Graduate Public Finance

Transcription

1 Graduate Public Finance Capital Taxes in a Spatial Setting Owen Zidar Princeton Fall 2017 Lecture 3 Thanks to Fullerton and Ta, David Albouy, Alan Auerbach, Raj Chetty, Kevin Murphy, 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 (Econ 523) Capital Taxes Lecture 3 1 / 104

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 in the capital market Brief aside on rental and asset markets for capital 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 (Econ 523) Capital Taxes Lecture 3 2 / 104

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 in the capital market Brief aside on rental and asset markets for capital 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 (Econ 523) Capital Taxes Lecture 3 3 / 104

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 (Econ 523) Capital Taxes Lecture 3 4 / 104

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 (Econ 523) Capital Taxes Lecture 3 5 / 104

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 (Econ 523) Capital Taxes Lecture 3 6 / 104

7 Level and distribution of capital income (2/2) Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 7 / 104

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 (Econ 523) Capital Taxes Lecture 3 8 / 104

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 in the capital market Brief aside on rental and asset markets for capital 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 (Econ 523) Capital Taxes Lecture 3 9 / 104

10 Impact of a Capital Tax Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 10 / 104

11 Impact of a Capital Tax Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 11 / 104

12 Impact of a Capital Tax (in Long Run) Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 12 / 104

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 (Econ 523) Capital Taxes Lecture 3 13 / 104

14 Aside on capital markets Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 14 / 104

15 Capital markets: 4 key equations We will use 4 equations to analyze capital markets 1 Stock Adjustment: the amount of capital today depends on how much there was yesterday, depreciation, and new investment 2 Asset pricing equilibrium: 1 the rental price of using an asset is simply the cost of buying the good and re-selling it after one period 3 Rental market equilibrium: the demand for using capital services is downward sloping 4 Investment market equilibrium: the supply of capital assets is upward sloping Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 15 / 104

16 Rental and asset markets are linked Use the link between rental and asset markets to analyze capital markets Rental Market Asset Market R t P t S(R t ) S(P t ) R* P* D(R t ) D(P t ) K* K t I* I t where R t is the rental price of using capital services K t and P t is the purchase price, which depends on the level of investment I t. Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 16 / 104

17 4 key equations 1 Stock Adjustment: K t = (1 δ)k t 1 + I t 2 Asset pricing equilibrium The rental cost of using an asset is simply the cost of buying the good and re-selling it after one period 3 Rental market equilibrium: K = D(R) 4 Investment market equilibrium: I = S(P) Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 17 / 104

18 2. Asset pricing equilibrium (without taxes) What is the relationship between rental and capital prices? The rental cost of using an asset is simply the cost of buying the good and re-selling it after one period R t = P t (1 δ)p t r r is the nominal rate of interest P t+1 is next year s price for the good Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 18 / 104

19 2. Asset pricing equilibrium: Housing example Suppose Suppose r =.10 and δ = 0 P t+1 =$ 110 K P t =$ 100 K What is R t? Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 19 / 104

20 2. Asset pricing equilibrium: Housing example Suppose Suppose r =.10 and δ = 0 P t+1 =$ 110 K P t =$ 100 K What is R t? R t = P t (1 δ)p t r R t = R t = 0 Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 19 / 104

21 2. Analyzing Rental Price We can rearrange the expression to show rental prices depend on three things: R t = rp t + δp t+1 + P t P t r 1 Interest cost 2 : rp t 2 Depreciation: δp t+1 3 Market re-evaluation: P t P t+1 Rental prices are higher, the higher is r, the greater is the physical rate of depreciation, and the faster the price of the asset is declining Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 20 / 104

22 2. Analyzing Rental Price: Car example R t = rp t + δp t+1 + P t P t r If cars lose their value quickly (i.e., P t >> P t+1 ), then rental prices will be pretty high Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 21 / 104

23 2. Analyzing Capital Prices We can also use the rental price expression to calculate the implied capital price P t = R t + R t+1(1 δ) (1 + r) + R t+2(1 δ) 2 (1 + r) This equation can be obtained by recursively substituting for future prices in the rental price equation This equation should look familiar to you (prices are PV of cash flow stream) Capital prices are higher when rental payments to the owner are large and soon Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 22 / 104

24 3. Rental Market Equilibrium for Housing Services K t = D(R t ) The demand for housing services depends on the flow cost of housing services (i.e., the rental rate R t ). R t is what I pay to use the asset Housing services are provided by the stock of housing K t The demand side of the market links the current rental price and the current stock Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 23 / 104

25 3. Rental Market Equilibrium Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 24 / 104

26 4. Investment Market Equilibrium I t = S(P t ) The supply of new construction, investment depends on its current price Think of this as a new car producer who decides how much to supply based on the current price Alternatively, housing construction firms see high house prices and build. They build more when prices are high. Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 25 / 104

27 4. Investment Market Equilibrium Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 26 / 104

28 4 key equations K t = (1 δ)k t 1 + I t (1) R t = P t (1 δ)p t r (2) K t = D(R t ) (3) I t = I (P t ) (4) 4 equations and 4 unknowns, but depends on past and the future. Where do past and future come in? Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 27 / 104

29 Market Equilibrium: Past and Future in Housing When we look at a market equilibrium for the housing market at any one point in time, we must realize that today s market is influenced by both the past and future The effect of the past comes through the effect of past production decisions on the stock of housing The effect of the future comes from the effect of future expected rental rates on the current price Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 28 / 104

30 What does the system look like in steady state? K = (1 δ) K + Ī R = P (1 δ) P 1 + r K = D( R) Ī = S( P) Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 29 / 104

31 What does the system look like in steady state? Ī = δ K R = P ( 1 K = D( R) Ī = S( P) ) (1 δ) 1 + r Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 30 / 104

32 What does the system look like in steady state? We can use the first two equations to plug into the second two equations and obtain the supply and demand in the use market. ( R 1 (1 δ) 1+r Ī }{{} δ K Ī = δ K ) = P K = D( R) = S( }{{} P ) ( R 1 (1 δ) ) 1+r Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 31 / 104

33 What does the system look like in steady state? K = D( R) K = 1 δ S ( R 1 (1 δ) 1+r ) This shows that we have a familiar supply and demand diagram where the quantity is K and the price is R Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 32 / 104

34 Capital Market Equilibrium R t S(R t ) R* D(R t ) K* K t Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 33 / 104

35 Earthquake Destroys part of capital stock Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 34 / 104

36 Earthquake Destroys part of capital stock The main impact is on the use market. Lower K increases R. Higher rental prices cause the asset price P to increase. However, since rental rates we decline as we rebuild capital stock, the increase in P is smaller than increase in R Investment follows P, so it will jump and slowly decline as we rebuild the stock Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 35 / 104

37 Earthquake Destroys part of capital stock Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 36 / 104

38 Speed of Adjustment What determines the speed of convergence to the steady state? 1 Elasticity of demand in the rental market ε D. For example, the more the rental price goes up following a destruction of the capital stock, the faster we will converge to steady state (since it will make the capital price go up more, and thereby also investments). With a higher elasticity (in absolute value), the rental price will go up more. 2 Elasticity of supply in the investment market ε S. This will make investment go up more when the capital price goes up. 3 The depreciation rate δ. This may be the most important aspect, since it puts a lower bound on the speed of convergence. The slowest rate at which the economy ever can return to the steady state is δ. Others examples: construction costs interest rates housing bubble? Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 37 / 104

39 Simple spatial model: One factor, two locations Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 38 / 104

40 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 (Econ 523) Capital Taxes Lecture 3 39 / 104

41 Initial equilibrium Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 40 / 104

42 Tax in west Causes capital to flee to east Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 41 / 104

43 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 (Econ 523) Capital Taxes Lecture 3 42 / 104

44 Welfare changes in each location Welfare in west falls by red amount Welfare in east increases Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 43 / 104

45 Net welfare changes in aggregate Net welfare loss in red Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 44 / 104

46 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 (Econ 523) Capital Taxes Lecture 3 45 / 104

47 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 in the capital market Brief aside on rental and asset markets for capital 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 (Econ 523) Capital Taxes Lecture 3 46 / 104

48 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 (Econ 523) Capital Taxes Lecture 3 47 / 104

49 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 (Econ 523) Capital Taxes Lecture 3 48 / 104

50 Fullerton and Ta (2017) Parameterized Harberger Model with Cobb Douglas Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 49 / 104

51 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 (Econ 523) Capital Taxes Lecture 3 50 / 104

52 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 (Econ 523) Capital Taxes Lecture 3 51 / 104

53 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 (Econ 523) Capital Taxes Lecture 3 52 / 104

54 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 (Econ 523) Capital Taxes Lecture 3 53 / 104

55 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 (Econ 523) Capital Taxes Lecture 3 54 / 104

56 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 (Econ 523) Capital Taxes Lecture 3 55 / 104

57 Equilibrium: closed form expressions Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 56 / 104

58 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 (Econ 523) Capital Taxes Lecture 3 57 / 104

59 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 (Econ 523) Capital Taxes Lecture 3 58 / 104

60 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 (Econ 523) Capital Taxes Lecture 3 59 / 104

61 Welfare Loss from Taxation Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 60 / 104

62 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 (Econ 523) Capital Taxes Lecture 3 61 / 104

63 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 (Econ 523) Capital Taxes Lecture 3 62 / 104

64 Understanding Equilibrium: Graphical and quantitative analysis Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 63 / 104

65 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 (Econ 523) Capital Taxes Lecture 3 64 / 104

66 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 ) (5) w r = (1 β) β ( Ly K y ) 1 (1 + τ K ) (6) Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 65 / 104

67 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 (7) This says that overall labor to capital ratio is a weighted average of the labor to capital ratio in both sectors Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 66 / 104

68 Factor Market Prices and Quantities We can invert 5 and 6 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 (8) 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 ) (9) (10) Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 67 / 104

69 Numerical Example X and Y produced given functions Identical households have utility: Fixed 3 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 (Econ 523) Capital Taxes Lecture 3 68 / 104

70 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 (Econ 523) Capital Taxes Lecture 3 69 / 104

71 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 (Econ 523) Capital Taxes Lecture 3 70 / 104

72 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 8 and dashed green line in next slide), so wage to rental ratio increases Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 71 / 104

73 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 (Econ 523) Capital Taxes Lecture 3 72 / 104

74 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 (Econ 523) Capital Taxes Lecture 3 73 / 104

75 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 (Econ 523) Capital Taxes Lecture 3 74 / 104

76 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 9 and 10 in which the (1 + τ k ) terms cancel) 3 (1 + τ k ) increases relative labor demand symmetrically in eq 8 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 (Econ 523) Capital Taxes Lecture 3 75 / 104

77 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 (Econ 523) Capital Taxes Lecture 3 76 / 104

78 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 (Econ 523) Capital Taxes Lecture 3 77 / 104

79 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 (Econ 523) Capital Taxes Lecture 3 78 / 104

80 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 9 and 10) 3 Lower capital allocation to producing X increases the weight of the non-corporate sector in labor and capital demand (see eq 8 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 (Econ 523) Capital Taxes Lecture 3 79 / 104

81 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 (Econ 523) Capital Taxes Lecture 3 80 / 104

82 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 (Econ 523) Capital Taxes Lecture 3 81 / 104

83 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 (Econ 523) Capital Taxes Lecture 3 82 / 104

