplifies the production function (3). The intuition behind the

derivation is that since all the marginal costs (φd, φr, wr) for automatable tasks are constant

across tasks, the parameter for the elasticity of substitution gets dropped. To derive it, we

need the following technical assumption.

Assumption 2.1. Every input for routine tasks, xd(i), xc(i) and n(i), has a positive measure

for assigned tasks. That is, xd(i) has a positive measure on i ∈ [0,m] and xc(j) and n(j)

have positive measures on j ∈ (m, 1].

With this assumption, the following proposition follows. For readability, define total

17In their model, tasks are not symmetric but vary by the needs for productivity. More precisely,the productivity demand increases with task index for labor but is constant for capital in theirmodel. In my model, tasks are dichotomized into routine tasks and the single non-routine task.Acemoglu and Restrepo (2018) were able to find many interesting results based on their rich settings.However, their focus is to analyze the effects of automation on growth patterns and find conditionsfor balanced growth. In contrast, my main focus is to examine the sign of intermediate good taxif not zero. Thus, for analytical tractability, I make an additional simplifying assumption on needsfor labor productivity across different tasks.

11

amounts of every input as

Xd =

∫ m

0

xd(i)di, Xc =

∫ 1

m

xc(i)di, Nr =

∫ 1

m

n(i)di.

Proposition 1. Let Assumption 2.1 hold. Then, we can simplify the production function as

Y = A

(Xd +

(βX

q−1q

c + (1− β)Nq−1q

r

) qq−1

)1−α

Nαn (5)

Note that my production function contains that from Guerreiro et al. (2020) as a special

case. To see this, let q −→∞ so that it becomes

Y = A (Xd + βXc + (1− β)Nr)1−αNα

n .

If φd = 1βφr, then the production function becomes equivalent to that of Guerreiro et al.

(2020).

2.3 Intermediate Good Producers

Both types of intermediate goods are produced by perfectly-competitive intermediate good

producers in the (global or domestic) market and are used in automatable tasks i ∈ [0, 1].

Following Guerreiro et al. (2020), I assume that tasks are symmetric—i.e., there is no dif-

ference among tasks in needs for productivity. Then, the price of intermediate goods type

k ∈ {d, r}, pk(i), is same across tasks—that is, pk(i) = pk ∀i— and is equal to marginal costs

of φk units of output18.

Following the literature, a change in the income flows to these producers in response

to intermediate good taxation will not be considered. If the welfare function includes such

18As in the previous studies, I assume that a tax on intermediate goods will not change the costof these intermediate goods. It is entirely possible that the cost of complementing intermediategoods changes after an intermediate good tax due to a cost increase of the supplier in the supplychain, and so does the cost of displacing intermediate goods. What’s important in this paper is thatthe sign of optimal intermediate-good tax does not alter as long as the after-tax cost of displacingintermediate goods is higher than the after-tax cost of complementing intermediate goods. Notethat if this is not true, then full automation will take place and there is no need for taxation ofintermediate goods.

12

income changes, then the optimal intermediate good tax is expected to be even higher. This

is because non-routine workers are expected to own more intermediate goods than routine

workers, making greater room for redistribution. Therefore, my main result is a lower bound

in this sense.

2.4 Government

In the main body for analytical results, while the government set tax rates, for clarity, it does

not provide any public good using the revenue and throws away the tax revenue to trash.

I bring back the public good provision in the appendix for general results and numerical

analysis (for close comparison with the existing studies) in which the government faces the

budget constraint:

G ≤ πrT (wrlr) + πnT (wnln) + τx

∫ 1

0

[φdxd (i) + φrxc(i)] di. (6)

Note that I can also include this budget constraint with an exogenously given government

spending target G, and all the theoretical results hold true.

2.5 Equilibrium

An equilibrium is defined as the collection of a set of allocations

{cr, lr, cn, ln, Nr, Xd, Xc, xd(i), xc(i), n(i),m}, prices {wr, wn, px}, and a tax system {T (.), τx}

such that: (i) given prices and taxes, allocations solve the households’ problem; (ii) given

prices and taxes, allocations solve the firms’ problem; and (iii) markets clear19.

The market-clearing conditions for routine and non-routine labor are given by

(1−m)n(i) = Nr = πrlr, (7)

Nn = πnln. (8)

19In the generalized version and numerical analysis, an equilibrium further includes governmentspending in the collection of a set of allocations and condition (iv) the government budget constraintis satisfied.

13

The market-clearing condition for displacing intermediate goods is

mxd(i) = Xd, (9)

while that for complementing robots is

(1−m)xc(i) = Xc, (10)

Similarly, the market-clearing condition for the output market is

πrcr + πncn ≤ Y −∫ 1

0

(φdxd(i) + φrxc(i)) di. (11)

Also, note that we get an interior solution only if for a given level of total task-output,

the total costs are the same between displacing intermediate goods and the composite of

routine workers and complementing intermediate goods. Equivalently,

(1 + τx)φd =wrn(i) + (1 + τx)φrxc(i)(

βxc(i)q−1q + (1− β)n(i)

q−1q

) qq−1

=wrNr + (1 + τx)φrXc(

βXq−1q

c + (1− β)Nq−1q

r

) qq−1

. (12)

Then, the constant marginal cost of each input implies that at an interior solution,

xd(i) =(βxc(i)

q−1q + (1− β)n(i)

q−1q

) qq−1

for every i. This implies the following relation at

an equilibrium with partial automation (m ∈ (0, 1)):

xd(i) =Xd

m=

(βX

q−1q

c + (1− β)Nq−1q

r

) qq−1

(1−m), for i ∈ [0,m]. (13)

Notice that using the fact that mxd(i) = Xd and xd(i) =(βxc(i)

q−1q + (1− β)n(i)

q−1q

) qq−1

for every task i, one can also express m as

m =Xd

Xd +

(βX

q−1q

c + (1− β)Nq−1q

r

) qq−1

. (14)

14

At an interior solution, by taking the first-order conditions of the profit function with respect

to the inputs and plugging in the equilibrium conditions above, we get

wn = αA1/α(1− α)

1−αα

[(1 + τx)φd]1−αα

, (15)

Xc =

1− β(φrβφd

)q−1− β

qq−1

πrlr, (16)

wr = (1− β)(1 + τx)φd

(1− β)(φrβφd

)q−1(φrβφd

)q−1− β

1q−1

. (17)

Note that when 0 <(φrβφd

)q−1< β, wr < 0. In other words, as long as φr

φd< β

qq−1 , wr > 0.

Then, we can derive

m = 1−(

(1 + τx)φd(1− α)A

) 1α πrlrπnln

β(1− β)(φrβφd

)q−1− β

+ (1− β)

qq−1

. (18)

2.6 Main Result: Asymmetric Information at Both HouseholdAnd Production Sides

In this section, the planner can observe neither household types nor intermediate good

types. What the planner observes are households’ reported income levels. For the reasons

highlighted in Guesnerie (1998), I focus on a linear robot tax. Note that in contrast to

Mirrlees (1971), the productivities of different agents are endogenous; they depend upon

τx. This feature of my model induces trade-offs between redistribution and production

efficiency by imposing a robot tax. To avoid degenerate cases, I focus on the case with

partial automation, 0 < m < 1. Furthermore, I assume that wn ≥ wr in an equilibrium.

In the Mirrlees’ settings, the policymaker’s problem is to choose allocations {cj, lj}j=r,n , G

and a uniform intermediate good tax rate τx to maximize the following utilitarian social

welfare:

15

πrωrUr + πnωnUn, (19)

where ωj are social weights for household of type j. These weights are normalized in a way

that πrωr + πnωn = 1. Following Guerreiro et al. (2020), I also focus on a case ωr ≥ 1, so

that the planner puts either equal or more weights on routine workers. The planner faces

the following resource constraint20:

πrcr + πncn ≤ πnwnln

(τx + α

α(1 + τx)

)+

πrwrlr(1 + τx)

. (20)

The planner also faces households’ incentive compatibility problems. For households of

type j to choose bundles {cj, lj}, the bundle must yield at least as high as the utility level of

any other arbitrary bundle choice {c, l} that satisfies the budget constraint c ≤ wjl−T (wjl).

This implies that u(cj, lj) ≥ u(c, l). In particular, routine workers must prefer their own-type

bundle, {cr, lr}, to the bundle that they would obtain from masquerading the non-routine

type by adjusting their hours of work, {cn, wnln/wr}. Also, non-routine workers must prefer

their own-type bundle, {cn, ln}, to the bundle that they would obtain from masquerading

the routine type by adjusting their hours of work, {cr, wrlr/wn}. Note that an important

assumption here is that mimicking the other type will not alter that worker’s productivity21.

We can write these as

u (cn, ln) ≥ u

(cr,

wrwnlr

), (21)

u (cr, lr) ≥ u

(cn,

wnwrln

). (22)

Recall that the wages of two types of households are given by (15) and (17). These are

necessary conditions for the household optimality and thus for an equilibrium.

In the following lemma, I show that the resource constraint (20) and the incentive com-

patibility constraints (21) and (22) are also sufficient conditions for an equilibrium. To

establish that, the planner can set an income tax schedule such that for example, the gov-

20To get this, plug in equilibrium conditions of endogenous variables to the output market-clearing condition. The details of the derivation are in the proof for Lemma 1.

21See Scheuer and Werning (2016) for discussion on this point.

16

ernment will appropriate the entire income if a household reports income levels other than

wrlr and wnln.

Lemma 1. The resource constraint (20) and the incentive compatibility constraints (21) and

(22) are necessary and sufficient conditions for an equilibrium.

Following Guerreiro et al. (2020), for analytical results, I first fix the level of τx and

obtain optimal allocations. Second, I search for the optimal level of τx. Let W (τx) be the

social welfare function (19) at the optimal allocation with a given level of τx. The social

planner’s problem at the second step is to find a maximizer τ ?x , subject to the resource and

incentive compatibility constraints. The optimal intermediate good tax level has to satisfy

the first-order condition of the objective function. Following Guerreiro et al. (2020), I focus

on cases where the IC constraint for non-routine workers binds, while that for routine workers

slacks. This is a scenario that Stiglitz (1982) calls a normal case. This holds in my numerical

exercises, as long as wn ≥ wr.

Note that the expression for the net output in the right-hand side of (20) can be rewritten

as:

τx + α

α (1 + τx)1/α

αA1/α(1− α)1−αα

φ1−αα

d

πnln + πrlr(1− β)φd

(1− β)(φrβφd

)q−1(φrβφd

)q−1− β

1q−1

(23)

Notice that the termτx + α

α (1 + τx)1/α

is equal to one when τx = 0 and strictly less than one when

τx 6= 0. This term measures the production inefficiency caused by the tax on intermediate

goods.

As discussed in Guerreiro et al. (2020), in the absence of complementing intermediate

goods, the key force to make the intermediate good tax positive is the reduction in the wage

gap. This, in turn, relaxes the planner’s information constraint from the non-routine workers’

incentive compatibility, and therefore the planner can improve welfare by this informational

gain, a similar intuition to Naito (1999) and Slavık and Yazici (2014). The problem with the

uniform intermediate good tax is that it may lower the wage of routine workers who have a

higher marginal utility of income. This negative effect can dominate the informational gain,

and thus it seems infeasible to determine the sign of the uniform tax with generality, at a

17

glance.

Recall the main intuition behind the main result mentioned in the introduction. Due to

the price differences between the two types of intermediate goods, a uniform intermediate

good tax indirectly imposes differential tax burdens on the two intermediate good types.

This additional key force makes the sign of the uniform tax to be always positive at an

interior solution.

Proposition 2. Let Assumption 2.1 hold. And suppose that the optimal allocation is such

that the incentive compatibility constraint for non-routine workers binds, but that for routine

workers does not bind. Then, at an interior solution (0 < m < 1 and lr > 0), the optimal

uniform linear intermediate good tax is strictly positive, regardless of the value of q ∈ [0, 1].

The optimal intermediate good tax rate generally satisfies

τx1 + τx

=α

1− απrlrπn

ωr (−ul (cr, lr))− (1− β)φd

(1− β)(φrβφd

)q−1(φrβφd

)q−1− β

1q−1

(24)

where ωr = ωr/µ, for µ being the Lagrange multiplier on the resource constraint.

Now, we analyze what a practically implementable labor income tax would look like with

this intermediate good taxation. In fact, the forms of marginal rates of substitution between

consumption and labor are the same as Guerreiro et al. (2020):

ul (cn, ln)

uc (cn, ln)= wn

τx + α

α (1 + τx),

and

ul (cr, lr) lruc (cr, lr)

=ωr − ηn uc(cr,wrlr/wn)uc(cr,lr)

ωr − ηnul(cr,wrlr/wn)

1wn

ul(cr,lr)1wr

wrlr1 + τx

,

where ηnπr is the multiplier for the IC constraint of non-routine workers. The properties of

labor income tax at optimum are same with Guerreiro et al. (2020) as well: (i) non-routine

workers are subsidized at the margin at partial automation, while (ii) routine workers are

taxed at the margin.

18

2.7 First-Best: Perfect Discrimination of Labor Types

To see the ideal outcome and have a reference point for the main result, I define two different

scenarios in this and the next section: the first-best and second-best outcomes. Following

Guerreiro et al. (2020), the first-best allocation maximizes the weighted average of utilities,

subject only to resource constraints without incentive compatibility constraints of house-

holds.

I denote such (positive) weights by ωr for routine households and by ωn for non-routine

households. These weights are normalized in a way that πrωr + πnωn = 1. The planner’s

problem is to choose {cr, lr, cn, ln,m, {xd(i), xc(i), n(i)}} to maximize utilitarian social welfare

of equation (19). Note that the first-best allocation implies production efficiency since robots’

markets are perfectly competitive. In the first-best scenario, the government can perfectly

distinguish and discriminate between the two labor types. Then, the government will whip

more productive non-routine workers to work more while increase the transfer amount from

non-routine households to routine households over time, so that the consumption levels are

equalized between the two households. Non-routine workers are more productive in a sense

that unlike routine workers, they are not replaceable. This is consistent with the first-order

conditions of the planner’s problem with respect to consumption amounts cj. This results

in a higher utility level for routine households. Since this creates incentives for non-routine

households to slack and imitate routine households, this outcome is an ideal but unrealistic

one.

I define the first-best allocation in the economy as the solution to the aforementioned

utilitarian social welfare function, absent informational constraints. The absence implies that

the planner can perfectly discriminate among two types of agents and enforce any allocation.

2.8 Second-best: Perfect Differentiation of intermediate good types

I now analyze a situation where the government cannot perfectly discriminate between two

types of labor but can perfectly distinguish between two types of intermediate goods. Denote

by τd as the intermediate good tax on displacing intermediate goods and by τr as that on

complementing intermediate goods. Both taxes are bounded from below but not above:

19

−1 ≤ τd <∞ and −1 ≤ τr <∞.

At a partial automation level 0 < m < 1, the differences from the main setting are

Xc =

1− β((1+τr)φrβ(1+τd)φd

)q−1− β

qq−1

πrlr (25)

wr = (1− β)(1 + τd)φd

(1− β)

β((1+τr)φrβ(1+τd)φd

)q−1− β

+ 1

1q−1

, (26)

and

wn = αA1/α(1− α)

1−αα

[(1 + τd)φd]1−αα

. (27)

Also, the parameter of automation m becomes

m = 1−(

(1 + τd)φd(1− α)A

) 1α πrlrπnln

. (28)

With this, in general, I obtain a result where the optimal intermediate good tax rate for

displacing intermediate goods is positive and that for complementing intermediate goods is

negative. The result is quite intuitive since the tax on displacing intermediate goods is similar

to the one in Guerreiro et al. (2020) and can be used as an instrument to reduce the wage gap

for the aforementioned informational gains, while that on complementing intermediate goods

can be used to boost the wage of routine labor at the expense of second-order production

loss. The proof can be found in the Appendix.

Proposition 3. Suppose that the optimal allocation is such that the incentive compatibility

constraint for non-routine workers binds, but that for routine workers does not bind. Then,

given partial automation (m < 1 and lr > 0), and given Assumption 2.1, the optimal tax for

displacing intermediate goods is always strictly positive (τ ?d > 0) if that for complementing

intermediate goods is zero (τr = 0), while that for complementing intermediate goods is

always strictly negative (τ ?r < 0) if that for displacing intermediate goods is zero (τd = 0),

20

regardless of the value of q ∈ [0, 1].

Note that the planner tends to be able to just use one of the two instruments for opti-

mality. Yet, the following corollary demonstrates some exceptions with the special case of

q −→ 0.

Corollary 1. Suppose that the optimal allocation is such that the incentive compatibility

constraint for non-routine workers binds, but that for routine workers does not bind. Suppose

also that q −→ 0. Then, given partial automation (m < 1 and lr > 0), and given Assumption

2.1, the optimal intermediate good tax rate on complementing intermediate goods is full

subsidy (τr = −1) and the optimal intermediate good tax rate on displacing intermediate

goods is strictly positive, unless one of the two instruments can solely achieve the first-best

allocations.

The reason for 100% subsidy for complementing intermediate goods comes from the Leon-

tief functional form of the joint production of routine labor and complementing intermediate

goods. Note that the wage rate of routine workers in this case becomes

wr = (1 + τd)φd − (1 + τr)φr.

We can see that the wage goes up by a linear rate of φr. The increase tends to dominate the

second-order production inefficiency by the subsidy.

3 Numerical Analysis

For numerical exercises, I mostly follow the settings used in the 2019 version of Guerreiro

et al. (2020). I assume the following utility functional form from Ales, Kurnaz, and Sleet

(2015) and Heathcote, Storesletten, and Violante (2017):

u(cj, lj) + v(G) = log(cj)− ζl1+νj

1 + ν+ χ log(G).

This utility functional form features two desirable properties. One is that this form is

consistent with balanced growth. The other is that the preferences are consistent with the

21

empirical evidence from Chetty (2006).

ζ is set at 10.63. ν is set at 4/3, so that the Frisch elasticity becomes 0.75 that is

consistent with the estimates discussed in Chetty et al. (2011). As discussed in Heathcote

et al. (2017), χ is chosen to be 0.233 that makes the optimal ratio of government spending

to output 18.9 percent, which is the weight observed in the U.S.

Furthermore, normalize A to one and choose α = 0.53 and πr = 0.55, which are estimates

from Chen (2016).

To capture the technological progress over time, I consider a sequence of static equilibria

in which the cost of intermediate goods geometrically falls over time, φk = φ0,ke−gφt for

k ∈ {d, r}. To allow my model to match the decline in task content estimated by Acemoglu

and Restrepo (2019) for the year 2000-2008, I set gφ = 0.0083. The period avoids the

financial crisis but captures the time period when the automation movement took off in the

U.S.

As for φk, Guerreiro et al. (2020) choose φd to be 0.4226, the lowest value that is consistent

with no automation in their status-quo equilibria22. For close comparison, I also choose

φd = 0.4226. As for the cost of complementing intermediate goods, the cost difference from

the cost of displacing intermediate goods varies across different industries; according to ad-

hoc documents, the price of cobots tends to be 55% of industrial robots, while that of cash

registers tends to be less than 5% of self-checkout registers23. For most of the numerical

analyses, I choose φr to be a third (33%) of φd. Meanwhile, I will demonstrate the optimal

intermediate good tax rate of the main setting with respect to the whole space of the cost

differences—that is, a matrix of different (φd, φr) composites.

3.1 First-best: Perfect Discrimination of Labor Types

Figure 1 shows the first-best outcome with the aforementioned parameter selections. Note

that while the full automation takes place only asymptotically, a large portion of routine

22The status-quo equilibria in their study are equilibria where there is no intermediate good tax,and the planner uses the income tax schedule estimated in Heathcote et al. (2017)

23See, for example, https://blog.robotiq.com/what-is-the-price-of-collaborative-robotsand a study by MIT http://web.mit.edu/2.744/www/Project/Assignments/humanUse/lynette/2-About%20the%20machine.html

22

tasks will be displaced in the early periods. The decrease in the cost of displacing in-

termediate goods over time results in the increasing wage gap between the two household

types. However, since in the first-best scenario, the government can perfectly distinguish

and discriminate between the two labor types, the government will whip more productive

non-routine workers to work more while increasing the transfer amount from non-routine

households to routine households over time, so that the consumption levels are equalized

between the two households. This is consistent with the first-order conditions of the plan-

ner’s problem with respect to consumption amounts cj. This results in a higher utility level

for routine households. Since this creates incentives for non-routine households to slack and

imitate routine households, this outcome is an ideal but unrealistic one.

3.2 Second-best: Perfect Identification of intermediate good types

Figure 2 and 3a, 3b, and 3c demonstrate the numerical results of the second-best outcome

with different functional choices of the joint production function involving routine workers

and complementing intermediate goods. Figure 2 shows the Leontief functional form version,

which is the case the most deviant from Guerreiro et al. (2020). As the corollary above

expected, the optimal tax rate of complementing intermediate goods is -1 (full subsidy)

while that of displacing intermediate goods is positive. The taxes and allocations are set at

the levels so that the incentive constraint for non-routine households binds, resulting in the

small utility difference between the two households.

Figure 3a, 3b, and 3c involve a Cobb-Douglas functional form assumption with different

values of share between the two inputs: 0.5, 0.45, and 0.55. Notice that the magnitude of

the optimal intermediate good tax rate becomes larger when the share parameter is lower24.

This is because the smaller share, the closer the economy becomes to the one in Guerreiro

et al. (2020) where there is no complementing intermediate goods. Furthermore, note that

these cases are the least deviant from Guerreiro et al. (2020). This is because Cobb-Douglas

is the least complementary case. Thus, we see much lower subsidy rates on complementing

intermediate goods than the Leontief case.

24Recall that the planner can just use either one of the intermediate good tax instruments in thesecond-best case.

23

3.3 Third-best: Main Settings

Figure 4 shows the third-best outcome that is the main setting outcome in this paper, with

a Leontief functional form assumption for the joint production of routine labor and comple-

menting intermediate goods. I choose this functional form since this is the most deviant case

from Guerreiro et al. (2020). Note that the optimal uniform intermediate good tax rate has

a decreasing trend over time. This is because the gains from the tax become smaller and the

costs from distorting production become larger as the cost of intermediate goods decreases

over time. More importantly, the cost difference between the two intermediate good types

becomes smaller since the technological progress rate is assumed to be the same between the

two. The more expensive displacing intermediate goods benefits disproportionately more

from technological progress. The reduced gap leads to a reduced rate of an increase in the

wage rate of routine workers by the intermediate good tax. We can see this from their

equilibrium wage rate in the Leontief assumption:

wr = (1 + τx)(φd − φr).

Then, over time, the redistribution force with the intermediate good tax becomes smaller,

leading to near zero optimal intermediate good tax rates asymptotically.

This motivates us to look into the optimal intermediate good tax rate in different coor-

dinates of (φd, φr). Figure 5 demonstrates this. As you can see in the figure, there is a hump

in the optimal intermediate good tax rate. When the gap between the two costs is large, the

routine worker wage is so high that there does not need to be much redistribution to make.

On the other hand, when there is only a small gap, then an increase in the intermediate good

tax rate will only marginally increase the routine labor wage. Therefore, there is a sweet

spot in between for redistribution. Note that this result partially addresses the issue of mag-

nitude. If the magnitude of the optimal intermediate good tax rate is negligible compared

to a case of Guerreiro et al. (2020) where the planner can tax only displacing intermediate

goods, then the theoretical finding on the sign of the optimal uniform intermediate good tax

may have little importance. Figure 5 partially addresses this concern since the magnitude

depends on the magnitude of the cost difference.

24

Furthermore, Figure 6a, b, c, and d provide a more direct answer. I look at the both

Cobb-Douglas and Leontief cases that are the closest and most deviant from Guerreiro

et al. (2020), respectively. I compare the welfare and routine households’ consumption

levels among all the scenarios of the first-best (“FB” in Figure 6a and 6b), the second-best

(“SB”), the main third-best (“TB”), and the outcome with τx = 0 (“Zero”). Recall that the

second-best allocations allow the planner to differentially tax displacing and complementing

intermediate goods, an analogous case to Guerreiro et al. (2020). Moreover, since the optimal

intermediate good tax becomes close to zero after 2050 in the Leontief case, I compare the

welfare levels between 2000 and 2050.

Figure 6a uses Cobb-Douglas with β = 0.5 and φr,0 = 0.33φd,0 same as with one of the

previous parameter value settings. As you can see, in this case, the difference between the

second-best and third-best is so small that it is almost negligible, compared to the difference

from the first-best and zero-intermediate goods-tax cases. This is because Cobb-Douglas is

the least gross-complement case, and thus the differential tax burden effect from the price

difference has relatively strong effects. Figure 6b is the same as Figure 6a except for the

y-axis to be consumption amounts of routine households. In the early periods, the uniform

intermediate good tax can dramatically increase the amounts up to 8.9% compared to the

zero intermediate good tax case.

As for the Leontief case, Figure 6c shows that the welfare improvement of the third-

best outcome is negligible, compared to that of the second-best especially in the beginning.

Figure 6d demonstrate that the consumption levels of routine households also increase but

only up to 0.6% in the early periods.

These results imply that while the magnitude of the welfare improvement by the optimal

uniform intermediate good taxation depends upon the degree of complementarity between

routine labor and complementing intermediate goods, the magnitude can be comparable to

the differential intermediate good taxation case.

25

4 Discussion on An Extension to A Dynamic Model

Note that Guerreiro et al. (2020) analyze a dynamic extension, focusing on their calibration-

based quantitative analysis of a dynamic version of their static model. In their dynamic

model, they incorporate an endogenous skill choice process with heterogeneous skill acqui-

sition costs. This addition introduces an additional negative effect of a positive robot tax

since it reduces incentives to acquire non-routine skill sets. Therefore, the optimality of a

robot tax becomes a quantitative question in the dynamic model. Given their result, the

optimality of an intermediate good tax in the dynamic model with the additional intermedi-

ate good types is also a quantitative question. For this reason and because of technical and

computational difficulties, I do not explore the dynamic extension in this paper. While my

work on the dynamic extension is available upon request, the analysis does not provide any

further insight and is thus omitted from the main text.

