Graduate Public Finance

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1 Graduate Public Finance Capital Taxes in a Spatial Setting Owen Zidar University of Chicago Lecture 3 Thanks to Fullerton and Ta, David Albouy, Alan Auerbach, Raj Chetty, Emmanuel Saez, Gabriel Zucman, and Treb Allen for providing their notes, some of which are reproduced here. Stephanie Kestelman provided excellent assistance making these slides. Graduate Public Finance Capital Taxes Lecture 3 1 / 71

2 Outline 1 Brief Introduction What is capital income? The level and distribution of capital income How is capital income taxed? 2 Capital Tax Incidence: simplest possible toy models Supply and demand Simple spatial model: One factor, two locations 3 Capital Tax Incidence: Harberger Fullerton and Ta (2017) Consumers and Producers Equilibrium Welfare loss from taxation Understanding equilibrium (graphical and quantitative analysis) Effect of Tax on Corporate Output Effect of Tax on Capital Effect of Tax on Corporate Capital Harberger Model (more general utility and technology) Graduate Public Finance Capital Taxes Lecture 3 2 / 71

3 Outline 1 Brief Introduction What is capital income? The level and distribution of capital income How is capital income taxed? 2 Capital Tax Incidence: simplest possible toy models Supply and demand Simple spatial model: One factor, two locations 3 Capital Tax Incidence: Harberger Fullerton and Ta (2017) Consumers and Producers Equilibrium Welfare loss from taxation Understanding equilibrium (graphical and quantitative analysis) Effect of Tax on Corporate Output Effect of Tax on Capital Effect of Tax on Corporate Capital Harberger Model (more general utility and technology) Graduate Public Finance Capital Taxes Lecture 3 3 / 71

4 Motivation Equity-efficiency tradeoffs of capital taxation seem especially stark 1 Efficiency Capital taxes reduce scale of economic activity Capital accumulation, which may be highly responsive to rates of return, is correlated with economic growth Capital mobility: taxes can lead to misallocation 2 Equity Distribution of capital income is much more unequal than labor income Capital mobility: burden may be shifted to labor 3 Evidence Empirical evidence/our understanding of capital taxation is less well developed than labor income taxation 4 Policy Relevance Future of fiscal policy (taxing robots, driverless cars, corp tax reform) Destination-based cash flow taxes, international reforms Graduate Public Finance Capital Taxes Lecture 3 4 / 71

5 What is capital income? Individuals derive market income (before tax) from labor and capital: z = wl + rk where w is wage, l is labor supply, k is wealth, r is rate of return on wealth 1 Labor income inequality is due to differences in working abilities (education, talent, physical ability, etc.), work effort (hours of work, effort on the job, etc.), and luck (labor effort might succeed or not) 2 Capital income inequality is due to differences in wealth k (due to past saving behavior and inheritances received), and in rates of return r (varies dramatically over time and across assets) Graduate Public Finance Capital Taxes Lecture 3 5 / 71

6 Level and distribution of capital income (1/2) Labor income wl 75% of national income z Capital income risk rk 25% of national income z (has increased in recent decades) Wealth stock k 400% 500% of national income z (is increasing) Rate of return on capital r 5% α = β r where α = rk/z share of capital income and β = k/z wealth to income ratio In GDP, gross capital share is higher (35%) because it includes depreciation of capital ( 10% of GDP) National Income = GDP depreciation of capital + net foreign income Graduate Public Finance Capital Taxes Lecture 3 6 / 71

7 Level and distribution of capital income (2/2) Graduate Public Finance Capital Taxes Lecture 3 7 / 71

8 How much is it taxed? In the US, total capital taxes can be decomposed into three categories of roughly equal importance: 1 Corporate tax = 3% of Y (around 20% of a 15% tax base) 2 Annual property rates = 3% of Y (around 1% of a 300% tax base) 3 Personal taxes on a capital income = 2.8% of Y (around 30% of a 15% x 60% = 9% tax base) + estates = 0.2% of Y (around 2% of a 10% tax base) Graduate Public Finance Capital Taxes Lecture 3 8 / 71

9 Outline 1 Brief Introduction What is capital income? The level and distribution of capital income How is capital income taxed? 2 Capital Tax Incidence: simplest possible toy models Supply and demand Simple spatial model: One factor, two locations 3 Capital Tax Incidence: Harberger Fullerton and Ta (2017) Consumers and Producers Equilibrium Welfare loss from taxation Understanding equilibrium (graphical and quantitative analysis) Effect of Tax on Corporate Output Effect of Tax on Capital Effect of Tax on Corporate Capital Harberger Model (more general utility and technology) Graduate Public Finance Capital Taxes Lecture 3 9 / 71

10 Impact of a Capital Tax Graduate Public Finance Capital Taxes Lecture 3 10 / 71

11 Impact of a Capital Tax Graduate Public Finance Capital Taxes Lecture 3 11 / 71

12 Impact of a Capital Tax (in Long Run) Graduate Public Finance Capital Taxes Lecture 3 12 / 71

13 Impact of a Capital Tax Who bears the capital tax in the long run? Who gets the triangle above R-pre-tax (i.e., consumer surplus in the typical S and D graph)? If firms don t earn profits, this all goes to workers in terms of higher wages or lower prices A key object is the elasticity of capital supply, is likely larger (and some think infinite) in the LR Note that the distortion in the capital market reduces surplus more than it increases tax revenues (as with most taxes) Finally, distortions due to capital taxation are often considered in a dynamic context in which the distortion compounds overtime (See Ivan Werning s recent paper on the classic Chamley-Judd results) Graduate Public Finance Capital Taxes Lecture 3 13 / 71

14 Impact of Capital Tax: One factor, two locations Setup 1 One factor (capital) 2 Two locations: east and west 3 Capital market in each location 4 Total K fixed in economy overall Graduate Public Finance Capital Taxes Lecture 3 14 / 71

15 Initial equilibrium Graduate Public Finance Capital Taxes Lecture 3 15 / 71

16 Tax in west Causes capital to flee to east Graduate Public Finance Capital Taxes Lecture 3 16 / 71

17 New allocation of capital K flows to east, lowering net returns in both Flows continue until after tax return is equalized across markets Graduate Public Finance Capital Taxes Lecture 3 17 / 71

18 Welfare changes in each location Welfare in west falls by red amount Welfare in east increases Graduate Public Finance Capital Taxes Lecture 3 18 / 71

19 Net welfare changes in aggregate Net welfare loss in red Graduate Public Finance Capital Taxes Lecture 3 19 / 71

20 What determines size of welfare loss in this toy example? 1 Size of tax change 2 Size of market being taxed (depends on fundamentals) 3 Elasticity of demand in both regions (quantity response more generally, which depends on S and D elasticities) 4 Strength of complementarities across markets (e.g., labor market) 5 Assumptions about effects/value of government spending (assumed to be zero here) 6 Presence of existing distortions Will formalize these ideas in the next section, but this example provides intuition for key forces in the Harberger model Graduate Public Finance Capital Taxes Lecture 3 20 / 71

21 Outline 1 Brief Introduction What is capital income? The level and distribution of capital income How is capital income taxed? 2 Capital Tax Incidence: simplest possible toy models Supply and demand Simple spatial model: One factor, two locations 3 Capital Tax Incidence: Harberger Fullerton and Ta (2017) Consumers and Producers Equilibrium Welfare loss from taxation Understanding equilibrium (graphical and quantitative analysis) Effect of Tax on Corporate Output Effect of Tax on Capital Effect of Tax on Corporate Capital Harberger Model (more general utility and technology) Graduate Public Finance Capital Taxes Lecture 3 21 / 71

22 Overview 1 Goals Characterize effects of corporate tax change in a GE model Who bears the burden of corporate taxes? (also capital, output taxes) 2 Two sectors (or locations) Corporate sector produces output X Non-corporate sector produces output Y 3 Markets Capital: prices r i, quantities K i where i {X, Y } Labor: prices w i, quantities L i Goods: prices p i, quantities X, Y 4 Agents Workers (representative, perfectly mobile, supply 1 unit of labor) Firm (representative, perfectly competitive, CRS) 5 Equilibrium Conditions Good and factor markets clear, factor price equalization Consumers max utility, firms earn zero profits Graduate Public Finance Capital Taxes Lecture 3 22 / 71

23 Comments 1 Harberger is workhorse analytical model: 2 sector and 2 factors 2 Fixed supply of capital and labor (short run, closed economy) 3 Key intuition is misallocation (magnitude depends on factor intensity, demand elasticities, etc) 4 Fullerton and Ta (2017) simplifies Harberger analysis (Cobb Douglas) 5 Similar to Hecksher-Ohlin model 6 When interpreting as locations not sectors, then implicitly assume no trade costs. Similarly, implicitly assumes no adjustment costs for capital and labor (so long run in that sense) 7 Abstracts from amenity or productivity effects of government spending (lump sum rebates or purchases in same share as consumers) Graduate Public Finance Capital Taxes Lecture 3 23 / 71

24 Fullerton and Ta (2017) Parameterized Harberger Model with Cobb Douglas Graduate Public Finance Capital Taxes Lecture 3 24 / 71

25 Consumers: Preferences and Budget Constraint Utility of representative worker is U = X γ Y 1 γ X is corporate sector output Y is non-corporate sector output Budget constraint is p x X + p y Y = I I is income, which is sum of labor and capital income p i is price of output in sector i where i {X, Y } Workers have fixed expenditure shares (e.g. I γ); demand for X and Y is: X = I γ Y = p x I (1 γ) p y N.B. note no labor supply or saving decision Graduate Public Finance Capital Taxes Lecture 3 25 / 71

26 Consumers: Indirect Utility Indirect utility is given ( ) γ ( ) 1 γ I γ I (1 γ) V (p x, p y, I ) = = I p ( ) γ ( ) 1 γ where p = px py γ 1 γ is the ideal price index p x Inverting indirect utility (i.e., V = Ī p ), gives the expenditure function I = E: So p is the price paid for each util p y E( p, U) = U p Graduate Public Finance Capital Taxes Lecture 3 26 / 71

27 Firms maximize profits Corporate sector solves: max(1 τ X )p x X (1 + τ K + τ KX )rk x wl x, where X = AKx α Lx 1 α K x,l x where τ X = tax on output of X τ K = tax on capital Non-corporate sector solves: τ KX = tax on capital in production of X max p y Y (1 + τ K )rk y wl y, where Y = BKy β Ly 1 β K y,l y Graduate Public Finance Capital Taxes Lecture 3 27 / 71

28 Firm optimization (and factor demand) FOCs: w = (1 τ X )p x (1 α)a w = p y (1 β)b ( ) β Ky L y ( ) α Kx L x and ( Lx (1 + τ K + τ KX )r = (1 τ X )p x αa (1 + τ K )r = p y βb K x ( ) 1 β Ly K y ) 1 α Graduate Public Finance Capital Taxes Lecture 3 28 / 71

29 Exogenous parameters Taxes: τ X, τ K, τ KX τ X is tax on corporate sector output (sales tax) τ K is tax on capital τ KX is tax on capital used in corporate sector Consumer Parameter: γ γ governs importance of corporate goods for utility 1 γ governs importance of non-corporate goods for utility Firm Parameters: α, β, A, B α is output elasticity of capital in sector X 1 α output elasticity of labor in sector X β output elasticity of capital in sector Y 1 β output elasticity of labor in sector Y A and B are productivity in corp and non-corp sectors Endowments; K, L K is total capital L is total labor Graduate Public Finance Capital Taxes Lecture 3 29 / 71