84 Harberger Model Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 83 / 104

85 Harberger (more general utility and technology) Ten equations needed for equilibrium are: p x X + p y Y = wl + rk (11) MRS XY = p x(1 + τ X ) p y (12) c x (w, r(1 + τ K + τ KX )) = p x (13) c y (w, r(1 + τ K )) = p y (14) w = p x F xl (15) w = p y F yl (16) r(1 + τ K + τ KX ) = p x (1 τ X )F xk (17) r = p y F yk (18) K = K x + K y (19) L = L x + L y (20) Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 84 / 104

86 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 (Econ 523) Capital Taxes Lecture 3 85 / 104

87 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 (Econ 523) Capital Taxes Lecture 3 86 / 104

88 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 (Econ 523) Capital Taxes Lecture 3 87 / 104

89 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 (Econ 523) Capital Taxes Lecture 3 88 / 104

90 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 (Econ 523) Capital Taxes Lecture 3 89 / 104

91 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 (Econ 523) Capital Taxes Lecture 3 90 / 104

92 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 (Econ 523) Capital Taxes Lecture 3 91 / 104

93 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 (Econ 523) Capital Taxes Lecture 3 92 / 104

94 Appendix: Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 93 / 104

95 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 (Econ 523) Capital Taxes Lecture 3 94 / 104

96 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 (Econ 523) Capital Taxes Lecture 3 95 / 104

97 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 (Econ 523) Capital Taxes Lecture 3 96 / 104

98 Increase the cost of new construction Suppose the economy is in steady state and suddenly the costs of new construction increase. Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 97 / 104

99 Increase the cost of new construction Suppose the economy is in steady state and suddenly the costs of new construction increase. The main impact is on the asset market. Supply curve shifts left. Lower supply decreases new construction The lower stock of capital causes rents to rise Rising rents and the prospects of future higher rents (remember long-run rents are higher) cause the price to rise As prices rise over time, construction rebounds Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 97 / 104

100 Increase the cost of new construction Back Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 98 / 104

101 Decline in the interest rate Suppose the economy is in steady state and suddenly the interest rate r declines. Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 99 / 104

102 Decline in the interest rate Suppose the economy is in steady state and suddenly the interest rate r declines. Back Lower r future streams of payments are more valuable, so P will jump up The jump in asset prices causes investment to jump up More investment increases the capital stock Higher capital stocks start to decrease rental rates Higher rental rates decrease asset prices We will end at a steady state with higher K and lower R Graduate Public Finance (Econ 523) Capital Taxes Lecture 3 99 / 104

103 Decline in the interest rate Back Graduate Public Finance (Econ 523) Capital Taxes Lecture / 104

104 US House Prices and Residential Investment Graduate Public Finance (Econ 523) Capital Taxes Lecture / 104

105 US House Prices and Residential Investment Was this an irrational bubble? Graduate Public Finance (Econ 523) Capital Taxes Lecture / 104

106 US House Prices and Residential Investment What did we see in the housing boom in the 2000s? Graduate Public Finance (Econ 523) Capital Taxes Lecture / 104

107 US House Prices and Residential Investment What did we see in the housing boom in the 2000s? Low interest rates and high capital prices Therefore, housing services are cheap to use and investment will increase as S(P) increases with P Suppose people expected higher future demand More downward pressure on rental prices, higher capital prices, more construction Graduate Public Finance (Econ 523) Capital Taxes Lecture / 104

108 US House Prices and Residential Investment What did we see in the housing bust in the late 2000s? Graduate Public Finance (Econ 523) Capital Taxes Lecture / 104

109 US House Prices and Residential Investment What did we see in the housing bust in the late 2000s? We ve build up a large housing stock Suppose now the anticipated increase in demand never comes Falling house prices Big decline in investment Also makes consuming housing services more expensive Things will adjust as housing stock goes back to the steady state Was this an irrational bubble? Observing a crash in the housing market does not tell us whether (1) there were rational expectations about future demand (coupled with low interest rates) or (2) a bubble (that could not be justified by expectations). Back Graduate Public Finance (Econ 523) Capital Taxes Lecture / 104