5 Conclusion & Limitations

This paper finds that despite the asymmetric information problem, the optimal uniform

intermediate good tax rate over different types of intermediate goods is strictly positive, as

long as the solution is interior. Here, I will acknowledge the limitations of this study. First,

my model focuses on short-term analyses and assumes away occupational choices between

routine and non-routine sectors for analytical tractability. As mentioned in the previous

section, the optimality of a robot tax becomes a quantitative question in the dynamic model

with endogenous skill acquisition. Given the focus of this study that highlights a new channel

of optimal taxation of intermediate goods in income redistribution, and given technical and

computational difficulties, I do not relax this assumption in this paper.

Another limitation is that I assume perfectly competitive external global markets for in-

termediate goods producers. One possible extension is to relax this assumption and to incor-

porate R&D process for intermediate goods producers along the lines of Aghion et al. (2013).

In this case, a uniform intermediate good tax that results in the differential tax burden on

different intermediate goods type would create a resource wedge in R&D between displacing

26

and complementing intermediate goods, leading to differential technological progress over

time. This could further drive up the optimal uniform intermediate good tax rate in the

long run.

27

References

Acemoglu, D., Manera, A., and Restrepo, P. (2020). “Does the us tax code favor automa-

tion?” Technical report, National Bureau of Economic Research.

Acemoglu, D. and Restrepo, P. (2018). “The race between man and machine: Implications

of technology for growth, factor shares, and employment”. American Economic Review,

108(6):1488–1542.

— (2019). “Automation and new tasks: How technology displaces and reinstates labor”.

Technical report, National Bureau of Economic Research.

— (2020). “The labor share, displacement and wage inequality”. Technical report, Mimeo.

Massachusetts Institute of Technology.

Aghion, P., Akcigit, U., and Fernandez-Villaverde, J. (2013). “Optimal capital versus labor

taxation with innovation-led growth”. Technical report, National Bureau of Economic

Research.

Atkinson, A. B. and Stiglitz, J. E. (1976). “The design of tax structure: direct versus indirect

taxation”. Journal of public Economics, 6(1-2):55–75.

Autor, D. H., Levy, F., and Murnane, R. J. (2003). “The skill content of recent technological

change: An empirical exploration”. The Quarterly journal of economics, 118(4):1279–1333.

Chen, Y. G. (2016). “The rise of the machines and the u.s. labor market”. Technical report,

mimeo, University of Maryland.

Chetty, R. (2006). “A new method of estimating risk aversion”. American Economic Review,

96(5):1821–1834.

Chetty, R., Guren, A., Manoli, D., and Weber, A. (2011). “Are micro and macro labor

supply elasticities consistent? a review of evidence on the intensive and extensive margins”.

American Economic Review, 101(3):471–75.

28

Costinot, A. and Werning, I. (2018). “Robots, trade, and luddism: A sufficient statistic ap-

proach to optimal technology regulation”. Technical report, National Bureau of Economic

Research.

Deaton, A. (1979). “Optimally uniform commodity taxes”. Economics Letters, 2(4):357–361.

Diamond, P. A. and Mirrlees, J. A. (1971a). “Optimal taxation and public production i:

Production efficiency”. The American Economic Review, 61(1):8–27.

— (1971b). “Optimal taxation and public production ii: Tax rules”. The American Economic

Review, 61(3):261–278.

Ding, L., Molina, J., et al. (2020). ““forced automation” by covid-19? early trends from

current population survey data”. Technical report, Federal Reserve Bank of Philadelphia.

Farhi, E. and Werning, I. (2013). “Insurance and taxation over the life cycle”. Review of

Economic Studies, 80(2):596–635.

Golosov, M., Kocherlakota, N., and Tsyvinski, A. (2003). “Optimal indirect and capital

taxation”. The Review of Economic Studies, 70(3):569–587.

Golosov, M., Troshkin, M., and Tsyvinski, A. (2011). “Optimal dynamic taxes”. Technical

report, National Bureau of Economic Research.

— (2016). “Redistribution and social insurance”. American Economic Review, 106(2):359–

86.

Golosov, M., Tsyvinski, A., Werning, I., Diamond, P., and Judd, K. L. (2006). “New dynamic

public finance: A user’s guide [with comments and discussion]”. NBER macroeconomics

annual, 21:317–387.

Gomes, R., Lozachmeur, J.-M., and Pavan, A. (2017). “Differential taxation and occupa-

tional choice”. The Review of Economic Studies, 85(1):511–557.

Guerreiro, J., Rebelo, S., and Teles, P. (2020). “Should robots be taxed?” Technical report,

National Bureau of Economic Research.

29

Guesnerie, R. (1998). A contribution to the pure theory of taxation. 25. Cambridge University

Press.

Heathcote, J., Storesletten, K., and Violante, G. L. (2017). “Optimal tax progressivity: An

analytical framework”. The Quarterly Journal of Economics, 132(4):1693–1754.

Kocherlakota, N. R. (2005). “Zero expected wealth taxes: A mirrlees approach to dynamic

optimal taxation”. Econometrica, 73(5):1587–1621.

Mirrlees, J. A. (1971). “An exploration in the theory of optimum income taxation”. The

review of economic studies, 38(2):175–208.

Naito, H. (1999). “Re-examination of uniform commodity taxes under a non-linear income

tax system and its implication for production efficiency”. Journal of Public Economics,

71(2):165–188.

Rothschild, C. and Scheuer, F. (2013). “Redistributive taxation in the roy model”. The

Quarterly Journal of Economics, 128(2):623–668.

Saez, E. (2001). “Using elasticities to derive optimal income tax rates”. The review of

economic studies, 68(1):205–229.

— (2002). “The desirability of commodity taxation under non-linear income taxation and

heterogeneous tastes”. Journal of Public Economics, 83(2):217–230.

— (2004). “Direct or indirect tax instruments for redistribution: short-run versus long-run”.

Journal of public economics, 88(3-4):503–518.

Scheuer, F. (2014). “Entrepreneurial taxation with endogenous entry”. American Economic

Journal: Economic Policy, 6(2):126–63.

Scheuer, F. and Werning, I. (2016). “Mirrlees meets diamond-mirrlees”. Technical report,

National Bureau of Economic Research.

Slavık, C. and Yazici, H. (2014). “Machines, buildings, and optimal dynamic taxes”. Journal

of Monetary Economics, 66:47–61.

30

Slemrod, J. and Kopczuk, W. (2002). “The optimal elasticity of taxable income”. Journal

of Public Economics, 84(1):91–112.

Slemrod, J. and Yitzhaki, S. (2002). “Tax avoidance, evasion, and administration”. In

“Handbook of public economics”, volume 3, pages 1423–1470. Elsevier.

Stantcheva, S. (2014). “Optimal income taxation with adverse selection in the labour mar-

ket”. Review of Economic Studies, 81(3):1296–1329.

— (2017). “Optimal taxation and human capital policies over the life cycle”. Journal of

Political Economy, 125(6):1931–1990.

Stiglitz, J. E. (1982). “Self-selection and pareto efficient taxation”. Journal of public eco-

nomics, 17(2):213–240.

Thuemmel, U. (2018). “Optimal taxation of robots”.

31

Appendices

A Proof of Proposition 1

By Assumption 2.1, we can differentiate the profit function (4) with respect to xd(i), xc(i),

and n(i). Then, the optimal choices of Nn, xd(i) for i ∈ [0,m], ni, xc(i) for i ∈ (m, 1] require

the following first-order conditions:

wn =αY

Nn

, (29)

(1 + τx)φd =(1− α)Y

(∫ m

0

xd (s)ρ−1ρ ds+

(∫ 1

m

[xc(s)

q−1q + n(s)

q−1q

] ρ−1ρds

) qq−1

)−1xd(i)

− 1ρ , for i ∈ [0,m], (30)

(1 + τx)φr =(1− α)Y

(∫ m

0

xd (s)ρ−1ρ ds+

(∫ 1

m

[βxc(s)

q−1q + (1− β)n(s)

q−1q

] ρ−1ρds

) qq−1

)−1(∫ 1

m

[βxc(z)

q−1q + (1− β)n(z)

q−1q

] ρ−1ρdz

) 1q−1

βxc(i)− 1q , for i ∈ (m, 1].

(31)

wr =(1− α)Y

(∫ m

0

xd (s)ρ−1ρ ds+

(∫ 1

m

[βxc(s)

q−1q + (1− β)n(s)

q−1q

] ρ−1ρds

) qq−1

)−1(∫ 1

m

[βxc(z)

q−1q + (1− β)n(z)

q−1q

] ρ−1ρdz

) 1q−1

(1− β)n(i)−1q , for i ∈ (m, 1]. (32)

From (30), it follows that in the m automated tasks, the optimal level of displacing robots

is equal across tasks. From (31) and (32), it also follows that in the 1 − m unautomated

tasks, the optimal level of routine labor and cobots are also the same across tasks. Hitherto,

whenever possible, we drop the task index i for simplicity. Note that the final producer will

32

choose m = 1 whenever the effective price of displacing intermediate goods is cheaper than

that of the combination of complementing intermediate goods and routine labor:

(1 + τx)φd >wrn+ (1 + τx)φrxc(

βxc(i)q−1q + (1− β)n(i)

q−1q

) qq−1

. (33)

They will choose m = 0 instead if the inequality above is the opposite25. We will observe 0 <

m < 1 if the inequality is equality. We are interested in this scenario of partial automation

since otherwise, as we will show, the intermediate good tax is always zero.

Then, as partial automation, we get xd(i) =[βxc(i)

q−1q + (1− β)n(i)

q−1q

]. Thus, at

partial automation, we can drop the substitution parameters for tasks, ρ. Then, we get

Y = A

[∫ m

0

xd(i)di+

(∫ 1

m

[βxc(i)

q−1q + (1− β)n(i)

q−1q

]di

) qq−1

](1−α)Nαn ,

α ∈ (0, 1) , ρ ∈ [0,∞) . (34)

Also, since the amount of xc(i) is the same across tasks, and since that of n(i) is the same

across tasks, we can write26

Y = A

[∫ m

0

xd(i)di+

∫ 1

m

[βxc(i)

q−1q + (1− β)n(i)

q−1q

] qq−1

di

](1−α)Nαn ,

α ∈ (0, 1) , ρ ∈ [0,∞) . (35)

What’s more, at partial automation, the final good producer is indifferent in the level of

automation m. In this case, we get

mxd(i) = Xd, for i ∈ [0,m], (1−m)xc(i) = Xc, and (1−m)n(i) = Nr,

, for i ∈ (m, 1], (36)

25Note that the denominator in (33) is a certain output level y(i)26It might be easier to see this with finite summation. If the sum is finite, then the integral

is(∑1

i=m xc(i)q−1

q +∑1

i=m n(i)q−1

q

) q

q−1

. Since xc(i) = xc(i′) and n(i) = n(i′) for i 6= i′, the sum

equals to(

[(1−m)xc(i)]q−1

q + [(1−m)n(i)]q−1

q

) q

q−1

= (1−m)[xc(i)q−1

q + n(i)q−1

q ]q

q−1

33

where Xd is the total amount of displacing robots, Xc is that of complementing robots,

and Nr is the total employment of routine labor, as defined in the main text27. Note that

by definition, we have m = Xd

Xd+

(βX

q−1q

c +(1−β)Nq−1q

r

) qq−1

if m > 0. Then, we can rewrite the

production function as

Y = A

(Xd +

(βX

q−1q

c + (1− β)Nq−1q

r

) qq−1

)1−α

Nαn

.

27Note that (1 − m)xc(i) = Xc, and (1 − m)n(i) = Nr implies (1 − m)[xq−1

qc + n

q−1

q ]q

q−1 =([(1−m)xc]

q−1

q + [(1−m)n]q−1

q

) q

q−1

=

(X

q−1

qc + N

q−1

qr

) q

q−1

.

34

B Proof of Lemma 1

A q −→ 1

Proof. The final good producer optimizes inputs and get

xc(i) =

πrlr(1−m)

(φrβφd

) 1β−1

, i ∈ (m, 1]

0, otherwise(37)

n(i) =

πrlr(1−m)

, i ∈ (m, 1]

0, otherwise(38)

xd(i) =

πrlr

(1−m)

(φrβφd

) ββ−1

, i ∈ [0,m]

0, otherwise(39)

m = max

{1−

((1 + τx)φd(1− α)A

) 1α(φrβφd

) ββ−1 πrlr

πnln, 0

}(40)

Y = A

[∫ m

0

xd(i)ρ−1ρ di+

∫ 1

m

[xc(i)

βn(i)1−β] ρ−1

ρ di

] ρρ−1

(1−α)

Nαn , α ∈ (0, 1) , ρ ∈ [0,∞) .

(41)

At an interior solution, while wn is the same in the main text,

wr = (1− β)(1 + τx)φd

(φrβφd

) ββ−1

. (42)

Combining these results, after some algebra, we can rewrite the resource constraint (15) as:

πrcr + πncn +G ≤ πnwnln

(τx + α

α(1 + τx)

)+πrwrlr1 + τx

(43)

This captures all the equilibrium conditions of the production side.

Given that households cannot change their types, the optimality of hosuehold choice gives

35

us

u (cj, lj) ≥ u(c, l), ∀(c, l) : c ≤ wjl − T (wjl) . (44)

Then, the following IC constraints are necessary conditions for the optimality:

u (cn, ln) ≥ u

(cr,

wrwnlr

)u (cr, lr) ≥ u

(cn,

wnwrln

)Notice that these two IC constraints are also sufficient conditions for the household side at

an equilibrium since the government can freely adjust the tax schedule T (.) so that for all

Y /∈ {Yn, Yr}, both household types receive worse allocations than their respective allocation.

The government can achieve this by setting the tax schedule to be:

T (y) = y −max

{c|u (ci, li) ≥ u

(c,y

wi

), for i = r, n

}(45)

�

B q −→ 0

Proof. The final good producer optimizes inputs and get

xc(i) =

πrlr(1−m)

, i ∈ (m, 1]

0, otherwise(46)

n(i) =

πrlr(1−m)

, i ∈ (m, 1]

0, otherwise(47)

xd(i) =

πrlr(1−m)

, i ∈ [0,m]

0, otherwise(48)

m = max

{1−

((1 + τx)φd(1− α)A

) 1α πrlrπnln

, 0

}(49)

36

Y = A

[∫ m

0

xd(i)ρ−1ρ di+

∫ 1

m

min[xc(i), n(i)]ρ−1ρ di

] ρρ−1

(1−α)

Nαn , α ∈ (0, 1) , ρ ∈ [0,∞) .

(50)

wr = (1 + τx)(φd − φr) (51)

Combining these results, again, we can rewrite the resource constraint (15) to

πrcr + πncn +G ≤ πnwnlnτx + α

α (1 + τx)+πrwrlr1 + τx

(52)

This captures all the equilibrium conditions of the production side.

The rest of the proof is the same with q −→ 1. �

C 0 < q < 1

Proof. The final good producer optimizes inputs and get

xc(i) =

πrlr

(1−m)

(1−β(

φrβφd

)q−1−β

) qq−1

, i ∈ (m, 1]

0, otherwise

(53)

n(i) =

πrlr(1−m)

, i ∈ (m, 1]

0, otherwise(54)

xd(i) =

πrlr

(1−m)

(β

(1−β(

φrβφd

)q−1−β

)+ (1− β)

), i ∈ [0,m]

0, otherwise

(55)

m = max

{1−

((1 + τx)φd(1− α)A

) 1α πrlrπnln

, 0

}(56)

37

Y = A

[∫ m

0

xd(i)ρ−1ρ di+

∫ 1

m

min[xc(i), n(i)]ρ−1ρ di

] ρρ−1

(1−α)

Nαn , α ∈ (0, 1) , ρ ∈ [0,∞) .

(57)

wr = (1− α)(1−m)Y

πrlr, (58)

Combining these results, after a lot of tedious algebra, we can rewrite the resource constraint

(15) to

πrcr + πncn +G ≤ πnwnlnτx + α

α (1 + τx)+πrwrlr1 + τx

(59)

This captures all the equilibrium conditions of the production side.

The rest of the proof is the same with q −→ 1. This concludes the whole proof. �

C Proof of Proposition 2

Proof. Define W (τx) = max πrωru (cr, lr) + πnωnu (cn, ln) + v(G). Then, the social planner’s

optimization problem is

maximizeτx

W (τx)

subject to

[ηrπr] u (cr, lr) ≥ u

(cn,

wnwrln

),

[ηnπn] u (cn, ln) ≥ u

(cr,

wrwnlr

),

[µ] πrcr + πncn +G ≤ πnwnln

(τx + α

α(1 + τx)

)+

πrwrlr(1 + τx)

.

38

Letting ηr = 0, we get

W ′ (τx) =− ηnπnul(cr,

wrwrlr

)d log (wr/wn)

d log (1 + τx)

1

1 + τx

wrlrwn

+ µ

[πnwnln

τx + α

α(1 + τx)2

(d logwn

d log(1 + τx)+

1− α(τx + α)

)+

πrwrlr(1 + τx)2

(d logwr

d log(1 + τx)− 1

)]. (60)

I separately prove the limit cases (q −→ 0 and q ←→ 1) and interior case (0 < q < 1).

A q −→ 0

With the equilibrium wages we have gotten above, we get:

wr = (1 + τx) (φd − φr)⇒d logwr

d log (1 + τx)= 1

wn = αA1/α(1− α)

1−αα

[(1 + τx)φd]1−αα

⇒ d logwnd log (1 + τx)

= −1− αα

wrwn

=(1 + τx)

1α

[φ

1αd − φrφ

1−αα

d

]αA1/α(1− α)

1−αα

⇒ d logwr/wnd log (1 + τx)

=1

α

Plugging these back into the envelope condition (60), we get:

W ′ (τx) = −ηnπnul(cr,

wrwrlr

)1

α (1 + τx)

wrlrwn

+ µπnwnlnτx + α

α (1 + τx)2

[−1− α

α+

1− ατx + α

]=

1

α (1 + τx)

[−ηnπnul

(cr,

wrwnlr

)wrlrwn− µπnwnln

τx1 + τx

1− αα

]Since µ > 0, if τx ≤ 0, if wr ≥ 0—that is, φr ≤ φd—, then we are guaranteed to have:

W ′(τx) > 0.

Thus, the social planner can always improve welfare by marginally increasing τx. Further-

more, the optimal level of τx implies W ′(τx) = 0, so we get:

39

τx1 + τx

=α

1− α

ηn

(−ul

(cr,

wrwnlr

)wrlrwn

)µwnln

(61)

The first order condition with respect to lr yields:

−ηnµul

(cr,

wrwrlr

)wrlrwn

= −

(ωrπrul (cr, lr) lr + πrwrlr

1+τx

πn

)

=πrlrπn

(φd − φr)[ωr (−ul (cr, lr))

φd − φr− 1

] (62)

Thus, we getτx

1 + τx=

α

1− απrlr

πnwnln(φd − φr)

[ωr (−ul (cr, lr))

φd − φr− 1

](63)

B q −→ 1 Case

Here, we analyze the Cobb-Douglas case. We have

Xc =

(φrβφd

) 1β−1

πrlr. (64)

Also,

m = 1−(

(1 + τx)φd(1− α)A

) 1α (πrlr)

(1−β)

πnlnXβc

= 1−(

(1 + τx)φd(1− α)A

) 1α (πrlr)

(1−β)

πnln

((φrβφd

) 1β−1

πrlr

)β

= 1−(

(1 + τx)φd(1− α)A

) 1α πrlrπnln

(φrβφd

) ββ−1

. (65)

40

Furthermore,

wr = (1− β)(1 + τx)φdXβc (πrlr)

−β

= (1− β)(1 + τx)φd

((φrβφd

) 1β−1

πrlr

)β

(πrlr)−β

= (1− β)(1 + τx)φd

(φrβφd

) ββ−1

. (66)

With the equilibrium wages we have gotten above, we get:

wr = (1 + τx) (1− β)φd

(φrβφd

) ββ−1

⇒ d logwrd log (1 + τx)

= 1

wn = αA1/α(1− α)

1−αα

[(1 + τx)φd]1−αα

⇒ d logwnd log (1 + τx)

= −1− αα

wrwn

=[(1 + τx)φd]

1α

αA1/α(1− α)1−αα

(φrβφd

) ββ−1

⇒ d logwr/wnd log (1 + τx)

=1

α

Plugging these back into the envelope condition (60), we get:

W ′ (τx) = −ηnπnul(cr,

wrwrlr

)1

α (1 + τx)

wrlrwn

+ µπnwnlnτx + α

α (1 + τx)2

[−1− α

α+

1− ατx + α

]=

1

α (1 + τx)

[−ηnπnul

(cr,

wrwnlr

)wrlrwn− µπnwnln

τx1 + τx

1− αα

]Since µ > 0, if τx ≤ 0, then we have:

W ′(τx) > 0.

Thus, the social planner can always improve welfare by marginally increasing τx. Further-

more, the optimal level of τx implies W ′(τx) = 0, so we get:

τx1 + τx

=α

1− α

ηn

(−ul

(cr,

wrwnlr

)wrlrwn

)µwnln

(67)

41

The first order condition with respect to lr yields:

−ηnµul

(cr,

wrwrlr

)wrlrwn

= −

(ωrπrul (cr, lr) lr + πrwrlr

1+τx

πn

)

=πrφdlrπn

(φrβφd

) ββ−1

φd(1− β)

ωr (−ul (cr, lr))(φrβφd

) ββ−1

φd(1− β)

− 1

(68)

Thus, we get

τx1 + τx

=α

1− απrlrπn

(φrβφd

) ββ−1

φd(1− β)

ωr (−ul (cr, lr))(φrβφd

) ββ−1

φd(1− β)

− 1

(69)

C 0 < q < 1 case

We showed that the statement holds in the limit cases of q −→ 0 and q −→ 1. Thus, we

are left with non-limit cases. The equilibrium conditions are stated in the main text for

0 < q < 1 case. Although we cannot solve for the closed form solutions of endogenous

variables wr and m, we can deduce the sign of the comparative statics with respect to τx.

Recall that wn is not a function of q. Recall that

wr = (1− β)(1 + τx)φd

(1− β)(φrβφd

)q−1(φrβφd

)q−1− β

1q−1

(70)

Letting ηr = 0, we get

W ′ (τx) =− ηnπnul(cr,

wrwrlr

)d log (wr/wn)

d log (1 + τx)

1

1 + τx

wrlrwn

+ µ

[πnwnln

τx + α

α(1 + τx)2

(d logwn

d log(1 + τx)+

1− α(τx + α)

)+

πrwrlr(1 + τx)2

(d logwr

d log(1 + τx)− 1

)]. (71)

With the equilibrium wages we have gotten above, we get:

42

wr = (1− β)(1 + τx)φd

(1− β)(φrβφd

)q−1(φrβφd

)q−1− β

1q−1

⇒ d logwrd log (1 + τx)

= 1

wn = αA1/α(1− α)

1−αα

[(1 + τx)φd]1−αα

⇒ d logwnd log (1 + τx)

= −1− αα

d logwr/wnd log (1 + τx)

=1

α

Plugging these back into the envelope condition (60), we get:

W ′ (τx) = −ηnπnul(cr,

wrwrlr

)1

α (1 + τx)

wrlrwn

+ µπnwnlnτx + α

α (1 + τx)2

[−1− α

α+

1− ατx + α

]=

1

α (1 + τx)

[−ηnπnul

(cr,

wrwnlr

)wrlrwn− µπnwnln

τx1 + τx

1− αα

]Since µ > 0, if τx ≤ 0, if wr ≥ 0—that is, φr ≤ φd—, then we are guaranteed to have:

W ′(τx) > 0.

Thus, the social planner can always improve welfare by marginally increasing τx.

Furthermore, the optimal level of τx implies W ′(τx) = 0, so we get:

τx1 + τx

=α

1− α

ηn

(−ul

(cr,

wrwnlr

)wrlrwn

)µwnln

(72)

43

The first order condition with respect to lr yields:

−ηnµul

(cr,

wrwrlr

)wrlrwn

= −

(ωrπrul (cr, lr) lr + πrwrlr

1+τx

πn

)

=πrlrπn

(1− β)φd

(1− β)(φrβφd

)q−1(φrβφd

)q−1− β

1q−1

ωr (−ul (cr, lr))

(1− β)φd

((1−β)

(φrβφd

)q−1

(φrβφd

)q−1−β

) 1q−1

− 1

Thus, we get

τx1 + τx

=α

1− απrlrπn

(1− β)φd

(1− β)(φrβφd

)q−1(φrβφd

)q−1− β

1q−1

ωr (−ul (cr, lr))

(1− β)φd

((1−β)

(φrβφd

)q−1

(φrβφd

)q−1−β

) 1q−1

− 1

�

44

D Proof of Proposition 3

Again, as in the proof of Proposition 2, I separately prove the limit cases (q −→ 0 and

q ←→ 1) and interior case (0 < q < 1).

A q −→ 1 case

The equilibrium levels of inputs will be

Xc =

((1 + τr)φr

(1 + τd)βφd

) 1β−1

πrlr. (73)

m = 1−(

(1 + τd)φd(1− α)A

) 1α (πrlr)

(1−β)

πnlnXβc

= 1−(

(1 + τd)φd(1− α)A

) 1α (πrlr)

(1−β)

πnln

(((1 + τr)φr

(1 + τd)βφd

) 1β−1

πrlr

)β

= 1−(

(1 + τd)φd(1− α)A

) 1α πrlrπnln

((1 + τr)φr

(1 + τd)βφd

) ββ−1

. (74)

Furthermore,

wr = (1− β)(1 + τd)φdXβc (πrlr)

−β

= (1− β)(1 + τd)φd

(((1 + τr)φr

(1 + τd)βφd

) 1β−1

πrlr

)β

(πrlr)−β

= (1− β)φd(1 + τd)1

1−β

((1 + τr)φr

βφd

) ββ−1

. (75)

Let τx = (τd, τr). Define W (τx) = max πrωru (cr, lr) + πnωnu (cn, ln) + v(G). Then, the

45

social planner’s optimization problem is

maximizeτx

W (τx)

subject to

[ηrπr] u (cr, lr) ≥ u

(cn,

wnwrln

),

[ηnπn] u (cn, ln) ≥ u

(cr,

wrwnlr

),

[µ] πrcr + πncn +G ≤ πnwnln

(τd + α

α(1 + τd)

)+ πrwrlr

(1 + τr − β(1 + τd)

(1 + τd)(1 + τr)(1− β)

).

Letting ηr = 0, and taking the partial derivative of the welfare function with respect to

τr, we get

∂W (τr)

∂τr=− ηnπnul

(cr,

wrwrlr

)d log (wr/wn)

d log (1 + τr)

1

1 + τr

wrlrwn

+ µ

[πrwrlr

1 + τr − β(1 + τd)

(1 + τd)(1 + τr)2(1− β)

(d logwr

d log(1 + τr)

)+

+ πrwrlrβ

(1− β)(1 + τr)2

]. (76)

With the equilibrium wages we have gotten above, we get:

wr = (1− β)φd(1 + τd)1

1−β

((1 + τr)φr

βφd

) ββ−1

⇒ d logwrd log (1 + τr)

=β

β − 1

wn = αA1/α(1− α)

1−αα

[(1 + τd)φd]1−αα

⇒ d logwnd log (1 + τr)

= 0

wrwn

=[(1 + τd)φd]

1−β+αβα(1−β)

αA1α (1− α)

1−αα

((1 + τr)φr

βφd

) ββ−1

⇒ d logwr/wnd log (1 + τr)

=β

β − 1

46

Plugging these back into the envelope condition above, we get:

∂W (τr)

∂τr= −ηnπnul

(cr,

wrwrlr

)β

β − 1

1

(1 + τr)

wrlrwn

+ µπrwrlrβ

β − 1

1 + τr − β(1 + τd)

(1 + τd)(1 + τr)2(1− β)

+ µπrwrlrβ

(1− β)(1 + τr)2

=1

(1 + τr)

β

β − 1

[− ηnπnul

(cr,

wrwnlr

)wrlrwn

+ µπrwrlrτr − τd

(1 + τd)(1 + τr)(1− β)

] .

Notice that since η > 0, the first term inside the closed bracket is positive, and since

µ > 0, the second term inside the closed parenthesis is positive only if τr > τd. This holds

true when τr = 0. Thus, when τr = 0 and when τd = 0, or when τr < 0 and when τd ≤ τr,

we are guaranteed to get∂W (τr)

∂(−τr)> 0. (77)

Therefore, the social planner can achieve greater welfare by imposing a positive comple-

menting intermediate goods subsidy, τr < 0, when τd ≤ τr.

Now, to get the optimal rate of τr, we equate the envelope condition to be zero and get

−ηnπnul(cr,

wrwnlr

)wrlrwn

+ µπrwrlrτr − τd

(1 + τd)(1 + τr)2(1− β)= 0

The FOC of lr implies

−ηnπnul(cr,

wrwrlr

)wrlrwn

= −(ωrπrul (cr, lr) lr + µπrwrlr

(1 + τr − β(1 + τd)

(1 + τd)(1 + τr)(1− β)

)).

(78)

Plugging this back into the condition, we get

µπrwrlrτr − τd

(1 + τd)(1 + τr)2(1− β)= ωrπrul (cr, lr) lr + µπrwrlr

(1 + τr − β(1 + τd)

(1 + τd)(1 + τr)(1− β)

)

That isµπrwrlr1 + τr

= −ωrπrul (cr, lr) lr. (79)

Next, taking the partial derivative of the welfare function with respect to τd, we get

47

∂W (τd)

∂τd=− ηnπnul

(cr,

wrwrlr

)d log (wr/wn)

d log (1 + τd)

1

1 + τd

wrlrwn

+ µ

[πnwnln

τd + α

α(1 + τd)2

(d logwn

d log(1 + τd)+

1− α(τd + α)

)+

πrwrlr1 + τr − β(1 + τd)

(1 + τd)2(1 + τr)(1− β)

d logwrd log(1 + τd)

−

πrwrlr1

(1 + τd)2(1− β)

]. (80)

With the equilibrium wages we have gotten above, we get:

wr = (1− β)φd(1 + τd)1

1−β

((1 + τr)φr

βφd

) ββ−1

⇒ d logwrd log (1 + τd)

=1

1− β

wn = αA1/α(1− α)

1−αα

[(1 + τd)φ]1−αα

⇒ d logwnd log (1 + τd)

= −1− αα

wrwn

=[(1 + τd)φd]

1−β+αβα(1−β)

αA1α (1− α)

1−αα

((1 + τr)φr

βφd

) ββ−1

⇒ d logwr/wnd log (1 + τd)

=1− β + αβ

α(1− β)

Plugging these back into the envelope condition above, we get:

∂W (τd)

∂τd= −ηnπnul

(cr,

wrwrlr

)1− β + αβ

α(1− β)

wrlrwn(1 + τd)

+

µ

(πnwnln

τd + α

α (1 + τd)2

[−1− α

α+

1− ατd + α

]+ πrwrlr

[β(τr − τd)

(1− β)2(1 + τd)2(1 + τr)

])=

1

α (1 + τd) (1− β)

[− ηnπnul

(cr,

wrwnlr

)wrlrwn

(1− β + αβ)−

µπnwnlnτd

1 + τd

(1− α)(1− β)

α+ µπrwrlr

αβ(τr − τd)(1− β)(1 + τr)(1 + τd)

]

Since µ > 0, if τd ≤ 0, and if τd ≤ τr then we have:

∂W (τd)

∂τd> 0.

Thus, the social planner can always improve welfare by marginally increasing τd. Further-

more, the optimal level of τd implies ∂W (τd)∂τd

= 0 for a given level of τr.

48

The FOC of lr implies

−ηnπnul(cr,

wrwrlr

)wrlrwn

= −(ωrπrul (cr, lr) lr + µπrwrlr

(1 + τr − β(1 + τd)

(1 + τd)(1 + τr)(1− β)

)).

(81)

Then, plugging this back into the envelope condition, we get

πrωrul(cr, lr)lr =− µπnwnlnτd

1 + τd

(1− α)(1− β)

α+ (82)

µπrwrlrβ(2 + τd − τr + α(2τr − 2))− 1− τd + (α− 2)β2(1 + τd)− τr

(1− β)(1 + τr)(1 + τd).

(83)

Then, we have two equations for two unknowns from (79) and (83).

B q −→ 0

Let τx = (τd, τr). Define W (τx) = max πrωru (cr, lr) +πnωnu (cn, ln) + v(G). Then, the social

planner’s optimization problem is

maximizeτx

W (τx)

subject to

[ηrπr] u (cr, lr) ≥ u

(cn,

wnwrln

),

[ηnπn] u (cn, ln) ≥ u

(cr,

wrwnlr

),

[µ] πrcr + πncn +G ≤ πnwnln

(τd + α

α(1 + τd)

)+ πrwrlr

φd − φr(1 + τd)φd − (1 + τr)φr

.

Letting ηr = 0, and taking the partial derivative of the welfare function with respect to

τr, we get

∂W (τr)

∂τr=− ηnπnul

(cr,

wrwnlr

)dwrdτr

lrwn. (84)

With the equilibrium wages we have gotten above, we get:

49

wr = (1 + τd)φd − (1 + τr)φr ⇒dwrdτr

= −φr

Plugging this back into the envelope condition above, we get:

∂W (τr)

∂τr= ηnπnul

(cr,

wrwrlr

)lrwnφr .

Notice that since η > 0, this term is negative. This holds true at all τr. Therefore, we get

∂W (τr)

∂(−τr)> 0. (85)

Therefore, the social planner can achieve greater welfare by imposing a positive complement-

ing intermediate goods subsidy, τr < 0.

Furthermore, the condition above implies that the optimal τr = −1. The intuition behind

this result is discussed in the main text.

Next, taking the partial derivative of the welfare function with respect to τd, we get

∂W (τd)

∂τd=− ηnπnul

(cr,

wrwrlr

)d log (wr/wn)

d log (1 + τd)

1

1 + τd

wrlrwn

+ µ

[πnwnln

τd + α

α(1 + τd)2

d log[wn

(τd+αα(1+τd)

)]d log(1 + τd)

]. (86)

With the equilibrium wages we have gotten above, we get:

wr = (1 + τd)φd − (1 + τr)φr ⇒d logwr

d log (1 + τd)=

φd(1 + τd)

(1 + τd)φd − (1 + τr)φr

wn = αA1/α(1− α)

1−αα

[(1 + τd)φd]1−αα

⇒ d logwnd log (1 + τr)

= −1− αα

wrwn

=(1 + τd)φd − (1 + τr)φr

αA1/α(1−α)

1−αα

[(1+τd)φd]1−αα

⇒ d logwr/wnd log (1 + τr)

=φd(1 + τd)

(1 + τd)φd − (1 + τr)φr+

1− αα

50

Plugging these back into the envelope condition above, we get:

∂W (τd)

∂τd= −ηnπnul

(cr,

wrwrlr

)wrlrwn

(φd

(1 + τd)φd − (1 + τr)φr+

1− αα(1 + τd)

)+

µ

(πnwnln

τd + α

α (1 + τd)2

[−1− α

α+

1− ατd + α

]).

