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32 (l + n)(l+0)2+p Thus for the economy to be on a balanced growth path with the capital stock at the goldenrule level, b must equal the difference between the total amount the young wish to save when = , . and the amount of that saving that must take the form of capital, kcR. By issuing quantity  1<gr does not depend on the assumption of logarithmic utility. Without logarithmic utility, the saving of the current periods young depends on the rate of return as well as on aftertax labor income. But the rate of return is determined by the next periods capitallabor ratio, which is not affected by government purchases in that period. Of course, T and G are related. The governments expenditures per unit of effective labor in period f are G, for purchases and [1 + f(.k,)]b, to retire the existing debt. The governments receipts are T, from taxes and (1 + + g)b,i from issuing new debt. Since expenditures and receipts must be equal, Tt equals + [1 + f{k,)]bt  (1 + n)(l + g)b,+i. The government can also do this through a social security program. See Problem 2.16. Tax Versus Bond Finance The possibiUty of the government using debt as well as taxes to finance its purchases requires that we change equation (2.69). First, some of the saving of the young takes the form of bonds instead of capital; thus the lefthand side of (2.69) becomes kt+i + bt+i, where b is the stock of bonds per unit of effective labor. (Paralleling our timing convention with capital, bfn refers to bonds purchased in period t. Thus to increase bt+i by 1 unit, the government must issue (1 i n)(l i g) bonds per unit of effective labor in f.) Second, taxes and purchases need not be equal; thus Tt replaces Gt on the righthand side of (2.69).* Moving the bti term over to the righthand side, we therefore have )2   1   Equation (2.70) shows that taxes and bonds have different effects on capital accumulation. When the government cuts taxes and issues bonds, the taxes to repay those bonds are levied on future generations. Thus the individuals currently alive are better off, and they therefore increase their consumption. Thus a switch from tax to bond finance reduces the capital stock. Since bonds represent net wealth in this economy, the government can use them to provide individuals with a way other than holding capital to transfer resources between youth and old age. Because of this, the government can use bonds to prevent the economy from accumulating too much capital. Consider an economy where the balanced growth path in the absence of a government involves k* > kcR. If the capital stock in some period, period t, equals its goldenrule level, the labor income of the young is (1  a)kgK, and they save fraction 1/(2 i p) of this. Thus kf+i + bt+i equals (la)kgj, sflGR. (2.71)
This policy involves no costs to the government, and no taxes, on the balanced growth path. When = ko, l + f(k) {l + Kg) (see Problem 2.14). Thus the amount the government needs, per unit of effective labor, to pay off its existing debt in period tis{i+n){l+g)b,. But the amount of new debt it issues m t is per unit of period( +1 effective labor; this is (1 + f7)(l +g)bt+\ per unit of penodf effective labor. When b is constant, these two quantities are equal, and so the new debt issues are just enough to pay off the outstanding debt. Finally, if the economy begins with > koR, the government needs to lev> taxes to move to its goldenrule level. Specihcally, suppose > < . If the govemment levies lumpsum taxes per unit of effective labor of amount (1 )  (1  a)k in period 0, the saving of the young per unit of period1 effective labor is [1/(1 + n)(l + 0)l[l/(2 1 p)](l  ) = OorWith b = acR Kr, this ensures that h = k(,R. The government can use the revenue from the taxes and the initial bond issue to increase the consumption of the penodO old. of bonds, the government can thus cause the balancedgrowthpath value of to equal its goldenrule value. Problems 2.1. Consider N firms each with the constant returns to scale production function = F{K,AL), or (using tht "ntensive form) = ALfik). Assume f(•) > 0, /"(•) < 0. