or property, and government imposes a tax on the stock of housing. Including property ... leisure, home work, market work and education. Allocating time to ...
Optimal Taxation in an Endogenous Growth Model with Government Supplied Educational Capital Paul Gomme Simon Fraser University and Research Center on Employment and Economic Fluctuations First Draft: December 1994 This Draft: May 1995
Abstract: Government expenditures are large when measured as a proportion of government spending or relative to GNP. This fact is incorporated into an endogenous growth model in which education produces new human capital. Capital in the education sector is provided by government and nanced by distortionary taxes. To incorporate property taxes, an important source of revenues for schooling, household production is incorporated into the model with household capital representing the property tax base. The model is calibrated to match key rst moment properties of the post-war U.S. economy. Di erent mixes of labor income, capital income and property taxes are analyzed with regard to their consequences for the welfare of the representative household. In general, taxes have two opposing e ects: higher taxes tend to retard growth and reduce welfare; however, when used to nance educational expenditures, taxes promote human capital accumulation and so growth which is welfare-enhancing. Keywords: endogenous growth, education, scal policy, calibration The nancial assistance of SSHRC is gratefully acknowledged.
1
1. Introduction Government expenditures on education are large. Over the period 1958{89, education expenses were 11.7% of total government expenses, and 3.9% of GNP.1 This spending is nanced, primarily, by distorting taxes on labor income, capital income, and property. While higher taxes, by themselves, tend to reduce both growth and welfare, using tax proceeds to subsidize education, and so human capital formation, increases growth and hence welfare. The net e ect is ambiguous. In general, there will be an optimal growth rate which is less than the maximum feasible growth rate. There are three key features of the model: (1) an education sector in which human capital is produced, (2) government levies distorting taxes to nance physical capital accumulation in the education sector, and (3) household production which is included to provide a rationale for property (household capital) which government may tax to nance part of its schooling expenditures. The education sector is broadly similar to the human capital accumulation models of Lucas (1988) and Arrow (1962?) in that private agents must allocate time to produce new units of human capital. Like King and Rebelo (1990) and Gomme (1993), capital is a necessary input to human capital production. However, unlike these earlier papers, government supplies the capital input used in the education sector. Government nances this schooling expense by imposing distorting taxes on labor income, capital income and property. Household production (Benhabib, Rogerson and Wright (1991) and Greenwood and Hercowitz (1991)) is introduced to provide a rationale for private agents to hold property. In home production models, agents consume both a market good (for example, restaurant meals) and a home good (for example, meals made at home). The home good requires both a time input and household capital. Here, household capital is interpreted as housing or property, and government imposes a tax on the stock of housing. Including property taxes is important because they are an important source of education funding; Fernandez 1 Government expenses are the sum of federal (Citibase variable GGFEX) and state and local expenses (GGSEX). Education expenses are the sum of government purchases of educational structures (GAGBE) and compensation of employees for state and local education (GAPGLE).