30 Endogenous Model Outcomes and Equilibrium Endogenous outcomes are K i, L i, p i, X, Y, w, r: Capital: prices r i, quantities K i where i {X, Y } Labor: prices w i, quantities L i Goods: prices p i, quantities X, Y Given τ X, τ K, τ KX, γ, α, β, A, B, K, L, equilibrium is defined by prices and quantities {w, r, p i, K x, K y, L x, L y, X, Y } such that good and factor markets clear and firms and workers optimize. Graduate Public Finance Capital Taxes Lecture 3 30 / 71

31 Equilibrium: closed form expressions Graduate Public Finance Capital Taxes Lecture 3 31 / 71

32 Ten Equations and ten unknowns (with taxes) In log terms, the equations are: ln X = ln I + ln γ ln p x ln Y = ln I + ln(1 γ) ln p y ln K x ln L x ln w + ln r = ln α ln(1 α) ln(1 + τ K + τ KX ) ln K y ln L y ln w + ln r = ln β ln(1 β) ln(1 + τ K ) ln X ln K x + ln p x ln r = ln(1 + τ K + τ KX ) ln α ln(1 τ X ) ln X + ln L x ln p x + ln w = ln(1 α) + ln(1 τ X ) ln Y ln K y + ln p y ln r = ln(1 + τ K ) ln β ln Y + ln K y ln p y + ln w = ln(1 β) where K = K x + K y and L = L x + L y. Equations without taxes Graduate Public Finance Capital Taxes Lecture 3 32 / 71

33 Solutions with taxation (1/2) Given taxes τ K, τ X, and τ KX, we have ( α(1 + τ K ) X = Aγ(1 τ X ) αγ(1 τ X )(1 + τ K ) + β(1 γ)(1 + τ K + τ KX ) K ( ) 1 α (1 α) (1 α)γ(1 τ X ) + (1 β)(1 γ) L ( β(1 + τ K + τ KX ) Y = B(1 γ) αγ(1 τ X )(1 + τ K ) + β(1 γ)(1 + τ K + τ KX ) K ( ) 1 β (1 β) (1 α)γ(1 τ X ) + (1 β)(1 γ) L αγ(1 τ X )(1 + τ K ) K x = αγ(1 τ X )(1 + τ K ) + β(1 γ)(1 + τ K + τ KX ) K β(1 γ)(1 + τ K + τ KX ) K y = αγ(1 τ X )(1 + τ K ) + β(1 γ)(1 + τ K + τ KX ) K (1 α)γ(1 τ X ) L x = (1 α)γ(1 τ X ) + (1 β)(1 γ) L ) β ) α Graduate Public Finance Capital Taxes Lecture 3 33 / 71

34 Solutions with taxation (1/2) (1 β)(1 γ) L y = (1 α)γ(1 τ X ) + (1 β)(1 γ) L ( I αγ(1 τx )(1 + τ K ) + β(1 γ)(1 + τ K + τ KX ) p x = A(1 τ X ) α(1 + τ K )K ( ) 1 α (1 α)γ(1 τx ) + (1 β)(1 γ) (1 α)l p y = I ( ) β αγ(1 τx )(1 + τ K ) + β(1 γ)(1 + τ K + τ KX ) B β(1 + τ K + τ KX )K ( ) 1 β (1 α)γ(1 τx ) + (1 β)(1 γ) (1 β)l w = I L [(1 α)γ(1 τ X ) + (1 β)(1 γ)] [ ] αγ(1 τx )(1 + τ K ) + β(1 γ)(1 + τ K + τ KX ) ) α r = I K (1 + τ K )(1 + τ K + τ KX ) Graduate Public Finance Capital Taxes Lecture 3 34 / 71

35 Welfare Loss from Taxation Graduate Public Finance Capital Taxes Lecture 3 35 / 71

36 Equivalent variation and Burden of taxation Equivalent variation EV is the change in wealth at initial prices that would be equivalent to the price change in terms of utility. EV = E( p 0, U 1 ) Ī = p 0 U 1 p 0 U 0 = p 0 (U 1 U 0 ) where p 0 and p 1 are the ideal prices in period 0 and 1 Use EV as a positive measure of tax burden, so EB = EV = p 0 (U 0 U 1 ) Amount that burden exceeds tax revenues is called excess burden (Auerbach and Hines, 2002) Graduate Public Finance Capital Taxes Lecture 3 36 / 71