Given that the optimal τr = −1, we get

∂W (τd)

∂τd= −ηnπnul

(cr,

wrwrlr

)wrlrwn

1

α(1 + τd)+

µ

(πnwnln

τd + α

α (1 + τd)2

[−1− α

α+

1− ατd + α

]).

Notice that the first and second terms are always positive. Note that the third term is

positive whenτd + α

α (1 + τd)2

[−1− α

α+

1− ατd + α

]> 0

This is true when τd ≤ 0. Given this condition, we get

∂W (τd)

∂τd> 0.

Thus, the social planner can always improve welfare by marginally increasing τd. Notice that

the form of the optimal τd is similar to the one in GRT, and we have

τd1 + τd

=α

1− απrlr

πnwnln(φd − φr)

[ωrµ

(−ul(cr, lr))φd − φr

− 1

]

C 0 < q < 1

We showed that the statement holds in the limit cases of q −→ 0 and q −→ 1. Thus, we

are left with non-limit cases. The equilibrium conditions are stated in the main text for

0 < q < 1 case. Although we cannot solve for the closed form solutions of endogenous

variables wr and m, we can deduce the sign of the comparative statics with respect to τx.

Recall that wn is not a function of q. It is useful to derive that at equilirbium, after some

51

algebra, we get

Y = A

(Xd +

(βX

q−1q

c + (1− β)(πrlr)q−1q

) qq−1

)1−α

(πnln)α (87)

=wnπnlnα

(88)

∫ 1

0

(φdxd + φrxc)di = φdm

1−m

(βX

q−1q

c + (1− β)(πrlr)q−1q

) qq−1

+ φrXc (89)

= φd

1((1+τd)φd(1−α)A

) 1α 1πnln

−(βX

q−1q

c + (1− β)(πrlr)q−1q

) qq−1

+ φrXc

(90)

Xc =

1− β((1+τr)φrβ(1+τd)φd

)q−1− β

qq−1

πrlr (91)

wr = (1− β)(1 + τd)φd

(βX

q−1q

c + (1− β)(πrlr)q−1q

) 1q−1

(πrlr)− 1q

= (1− β)(1 + τd)φd

(1− β)

β((1+τr)φrβ(1+τd)φd

)q−1− β

+ 1

1q−1

(92)

Note that the government problem with the income taxation specified in Lemma 1 is the

52

following:

maximizeτx

W (τx)

subject to

[ηrπr] u (cr, lr) ≥ u

(cn,

wnwrln

),

[ηnπn] u (cn, ln) ≥ u

(cr,

wrwnlr

),

[µ] πrcr + πncn +G ≤ πnwnln−

φdπnln

((1− α)A

(1 + τd)φd

) 1α

+ φd

(βX

q−1q

c + (1− β)Nq−1q

r

) qq−1

− φrXc,

where W (τx) = max πrωru (cr, lr) + πnωnu (cn, ln) + v(G).

Then, for fixed τd, the envelope condition with respect to τr gives

W ′(τr) =− ηnπnul(cr,

wrwrlr

)dwr

d (1 + τr)

lrwn

+ µ

[φd

d log

((βX

q−1q

c + (1− β)(πrlr)q−1q

) qq−1

)d log(1 + τr)(

βXq−1q

c + (1− β)(πrlr)q−1q

) qq−1

1 + τr− φr

dXc

d(1 + τr)

]. (93)

Note that

wrd(1 + τr)

= β(1− β)(1 + τd)φd(πrlr)− 1qX− 1q

cdXc

d(1 + τr)

(βX

q−1q

c + (1− β)Nq−1q

r

) 2−qq−1

,

d log

((βX

q−1q

c + (1− β)(πrlr)q−1q

) qq−1

)d log(1 + τr)

= βX− 1q

cdXc

d(1 + τr)

1 + τr

βXq−1q

c + (1− β)(πrlr)q−1q

,

and recall that from the first order condition of Xc on the production side equation (??), we

53

have

(1 + τx)φr = β(1 + τx)φd

(βX

q−1q

c + (1− β)(πrlr)q−1q

) 1q−1

X− 1q

c .

That is,

X− 1q

c =φrβφd

(βX

q−1q

c + (1− β)(πrlr)q−1q

) −1q−1

. (94)

Then, we get

W ′(τr) =− ηnπnul(cr,

wrwrlr

)β(1− β)(1 + τd)φd(πrlr)

− 1qX− 1q

cdXc

d(1 + τr)(βX

q−1q

c + (1− β)Nq−1q

r

) 2−qq−1 lr

wn. (95)

Notice that since η > 0 and the derivative of Xc with respect to τr is negative, the whole

term is negative. Therefore, we have

dW (τr)

d(−τr)> 0

Now, we fix τr and determine the sign of optimal τd.

W ′(τd) =− ηnπnul(cr,

wrwrlr

)d log (wr/wn)

d log (1 + τd)

1

1 + τd

wrlrwn

+ µ

[πnwnln

1

α(1 + τd)

d logwnd log(1 + τd)

− φd(

(1− α)A

(1 + τd)φd

) 1α

[d

d log(1 + τd)log

(((1− α)A

(1 + τd)φd

) 1α

)]πnln

1 + τd

+ φd

d log

((βX

q−1q

c + (1− β)(πrlr)q−1q

) qq−1

)d log(1 + τd)(

βXq−1q

c + (1− β)(πrlr)q−1q

) qq−1

1 + τd− φr

dXc

d(1 + τd)

]. (96)

54

Note that

d logwrd log(1 + τd)

= 1 + βX− 1q

cdXc

d(1 + τd)

1 + τd

βXq−1q

c + (1− β)(πrlr)q−1q

d log wrwn

d log(1 + τd)= 1 + βX

− 1q

cdXc

d(1 + τd)

1 + τd

βXq−1q

c + (1− β)(πrlr)q−1q

+1− αα

d log

((βX

q−1q

c + (1− β)(πrlr)q−1q

) qq−1

)d log(1 + τd)

= βX− 1q

cdXc

d(1 + τd)

1 + τd

βXq−1q

c + (1− β)(πrlr)q−1q

,

and again recall that

X− 1q

c =φrβφd

(βX

q−1q

c + (1− β)(πrlr)q−1q

) −1q−1

. (97)

Then, we get

W ′(τd) =− ηnπnul(cr,

wrwrlr

)1 +X− 1q

cdXc

d(1 + τd)

1 + τd

βXq−1q

c + (1− β)(πrlr)q−1q

+1− αα

1

1 + τd

wrlrwn

+ µ

[α− 1

α2

πnwnln1 + τd

+ φd

((1− α)A

(1 + τd)φd

) 1α πnln

1 + τd

](98)

Notice that the derivative of Xc with respect to τd is positive, so the first term is guaranteed

to be positive. The second term is always negative (since 0 < α < 1), while the third term

is positive. Then, notice

α− 1

α2

πnwnln1 + τd

+ φd

((1− α)A

(1 + τd)φd

) 1α πnln

1 + τd≥ 0

when

−(1 + τd)φd + φd ≥ 0.

Thus, when τd ≤ 0, the inequality holds true. Therefore, when τd ≤ 0, W ′(τd) > 0.

55

E General Model with Government Spending

The results of this paper carry through with the incorporation of government spending in

the model. First, the household problem becomes

maximizecj ,lj

Uj = u(cj, lj) + v(G)

subject to cj ≤ wjlj − T (wjlj),

where v′(G) > 0, v′′(G) < 0.

Accordingly, the output market equilibrium condition becomes

πrcr + πncn +G ≤ Y −∫ 1

0

(φdxd(i) + φrxc(i)) di. (99)

Notice that these additions will not affect any of the derivation and proofs for the propositions

above.

56

F Figures

A Figure 1: First-best Outcome

57

B Figure 2: Second-best Outcome—Leontief

58

C Figure 3x: Second-best Outcome—Cobb-Douglas

C.1 Figure 3a: β = 0.5

59

C.2 Figure 3b: β = 0.45

60

C.3 Figure 3c: β = 0.55

61

D Figure 4: Third-best Outcome–Leontief

62

E Figure 5: Third-best Outcome–Relation between τx and inter-

mediate goods Cost Difference

63

F Figure 6x: Welfare and Consumption Amount Comparison of

First-best, Second-best, and Third-best Outcomes

F.1 Figure 6a: Cobb-Douglas (β = 0.5)

64

F.2 Figure 6b: Routine Household Consumption—Cobb-Douglas (β = 0.5)

65

F.3 Figure 6c: Leontief

66

F.4 Figure 6c: Routine Household Consumption—Leontief

2000 2010 2020 2030 2040 2050

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

cr

Comparison of Routine Household Consumption

FB

SB

TB

Zero

67