2 and Rogerson (1993) report that 45% of U.S. primary and secondary education is nanced through property taxes. The model is broadly similar to Shell (1967) in that distorting taxes are used to foster accumulation of knowledge. There are, however, several key di erences. First, Shell assumes that knowledge is a purely public good and so its competitively determined price is zero. This motivates government funding of knowledge accumulation. Here, human capital is bundled with an agent's time which implies that the agent can capture the returns to accumulating knowledge. The model does not provide a strong reason for government to intervene in the education sector; the only justi cation is the empirical reality that governments do intervene in this market. Second, Shell considers only a single, income tax while in the model analyzed below, labor income, capital income and property taxes are all instruments of government policy. A more distantly related literature includes the overlapping generations models of Fernandez and Rogerson (1993) and Glomm and Ravikumar (1993). Fernandez and Rogerson model property taxes as the sole source of revenue for schooling, and focus on the disparity of educational expenses per student across communities. Glomm and Ravikumar, on the other hand, allow for only income taxes to fund school expenditures. Their goal is to match observations on per capita income, years of schooling, public expenditures on education and student-teacher ratios. In both these papers, the quality of an o spring's education enters directly in the utility function of the parent, and neither paper considers growth. The model is described in Section 2. There are three groups of agents in the economy: households, rms and government. The representative household allocates its time between leisure, home work, market work and education. Allocating time to education augments the consumer's stock of human capital. The household also operates its home production function and rents capital to rms. Firms use labor and physical capital to produce a composite market good which can be used as either consumption or investment in physical capital. Finally, government levies taxes on labor and capital income and on property. Government spending is divided between exogenous spending and nancing the education sector. The model is suciently complex that it must be solved computationally. Thus, the model is calibrated in Section 3. Following in the real business cycle tradition, the model
3 is parameterized so as to match key long run observations for the U.S. economy, like the growth rate, capital{output ratio, and so on. A series of tax experiments are conducted in Section 4. To start, each of the labor income, capital income and property tax rates are varied, holding the other tax rates at their benchmark values. The nal experiment looks for an optimal mix of taxes. A common feature of these experiments is that the representative agent's welfare is maximized at a set of taxes which do not maximize the economy's growth rate. Intuitively, there is a tradeo between the distortionary aspects of higher taxes and the welfare-enhancing e ects of higher growth. Finally, Section 5 concludes.
2. The Economic Environment 2.1. Preferences
The consumer side of the model builds on the home production work of Benhabib, Rogerson and Wright (1991) and Greenwood and Hercowitz (1991). The representative agent has preferences over time dated composite consumption, Ct, and leisure, `t, given by an additively time separable utility function, 1 X U0 = tu(Ct; `t) 0 < < 1 t=0
(1)
cHt = H (kHt ; htnHt)
(2)
The function u is assumed to be homogeneous of degree 1 ? in Ct. Consumption, Ct, is a composite of the consumption of market goods, cMt, and of goods produced in the home, cHt: Ct = C (cMt; cHt) where C is homogeneous of degree 1. Home production is governed by
where kHt is capital used in the household sector, nHt is time allocated to home production, and ht is human capital. Throughout, time and human capital are bundled together so that an allocation of time implies an allocation of human capital to an activity. The function H is constant-returns-to-scale.
4 The household also allocates time to producing new human capital, nEt. The evolution of a household's human capital is:
ht+1 = G(kEt ; htnEt) + (1 ? h )ht
(3)
where kEt is government supplied physical capital used in producing new human capital (education) and h is the depreciation rate of human capital. G is assumed to be homogeneous of degree 1. The individual also faces the following laws of motion on physical capital:
kMt+1 = (1 ? M )kMt + iMt
(4)
kHt+1 = (1 ? H )kHt + iHt
(5)
where iMt (iHt ) is investment in market (household) capital and M (H ) is the depreciation rate of market (household) capital. The budget constraint of the representative household is
cMt + iMt + iHt = (1 ? n)wthtnMt + (1 ? k )rt kMt + M k kMt ? H kHt
(6)
Above, n, k and H are the tax rates on labor income, capital income and household capital, respectively. The rst term on the right hand side of (6) is after-tax labor income, where market time, nMt, is combined with human capital to provide e ective units of labor. Capital income is given by the second term on the right hand side while the third term represents a capital consumption allowance. Finally, the fourth term on the right hand side is the tax payable on household capital. This term is intended to capture the e ect of property taxes collected in the U.S. Finally, the household's time allocation is normalized to unity. Its time constraint is
`t + nMt + nHt + nEt 1
(7)
5 2.2. Firms
The typical rm faces a sequence of static, one period problems: choose capital, kMt, and labor input, htnMt, to maximize period pro ts:
F (kMt; htnMt) ? rtkMt ? wthtnMt
(8)
where F is constant-returns-to-scale. The rm's eciency conditions are: and
rt = F1(kMt ; htnMt)
(9)
wt = F2(kMt; htnMt)
(10)
2.3. Government
To start, government must satisfy its budget constraint,
Gt = nhtnMtwt + k kMtrt ? M k kMt + H kHt (11) Total government expenditures, Gt, are the sum of exogenous spending, Gxt, and investment in capital used in the education sector, iEt. The law of motion for educational capital is
kEt+1 = (1 ? E )kEt + iEt
(12)
2.4. Competitive Equilibrium
De nition: A competitive equilibrium consists of sequences, fcMt; cHt; Ct; `t; nMt; nHt; nEt; ht+1; kMt+1; kHt+1; kEt+1 ; rt; wt; Gxt; iEtg1t=0 such that: 1. fcMt; cHt; Ct; `t; nMt ; nHt; nEt ; ht+1; kMt+1 ; kHt+1g solve the consumer's problem of maximizing (1) subject to (2){(7) taking as given frt; wt; Gxt; kEt+1 ; iEtg; 2. fkMt; htnMtg solve the rm's problem with factor prices, rt and wt, satisfying (9) and (10); 3. fkEt+1; iEt ; Gxtg satisfy the government's constraints, (11) and (12); and 4. feasibility or goods market clearing:
cMt + iMt + iHt + iEt + Gxt = Yt
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3. Calibration In this section, speci c functional forms for the utility, production and other functions are speci ed. The parameters which are thus introduced are chosen to match key long run facts for the U.S. economy. To start, the period utility function is given by (
[Ct! `1t ?! ]1? ; 1?
0 < < 1, > 1 : ! ln Ct + (1 ? !) ln `t; = 1 The home production function is speci ed as:
u(Ct; `t) =
(h n )1? cHt = H (kHt ; htnHt) = AhkHt t Ht
while aggregated consumption is 1? : Ct = C (cMt; cHt) = cMtcHt
Market production is given by the Cobb-Douglas function, (N Ht )1? Y = F (kM ; hnM ) = Af KMt Mt
while the human capital production is given by
G(kE ; hnE ) = Ag (Ktg ) [NEtHt]1? : Parameter values are chosen to match key long run averages observed in the U.S. economy. Greenwood, Rogerson and Wright (1995) give capital:output ratios for market and household capital as: km = 4 (13) Y
kh = 5: Y
(14)
kE = 0:4: Y
(15)
Capital Estimates for the United States indicates that capital in the education sector is about 1=10 that of the market sector; thus,
7 Greenwood, Rogerson and Wright also cite evidence that approximately 11.8% of output is allocated to investment in market capital while 13.5% is allocated to household capital. Thus, along the balanced growth path,
K km = 0:118 Y H kh = 0:135 Y
(16) (17)
Greenwood, Rogerson and Wright summarize survey evidence on household time use. Approximately 20% of time is allocated to market activity and 33% to household time. The survey they cite ignores time of the young, which is when most people accumulate human capital. As a rough guess, individuals probably spend 5% of their time in formal education, or about 1=4 that allocated to market activity. This evidence gives:
nM = 0:20
(18)
nH = 0:33
(19)
nE = 0:05:
(20)
Leisure can, then, be computed as ` = 1 ? nM ? nH ? nE . The discount factor, , is assigned a value which yields a 1% real return per quarter, or 4% per annum. Mehra and Prescott (1985) surveyed the micro evidence on the coecient of relative risk aversion and found that most estimates are between 1 and 2. Here a value of 1 1=2 is used: = 1 1=2. The tax rates on capital, k , and labor income, n, are set to the values used by Greenwood, Rogerson and Wright (1995): k = 0:70 and n = 0:25. The tax rate on housing (household capital), h, is assigned a value of 0.001. The production function constants Af and Ah have little e ect on the calibration and are assigned values of 1.0. The depreciation rate of human capital, h, is set to 0. A nal piece of evidence is that the quarterly per capita real growth rate of U.S. output has been 0.35% since the Korean War. There are, now, enough restrictions to uniquely determine all the parameters in the model. Their values are given in Table 1.
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4. Taxation Experiments The presence of taxes distorts the competitive equilibrium away from the Pareto optimum. Higher taxes, used to nance the education sector, have two opposing e ects on individual welfare: 1. Higher taxes impose a greater distortion on individual decisions, and thus have a negative e ect on welfare. 2. Higher growth rates, resulting from greater human capital accumulation, makes the representative agent better o . The net e ect is, of course, ambiguous. Four sets of tax experiments are conducted. The rst three vary k , n and h individually with changes in government revenue impacting on its nancing of education. The last experiment varies all taxes in order to see the optimal mix of taxes.2 4.1. Labor Tax Experiment
The results of this experiment are summarized in Fig. 1 which plots the response of di erent variables to labor taxes in the range of 20% to 90%. A theme which is repeated through the tax experiments is that the representative agent does not wish to maximize the economy's growth. Here, for example, the growth rate would be maximized by taxing labor income at a rate of 80% while lifetime utility is maximized at a rate of around 50%. The associated welfare bene t of increasing the tax on labor income from 25% to 50% is just over 16% of output. As stated earlier, an increase in taxes has two e ects. Higher taxes induce a greater distortion and remove purchasing power from the individual and hence reduces welfare. O setting this is a growth rate e ect induced by the use of the additional tax revenues in nancing education capital. Clearly, the growth rate e ect dominates only up to a tax rate of 50%. The explanation for why there is a maximum growth rate can be obtained by looking at government tax revenue. Tax collections are maximized at a rate of 80%. Consequently, 2 Since attention is focused on the balanced growth path, tax rates are implicitly time consistent. Thus, the problems of Chamley (19xx) are avoided.
9 educational capital is also maximized at a tax rate of 80%, and given the complementarity between capital and time in the education sector, so is time allocated to education. Consider, now, the e ect of raising the labor tax rate to its welfare maximizing level of 50%. Relative to the benchmark economy, educational capital rises by a factor of 4 while time allocated to education rises from 5% to around 7.4% of the total. If the average length of schooling in the U.S. were 12 years, this would correspond to an increase of 6 years. Finally, the annualized per capita growth rate would rise from 1.4% to 3.3%. 4.2. Capital Tax Experiment
There is a modest welfare bene t of 0.7% of output associated with lowering the capital tax from its benchmark value of 70% to 64%. This results in a slight decline in the per capita growth rate from 1.4% per annum to 1.3%. As before, welfare is maximized at less than the maximum growth rate. The remainder of the results are qualitatively similar to those above. 4.3. Housing Tax Experiment
This is, perhaps, the most natural experiment in that a large proportion of property taxes are used to nance education. Lifetime would be increased by raising the tax rate on household capital (housing) to 5% from its benchmark value of 0.65%. The measured welfare bene t of such a policy would be almost 14% of output, and the per capita growth rate would rise to 3% per year from its benchmark value of 1.4%. In this case, a tax rate which yielded a maximum growth rate could not be found, even for tax rates in excess of 100%. 4.4. Optimal Tax Mix
The experiment here is to simultaneously vary the tax rates, K , N and H , so as to maximize lifetime utility. The results are summarized in Table 2 which also includes values for the benchmark model. The rst observation is that the optimal tax mix calls for a near doubling of the labor tax rate (from 25% to 48%), a substantial increase in the housing tax (from 0.65% to 4.2%) and over a 10 fold decline in the tax rate on capital income (from 70% to 4.7%). These
10 results are similar in nature to those typically found in the optimal tax literature: capital should be taxed at a very low rate while labor should be taxed at a relatively much higher rate. Equally dramatic are the implications for the relative sizes of the capital stocks. In the benchmark economy, household capital is 56% of the total capital stock; with an optimal tax mix, this falls to 23%. The share of education capital rises sharply from 4.5% to 17% while that of market capital rises form 40% to 59%. Taking these results at face value, too much capital is currently tied up in housing, and the education sector is underfunded. With regards to time allocations, the optimal tax mix leads to small declines in time allocated to home production and leisure, and a much larger fall in market time. Education time rises over 55% (from 5% of the total to 7.8%). Market consumption as a share of output falls from 38% to 30%. Relative to market output, home production falls 20% with a similar decline in aggregated consumption. O setting the falls in consumption and leisure associated with adopting an optimal tax mix is an increase in the annual growth rate form 1.4% to 3.8%. The measured welfare gain is just over 46% of output. This gure is in the same ball park as the welfare gains associated with lower tax rates computed by King and Rebelo (1990).
5. Conclusions In the model analyzed in this paper, government supplies capital to the education sector which produces human capital, the engine of growth in this model. Government has access to labor income, capital income and property taxes. Household production was introduced to the model to provide a reason for households to hold property (household capital) since, in the U.S., property taxes are an important source of education funding. Qualitatively, the results suggest that labor income and property tax rates are too low while the tax rate on capital income is too high. Quantitatively the optimal tax mix requires a doubling of the labor tax rate, nearly a 10 fold increase in property tax rates, and over a 10 fold decrease in the tax on capital income. These results are similar to those seen in the optimal taxation literature. In light of these results, it is not surprising that the model model predicts that the optimal tax mix requires substantially less home capital, and increased funding of the education sector.
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6. References Benhabib, Jess, Richard Rogerson and Randall Wright, 1991, Homework in Macroeconomics: Household Production and Aggregate Fluctuations, The Journal of Political Economy 99, 1166. Fernandez, R. and R. Rogerson, 1993, Keeping People Out: Income Distribution, Zoning and the Quality of Public Education, National Bureau of Economic Research Working Paper 4333. Glomm, Gerhard and B. Ravikumar, 1993, Endogenous Expenditures on Public Schools and Persistent Growth, Minneapolis: Institute for Empirical Macroeconomics Discussion Paper 85. Gomme, Paul, 1993, Money and Growth Revisited: Measuring the Costs of In ation in an Endogenous Growth Model, Journal of Monetary Economics 32, 51{77. Greenwood, Jeremy and Zvi Hercowitz, 1991, The Allocation of Capital and Time over the Business Cycle, The Journal of Political Economy 99, 1188. Greenwood, Jeremy, Richard Rogerson and Randall Wright, 1995, Household Production in Real Business Cycle Theory, in: Frontiers of Business Cycle Research, ed., Thomas F. Cooley (Princeton, NJ: Princeton University Press), 157{174. King, Robert G. and Sergio Rebelo, 1990, Public Policy and Economic Growth: Developing Neoclassical Implications, The Journal of Political Economy 98, S126. Lucas, Robert E., Jr., 1988, On the Mechanics of Economic Development, Journal of Monetary Economic 22, 3{42. Mehra, Rajnish and Edward C. Prescott, 1985, The Equity Premium: A Puzzle, Journal of Monetary Economics 15, 145{161. Shell, Karl, 1967, A Model of Inventive Activity and Capital Accumulation, in: Essays on the Theory of Optimal Economic Growth, ed., Karl Shell (Cambridge, Massachusetts: The MIT Press), 67{85.
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Appendix A. Dynamic Programming 1. Households The typical household's problem can be cast in terms of the following Bellman equation:
V (kMt; kHt; ht; ) max fu(Ct; `t) + V (kMt+1; kHt+1; ht+1; )g
(A:1)
Ct = C (cMt; cHt)
(A:2)
cHt = H (kHt ; htnHt)
(A:3)
subject to
ht+1 = G(kEt ; htnEt) + (1 ? h )ht cMt + kMt+1 + kHt+1 = (1 ? n)wthtnMt + (1 ? k )rt kMt + M k kMt ? H kHt + (1 ? M )kMt + (1 ? H )kHt `t + nMt + nHt + nEt = 1
(A:4) (A:5) (A:6)
Alternatively, write the Bellman equation as:
V (kMt ; kHt; ht; ) max u(Ct; `t) + V (kMt+1 ; kHt+1; ht+1; ) + 1t[C (cMt; cHt) ? Ct] + 2t[H (kHt ; htnHt) ? cHt] + 3t[G(kEt ; htnEt) + (1 ? h )ht ? ht+1] + 4t[(1 ? n)wthtnMt + (1 ? k )rt kMt + M k kMt ? cMt ? kMt+1 ? kHt+1] + 5t[1 ? `t ? nMt ? nHt ? nEt] The rst-order conditions are:
(A:7)
u1(Ct; `t) = 1t
(A:8)
u2(Ct; `t) = 5t
(A:9)
1tC1(cMt; cHt) = 4t
(A:10)
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1tC2(cMt; cHt) = 2t
(A:11)
4t(1 ? n)wtht = 5t
(A:12)
2thtH2 (kHt; htnHt) = 5t
(A:13)
3thtG2(kEt ; htnEt) = 5t
(A:14)
V1(kMt+1; kHt+1; ht+1; ) = 4t
(A:15)
V1(kMt; kHt; ht; ) = 4t[(1 ? k )rt + M k + 1 ? M ]
(A:16)
V2(kMt+1; kHt+1; ht+1; ) = 4t
(A:17)
V2(kMt ; kHt; ht; ) = 2tH1 (kHt; htnHt) + 4t (1 ? H )
(A:18)
V3(kMt+1; kHt+1; ht+1; ) = 3t V3(kMt ; kHt; ht; ) = 2tnHtH2 (kHt; htnHt) + 3t [nEtG2(kEt ; htnEt ) + 1 ? h] + 4t (1 ? n)wtnMt
(A:19)
u1(Ct; `t)C2(cMt; cHt)htH2(kHt; htnHt) = u2(Ct; `t)
(A:21)
u1(Ct; `t)C1(cMt; cHt)(1 ? n)wtht = u2(Ct; `t)
(A:22)
(A:20)
The associated eciency conditions are:
u1(Ct; `t)C1 (cMt; cHt) = u1(Ct+1; `t+1)C1(cMt+1 ; cHt+1)[(1 ? k )rt+1 + M k + 1 ? M ] (A:23) u1(Ct; `t)C1(cMt; cHt) = u1(Ct+1; `t+1)[C2(cMt+1; cHt+1)H1 (kHt+1; ht+1nHt+1) (A:24) + C1(cMt+1; cHt+1)(1 ? H )] u2(Ct; `t) = u (C ; ` )C (c ; c )n H (k ; h n ) 1 t+1 t+1 2 Mt+1 Ht+1 Ht+1 2 Ht+1 t+1 Ht+1 htG2 (kEt; htnEt) + h Gu(2k(Ct+1;; h`t+1)n ) [nEt+1G2(kEt+1 ; ht+1nEt+1) + 1 ? h ] t+1 2 Et+1 t+1 Et+1
+ u1(Ct+1; `t+1)C1(cMt+1; cHt+1)(1 ? n)wt+1nMt+1
(A:25) Equations (A.21) and (A.22) govern the allocation of time to education and the market, respectively. In (A.21), u1(t)C2 (t) is the marginal utility of home consumption while
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htH2(t) is the marginal product of labor in the education sector. The product of these
must just equal the marginal utility of leisure. (A.22) has a similar interpretation except with respect to market time. The accumulation of market and household capital is determined by (A.23) and (A.24). In (A.23), the marginal rate of substitution between market consumption at t and t + 1, u1(t)C1(t)=[ u1(t + 1)C1(t + 1)], must equal the return earned by market capital which is given by the term in square brackets in (A.23). Equation (A.24) has a similar interpretation except that it applies to household capital. Finally (A.25) governs human capital accumulation. The three terms in the brace brackets represent the return to accumulating an extra unit of human capital. The sources of these returns are determined by the allocation of time at t +1 between home production, education and market activity. As with physical capital, this return must equal a marginal rate of substitution, in this case between leisure at dates t and t + 1.
2. Firms The typical rm solves max
kMt; htnMt F (kMt; htnMt) ? rt kMt ? wthtnMt
(A:26)
The associated eciency conditions are:
rt = F1(kMt ; htnMt)
(A:27)
wt = F2(kMt; htnMt)
(A:28)
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3. Balanced Growth Transformation In the economy described above, growth results from human capital accumulation. This would occur absent government investment in education capital; see Lucas (1988). Asymptotically, all variables either grow at some common rate (for example, consumption, investment, the various capital stocks) or not at all (for example, time allocations and marginal products). The task at hand is to perform a balanced growth transformation which renders all variables stationary. To start, de ne x^t xt=ht|that is, variables with a \hat" have been transformed by dividing through by human capital. Recall that the functions C , F , G and H are all homogeneous of degree 1 while u is homogeneous of degree 1 ? in Ct. Using (A.27) and (A.28) to substitute out for rt and wt, and replacing (A.5) with the feasibility constraint yields the following set of equations:
C^t = C (^cMt; c^Ht)
(A:29)
c^Ht = H (k^Ht; nHt)
(A:30)
ht+1 = G(k^ ; h n ) + 1 ? Et t Et h ht h t +1 ^ ^ ^ c^Mt + h kMt+1 + kHt+1 + kEt+1 + G^ xt t ^ = F (kMt; nMt) + (1 ? M )k^Mt + (1 ? H )k^Ht + (1 ? E )k^Et `t + nMt + nHt + nEt = 1
(A:31)
u1(C^t; `t)C2(^cMt; c^Ht)H2 (k^Ht; nHt) = u2(C^t; `t)
(A:34)
(A:32) (A:33)
u1(C^t; `t)C1 (^cMt; c^Ht)(1 ? n)F2 (k^Mt; nMt ) = u2(C^t; `t) (A:35) ? h t +1 ^ u1(C^t+1; `t+1)C1(^cMt+1; c^Ht+1) u1(Ct; `t)C1(^cMt; c^Ht) = h (A:36) t [(1 ? k )F1 (k^Mt+1; nMt+1) + M k + 1 ? M ] ? h t +1 u1(C^t; `t)C1 (^cMt; c^Ht) = h u1(C^t+1; `t+1)[C2(^cMt+1; c^Ht+1)H1 (k^Ht+1; nHt+1) t + C1(^cMt+1; c^Ht+1)(1 ? H )] (A:37)
16
u2(C^t; `t) = ht+1 ? u (C^ ; ` )C (^c ; c^ )n H (k^ ; n ) 1 t+1 t+1 2 Mt+1 Ht+1 Ht+1 2 Ht+1 Ht+1 G2(kEt ; htnEt) ht ^ ; `t+1) [nEt+1G2 (kEt+1; ht+1nEt+1) + 1 ? h ] + G (ku2(Ct+1 2 Et+1 ; ht+1 nEt+1 ) ^ ^ + u1(Ct+1; `t+1)C1(^cMt+1; c^Ht+1)(1 ? n)F2 (kMt+1; nMt+1)nMt+1 (A:38) G^ xt + ^iEt = G^ t = nnMtF2(k^Mt; nMt) + k k^MtF1(k^Mt; nMt) ? k k^Mt + H k^Ht (A:39) Notice that since Gxt is speci ed to be a xed fraction of output, Gxt will grow at the same rate as output and so G^ xt will be stationary.
Parameter
!