37 Average and Marginal Excess Burden Average Excess Burden (AEB) is the total welfare loss from the tax divided by the total revenue collected by the government: AEB = EB R where p 0 and p 1 are the ideal prices in period 0 and 1 Marginal excess burden (MEB) measures the effects of a small change in the tax rate on burden: EB = p 0 (EB 1 EB 2 ) MEB = EB R N.B. See Hendren s recent TPE paper for more detailed discussion Graduate Public Finance Capital Taxes Lecture 3 37 / 71

38 Understanding Equilibrium: Graphical and quantitative analysis Graduate Public Finance Capital Taxes Lecture 3 38 / 71

39 Understanding Equilibrium: Graphs and numerical example There are a lot of moving parts Helpful to think about relative factor markets (relative prices and relative quantities) in the two sectors Will start with demand side, then supply side, then analyze equilibrium graphically pre and post taxes Will work with a calibrated version of the model to do quantitative analysis Graduate Public Finance Capital Taxes Lecture 3 39 / 71

40 Relative factor demand Taking ratios of each sector s FOCs gives: w r = (1 α) α ( Lx K x ) 1 (1 + τ K + τ KX ) (1) w r = (1 β) β ( Ly K y ) 1 (1 + τ K ) (2) Graduate Public Finance Capital Taxes Lecture 3 40 / 71

41 Relative factor supply Recall L = L x + L y K = K x + K y Thus, the economy-wide labor capital ratio is: L K = L x K + L y K L K = L ( ) x Kx + L ( ) y Ky K x K K y K (3) This says that overall labor to capital ratio is a weighted average of the labor to capital ratio in both sectors Graduate Public Finance Capital Taxes Lecture 3 41 / 71

42 Factor Market Prices and Quantities We can invert 1 and 2 to get L x /K x and L y /K y as functions of w/r. Then L K = ( w r ) ( ( 1) (1 + τk + τ kx )(1 α) K x α K + (1 + τ k)(1 β) β In equilibrium, we found ) K y K (4) K x K = αγ(1 τ X )(1 + τ K ) αγ(1 τ X )(1 + τ K ) + β(1 γ)(1 + τ K + τ KX ) K y K = β(1 γ)(1 + τ K + τ KX ) αγ(1 τ X )(1 + τ K ) + β(1 γ)(1 + τ K + τ KX ) (5) (6) Graduate Public Finance Capital Taxes Lecture 3 42 / 71

43 Numerical Example X and Y produced given functions Identical households have utility: Fixed 1 level of income I = 2, 400 Demand for X and Y is given by: X = AK x.6 L.4 x Y = BK y.2 L.8 y U = X.5 Y.5 X = 2400(.5) = 1200 p x p x Y = 2400(.5) = 1200 p y p y Assume unity of prices in the initial state (p x = p y = r = w = 1) Graduate Public Finance Capital Taxes Lecture 3 43 / 71

44 Initial quantities and prices (τ K = τ KX = τ X = 0) With this parameterization of utility and technology, we have: Value L x 480 L y 960 K x 720 K y 240 X 1200 Y 1200 p x 1 p y 1 r 1 w 1 Use the values above to derive A 1.96 and B Graduate Public Finance Capital Taxes Lecture 3 44 / 71

45 Initial Factor Market Equilibrium (τ K = τ KX = τ X = 0) w/r Lx/Kx(0) L/K(0) Ly/Ky(0) L/K X Overall Y Figure: Wage to Rent Ratio in both sectors and economy overall L x K x = 2 3, Kx K = 3 4, Ly K y = 4, and Ky K = 1 4, so L K = = 1.5. Graduate Public Finance Capital Taxes Lecture 3 45 / 71

