On the Optimal Taxation of Capital Income

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110 JONES, MANUELLI, AND ROSSI If s.t. c t+k t+1+g t u^ c(0) { k 0 F k(k 0,1)k 0+: max : ; t u(ct) t=0 k 0>0 given t=0 F(k t,1)+(1&$)k t ; t u^ c( t ) { n F n( t )< : t=0 ; t u^ c(t)g t, (8) then the solution to the planner's problem is such that *>0. Before we present the proof of the proposition it is useful to interpret (8). The left hand side is revenue from taxation of capital at time zero plus the present value of labor income taxation. Since both k0 and labor are in fixed supply, taxation of these two factors is equivalent to lump sum taxation. The right hand side is the present value of government spending. Thus, (8) says that, at the prices implied by the first best allocation, revenue from lump sum taxes falls short of expenditures. This is of course an arbitrary assumption. It is equivalent to ruling out lump sum taxation and without it the problem becomes uninteresting as no distortionary taxes are used. It seems a reasonable assumption in real world applications. Proof. Consider the following less restrictive version of the Ramsey problem, s.t. : t=0 max : s.t. c t+g t+k t+1 t=0 ; t u(c t) F(k t,1)+(1&$)k t ; t u c(t)[c t&(1&{ n ) F n( t )]&u c(0)[F k(0)(1&{ k 0)+1&$]k 0 and let be the Lagrange multiplier associated with the second constraint. Suppose first that *=0. In this case the solution is ``first best'' and given by (c^, k ). Note that, : t=0 ; t u^ c( t )(c^ t&(1&{ n ) F n( t )) = : t=0 ; t u^ c ( t )[(F k(t)+1&$)k t+{ n F n( t )&k t+1&g t] =u^ c(0)(F k(0)+1&$)k 0+ : t=0 ; t u^ c( t ) { n F n( t )& : t=0 ; t u^ c( t ) g t , 0,
where we used the property that u^ c( t )=u^ c( t +1) ;[F k(t+1)+1&$] for t 0. Using this equality in the second constraint of the less restrictive problem we get or u^ c(0)(F k(0)+1&$)k 0+ : & : t =0 t=0 ; t u^ c( t ) { n F n( t ) ; t u^ c( t ) g t & u^ c(0)[(1&{ k 0 ) F k(0)+1&$]k 0 u^ c(0) { k 0 F k(0)k 0+ : CAPITAL INCOME TAXATION t=0 ; t u^ c( t ) { n F n( t ) : t=0 This contradicts (8) and hence it shows that *>0. K ; t u^ c( t ) g t . An alternative, more intuitive argument that shows that *>0 can be constructed as follows: Denote by V* the maximized value of the planner's objective function, and note that the change in V* due to an increase in { k 0 is given by *u c*(0) F k*(0)k 0. Since increases in { k 0 are equivalent to increases in lump sum taxes and since the higher the level of lump sum taxes the higher the value of utility, it follows that * must be positive. Our findings for the case of factor income taxation are reminiscent of the results of Diamond and Mirrlees [10] and Stiglitz and Dasgupta [25] that also show that the existence of pure profits affects the optimal commodity tax schedule. (b) Two Types of Labor with Equal Taxes The previous example discussed a particular form of violation of the assumptions that give rise to the Chamley Judd result: in terms of the pseudo-utility function W of the planner's problem, the term Wk is not zero. Here, we consider the alternative possibility that the constraints , i are binding in the steady state. A simple example of such a case is when the planner cannot distinguish between income from two types of labor. Because of this, we will require that the tax rate on the two types of labor be equal. This is a convenient way of modeling a more general type of constraint: restrictions on tax rates. The example we will consider features one household that sells two types of labor to the market, n1 and n2. This is in the spirit of Stiglitz [27] where an example is analyzed with two types of consumers each with a different type of labor. He analyzes a planner's problem with a weighted average of the two consumer's utility functions and shows that the limiting tax rate on capital income is not zero. He ascribes this finding to redistributional motives that arise due to the 0 111
Page 1 and 2: journal of economic theory 73, 93 1
Page 3 and 4: CAPITAL INCOME TAXATION of feasible
Page 5 and 6: We consider the simplest infinitely
Page 7 and 8: Using (1.e) and rearranging we obta
Page 9 and 10: The first constraint is similar to
Page 11 and 12: From the planner's problem, Hence,
Page 13 and 14: CAPITAL INCOME TAXATION choosing th
Page 15 and 16: Letting + t be the Lagrange multipl
Page 17: function of the after-tax real wage
Page 21 and 22: non-zero result we have to assume t
Page 23 and 24: CAPITAL INCOME TAXATION 4. EXTENSIO
Page 25: CAPITAL INCOME TAXATION 13. L. E. J

On the Optimal Taxation of Capital Income

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