M H E h Af Ah Ag K N H
Table 1: Model Parameters
Interpretation capital share, home production capital share, education capital share, market production utility function weight coecient of relative risk aversion depreciation rate of market capital depreciation rate of household capital depreciation rate of education capital depreciation rate of human capital market production function constant home production function constant education production function constant tax rate on capital tau rate on labor tax rate on housing discount factor weight in consumption aggregator
Value 0.218 0.631 0.298 0.575 2.500 0.033 0.027 0.081 0.000 1.000 1.000 0.037 0.700 0.250 0.006 0.992 0.256
Table 2: Optimal Tax Mix K N H
Market good consumption Home good consumption Market capital Home Capital Education Capital Market time Home time Education time Leisure time Annual growth rate (%) Aggregated Consumption Market output
Benchmark 0.700 0.250 0.006 0.132 0.473 1.240 1.722 0.138 0.200 0.330 0.050 0.420 1.407 0.341 0.344
Optimal 0.047 0.480 0.042 0.104 0.385 1.743 0.690 0.513 0.179 0.327 0.078 0.416 3.826 0.275 0.352
Figure 1a: Labor Tax Experiments Growth Rate
Tax Revenue
1.012
0.17
1.01
0.16 0.15
1.008
0.14
1.006
0.13
1.004
0.12
1.002
0.11
1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Lifetime Utility
Welfare Benefit 20 15 10 5 0 -5 -10 -15 -20 -25 -30 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-220 -240 -260 -280 -300 -320 -340 -360 -380 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Educational Capital 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Educational Time 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 1b: Labor Tax Experiments Market Consumption 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Market Capital 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Leisure 0.445 0.44 0.435 0.43 0.425 0.42 0.415 0.41 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Market Time 0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Household Capital
Household Time
3
0.35
2.5
0.345
2
0.34
1.5 1 0.5 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.335 0.33 0.325 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 2a: Capital Tax Experiments Growth Rate
Tax Revenue
1.0038
0.116
1.0036
0.115
1.0034 0.114
1.0032 1.003
0.113
1.0028
0.112
1.0026 0.111
1.0024 1.0022 0.5
0.6
0.7
0.8
0.9
0.11 0.5
Lifetime Utility -255 -260 -265 -270 -275 -280 -285 -290 -295 -300 -305 0.5
0.7
0.8
0.9
Welfare Benefit 5 0 -5 -10 -15 -20
0.6
0.7
0.8
0.9
-25 0.5
Educational Capital
0.6
0.7
0.8
0.9
Educational Time
0.15
0.052
0.14
0.05
0.13
0.048 0.046
0.12
0.044
0.11
0.042
0.1
0.04
0.09
0.038
0.08 0.5
0.6
0.6
0.7
0.8
0.9
0.036 0.5
0.6
0.7
0.8
0.9
Figure 2b: Capital Tax Experiments Market Consumption
Leisure
0.16
0.9
0.427 0.426 0.425 0.424 0.423 0.422 0.421 0.42 0.419 0.418 0.417 0.5
0.9
0.209 0.208 0.207 0.206 0.205 0.204 0.203 0.202 0.201 0.2 0.199 0.5
0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.5
0.6
0.7
0.8
Market Capital 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.5
0.6
0.7
0.8
0.7
0.8
0.9
Market Time
Household Capital 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.5
0.6
0.6
0.7
0.8
0.9
Household Time 0.336 0.335 0.334 0.333 0.332 0.331 0.33 0.329
0.6
0.7
0.8
0.9
0.328 0.5
0.6
0.7
0.8
0.9
Figure 3a: Housing Tax Experiments Growth Rate
Tax Revenue
1.011 1.01 1.009 1.008 1.007 1.006 1.005 1.004 1.003 1.002 1.001
0.16 0.155 0.15 0.145 0.14 0.135 0.13 0.125 0.12 0.115 0.11 0.105 0
0.1
0.2
0.3
0.4
0
Lifetime Utility
0.1
0.2
0.3
0.4
Welfare Benefit
-240
15
-245 10
-250 -255
5
-260 0
-265 -270
-5
-275 -280
-10 0
0.1
0.2
0.3
0.4
0
Educational Capital
0.1
0.2
0.3
0.4
Educational Time
0.6
0.09
0.5
0.08 0.07
0.4
0.06
0.3
0.05
0.2
0.04
0.1
0.03
0
0.02 0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
Figure 3b: Housing Tax Experiments Market Consumption
Leisure
0.145 0.14 0.135 0.13 0.125 0.12 0.115 0.11 0.105 0.1
0.435 0.43 0.425 0.42 0.415 0.41 0.405 0.4 0.395 0
0.1
0.2
0.3
0.4
0
Market Capital
0.1
0.2
0.3
0.4
Market Time
1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7
0.206 0.205 0.204 0.203 0.202 0.201 0.2 0.199 0
0.1
0.2
0.3
0.4
0
Household Capital
0.1
0.2
0.3
0.4
Household Time
2.5
0.34
2
0.335 0.33
1.5
0.325 1
0.32
0.5
0.315
0
0.31 0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