46 Effect of Tax on Corporate Output (τ X =.3) 1 τ X reduces demand for X 2 We will have factors move from producing X to producing Y until prices and quantities re-equilibrate 3 Specifically, since w x = w y (1 + τ x )p x MPL x = p y MPL y, we need a combination of lower p x and higher MPL x (and thus lower factor demand in x) and/or higher p y and lower MPL y 4 The movement of both factors to Y increases the weight of the non-corporate sector in labor and capital demand (see eq 4 and dashed green line in next slide), so wage to rental ratio increases Graduate Public Finance Capital Taxes Lecture 3 46 / 71

47 Tax on Corporate Output w/r Lx/Kx(t_x) L/K(t_x) 2 Ly/Ky(t_x) 4 6 L/K X(0) Overall(0) Y(t_x) Y(0) X(t_x) Overall(t_x) Figure: Relative Factor market equilibrium with τ X =.3, τ K = τ KX = 0 Graduate Public Finance Capital Taxes Lecture 3 47 / 71

48 Tax on Corporate Output: Prices and Quantities Panel A: Allocations and Prices t X = 0 t X =.3 t X =.31 L x L y 960 1, , K x K y X 1,200 1, , Y 1,200 1, , p x p y r w w/r L x /K x L y /K y L/K Graduate Public Finance Capital Taxes Lecture 3 48 / 71

49 Tax on Corporate Output: Welfare and Burden Measures Panel B: Exact Measures of Welfare t X = 0 t X =.3 t X =.31 p 0 2 p U 1,200 1, , EB R AEB MEB Graduate Public Finance Capital Taxes Lecture 3 49 / 71

50 Effect of Tax on Capital 1 Suppose a tax on all capital: τ K =.3, and τ KX = τ X = 0 2 Both sectors face tax on capital, so capital allocation across sectors does not change (see 5 and 6 in which the (1 + τ k ) terms cancel) 3 (1 + τ k ) increases relative labor demand symmetrically in eq 4 in both sectors (i.e., it shifs up L i /K i ), so factor allocation stays constant and all adjustment is through relative prices 4 In this case, capital fully bears the burden of the tax (i.e., w/r rises by 30% to offset tax increase) N.B. remember that in these examples, the overall stock of capital is fixed. In practice, investment and firm creation respond to taxes. A key question is how much they respond Graduate Public Finance Capital Taxes Lecture 3 50 / 71

51 Tax on Capital w/r Lx/Kx(0) = L/K(0) = 2 4 Ly/Ky(0) = 6 Lx/Kx(t_k) L/K(t_k) L/K Ly/Ky(t_k) X(0) Overall(0) Y(t_k) Y(0) X(t_k) Overall(t_k) Figure: Relative Factor market equilibrium with τ K =.3, τ KX = τ X = 0 Graduate Public Finance Capital Taxes Lecture 3 51 / 71

52 Tax on Capital: Allocation and Quantities Panel A: Allocations and Prices t K =0 t K =.3 t K =.31 L x L y K x K y X 1,200 1,200 1,200 Y 1,200 1,200 1,200 p x p y r w w/r L x /K x L y /K y L/K Graduate Public Finance Capital Taxes Lecture 3 52 / 71

53 Tax on Capital: Welfare and Burden Measures Panel B: Exact Measures of Welfare t K =0 t K =.3 t K =.31 p 0 2 p U 1,200 1,200 1,200 EB R AEB 0 0 MEB 0 Graduate Public Finance Capital Taxes Lecture 3 53 / 71

54 Effect of Tax on Corporate Capital 1 Now suppose a tax on corporate capital, τ KX =.3 px MPKx 2 Corporate sector demands less capital (r x = 1+τ KX ), so capital flows from corporate to non-corporate sector (see eq 5 and 6) 3 Lower capital allocation to producing X increases the weight of the non-corporate sector in labor and capital demand (see eq 4 and dashed green line in next slide) 4 Causes misallocation (too much K y and thus, too much Y, not enough X ), which reduces welfare as in prior example Graduate Public Finance Capital Taxes Lecture 3 54 / 71

55 Tax on Corporate Capital w/r Lx/Kx(t_kx) 0 L/K(t_kx) 2 Ly/Ky(t_kx) 4 6 L/K X(0) Overall(0) Y(t_kx) Y(0) X(t_kx) Overall(t_kx) Figure: Relative Factor market equilibrium with τ KX =.3, τ K = τ X = 0 Graduate Public Finance Capital Taxes Lecture 3 55 / 71

56 Tax on Corporate Capital: Allocation and Quantities Panel A: Allocations and Prices t KX = 0 t KX =.3 t KX =.31 L x L y K x K y X 1,200 1, , Y 1,200 1, , p x p y r w w/r L x /K x L y /K y L/K Graduate Public Finance Capital Taxes Lecture 3 56 / 71

57 Tax on Corporate Capital: Welfare and Burden Measures Panel B: Exact Measures of Welfare t KX = 0 t KX =.3 t KX =.31 p 0 2 p U 1,200 1, , EB R AEB MEB Graduate Public Finance Capital Taxes Lecture 3 57 / 71

58 Harberger Model Graduate Public Finance Capital Taxes Lecture 3 58 / 71

59 Harberger (more general utility and technology) Ten equations needed for equilibrium are: p x X + p y Y = wl + rk (7) MRS XY = p x(1 + τ X ) p y (8) c x (w, r(1 + τ K + τ KX )) = p x (9) c y (w, r(1 + τ K )) = p y (10) w = p x F xl (11) w = p y F yl (12) r(1 + τ K + τ KX ) = p x (1 τ X )F xk (13) r = p y F yk (14) K = K x + K y (15) L = L x + L y (16) Graduate Public Finance Capital Taxes Lecture 3 59 / 71

60 Definitions Share of income spent on X and Y : s x p x X p x X + p y Y, s y Share of income from labor and capital: s w wl wl + rk, s r Cost shares in production of X and Y : p y Y p x X + p y Y, s x + s y = 1 rk wl + rk wl x rk x θ L, θ K, θ x + θ y = 1 wl x + rk x wl x + rk x wl y rk y φ L, φ K, φ x + φ y = 1 wl y + rk y wl y + rk y Graduate Public Finance Capital Taxes Lecture 3 60 / 71

61 Definitions Share of labor and capital used to produce X : λ L L x L, λ K K x K By Euler s Theorem and CRS, we also have: p x X = wl x + rk x, p y Y = wl y + rk y s x θ L s x (1 θ K ) λ L = = s x θ L + s y φ L 1 s x θ K s y φ K s x θ K s x (1 θ L ) λ K = = s x θ K + s y φ K 1 s x θ L s y φ L Graduate Public Finance Capital Taxes Lecture 3 61 / 71

62 Log-Linearization s x ( ˆp x + ˆX ) + s y ( ˆp y + Ŷ ) = s w ŵ + s r ˆR ˆX Ŷ = σ D( ˆp y ˆp x dτ X ) ˆp x = θ L ŵ + θ K (ˆr + dτ K + dτ KX ) ˆp y = φ L ŵ + φ K (ˆr + dτ K ) λ L ˆL x + (1 λ L ) ˆL y = 0 λ K ˆK x + (1 λ K ) ˆK y = 0 ˆL x = ˆX + θ K σ X (ˆr + dτ K + dτ KX ŵ ˆK x = ˆX + θ L σ X (ŵ dτ K dτ KX ˆr ˆL y = Ŷ + φ K σ Y (ˆr + dτ K ŵ) ˆK y = Ŷ + φ L σ Y (ŵ dτ K ˆr) Graduate Public Finance Capital Taxes Lecture 3 62 / 71

63 Matrix Form of the System of Linear Equations σ D θ L θ K φ L φ K λ L 0 1 λ L λ K 0 1 λ K 0 θ K σ X θ K σ X θ L σ X θ L σ X φ K σ Y φ K σ Y φ L σ Y φ L σ Y ˆp x ŵ ˆr ˆX Ŷ ˆL x ˆK x ˆL y ˆK y = σ X dτ X + 0 θ K θ K σ X θ L σ X 0 0 dτ KX where we eliminated equation 1 by Walras law and normalized p y = 1, so ˆp y = 0 Graduate Public Finance Capital Taxes Lecture 3 63 / 71

64 Two Main Effects of Taxing K x 1 Substitution effects: capital bears incidence 2 Output effects: capital may not bear all incidence Graduate Public Finance Capital Taxes Lecture 3 64 / 71

65 Substitution effects Tax on K x shifts production in X away from K so aggregate demand for K goes down Because total K is fixed, r falls K bears some of the burden Another intuition for this is that capital is misallocated across sectors, which lowers r and rk Graduate Public Finance Capital Taxes Lecture 3 65 / 71

66 Output effects Tax on K x makes X more expensive Demand shifts to Y Case 1: K x /L x > K y /L y (X: cars, Y: bikes) X more capital intensive lower aggregate demand for K Output + subst. effect: K bears the burden of the tax Case 2: K x /L x < K y /L y (X: bikes, Y: cars) X less capital intensive higher aggregate demand for K Subst. and output effects have opposite signs labor may bear some the tax Graduate Public Finance Capital Taxes Lecture 3 66 / 71

67 Takeaways Harberger showed that under a variety of reasonable assumptions, capital bears exactly 100 percent of the tax. Note that this is the burden on all capital as capital flees the corporate sector, it depresses returns in the noncorporate sector as well. Both the realism of the model and the characterization of the corporate income tax as an extra tax on capital in the corporate sector are subject to question, as discussed in considerable detail by the subsequent literature on the effects of the corporate tax. Alan Auerbach See Auerbach TPE paper on who bears the corporate tax for more details Graduate Public Finance Capital Taxes Lecture 3 67 / 71

68 Appendix: Graduate Public Finance Capital Taxes Lecture 3 68 / 71

69 Ten equations and ten unknowns (without taxes) In log terms, the equations are: ln X = ln A + α ln K x + (1 α) ln L x = ln I + ln γ ln p x ln Y = ln B + β ln K y + (1 β) ln L y = ln I + ln(1 γ) ln p y ln w = ln p x + ln(1 α) + ln A + α(ln K x ln L x ) = ln p y + ln(1 β) + ln B + β(ln K y ln L y ) ln r = ln p x + ln α + ln A + (1 α)(ln L x ln K x ) = ln p y + ln β + ln B + (1 β)(ln L y ln K y ) where K = K x + K y and L = L x + L y. Graduate Public Finance Capital Taxes Lecture 3 69 / 71

70 Solutions in initial equilibrium without taxes (1/2) Solving for the system of equations, quantities are: ( ) α ( ) 1 α αk (1 α)l X = Aγ αγ + β(1 γ) (1 α)γ + (1 β)(1 γ) ( ) β ( βk (1 β)l Y = B(1 γ) αγ + β(1 γ) (1 α)γ + (1 β)(1 γ) αγ K x = αγ + β(1 γ) K β(1 γ) K y = αγ + β(1 γ) K (1 α)γ L x = (1 α)γ + (1 β)(1 γ) L (1 β)(1 γ) L y = (1 α)γ + (1 β)(1 γ) L ) 1 β Graduate Public Finance Capital Taxes Lecture 3 70 / 71

71 Solutions in initial equilibrium without taxes (2/2) Prices are: p x = I ( ) α ( αγ + β(1 γ) (1 α)γ + (1 β)(1 γ) A αk (1 α)l ( ) β ( αγ + β(1 γ) (1 α)γ + (1 β)(1 γ) ) 1 α ) 1 β p y = I B βk (1 β)l w = I ((1 α)γ + (1 β)(1 γ)) L r = I K (αγ + β(1 γ)) Back Graduate Public Finance Capital Taxes Lecture 3 71 / 71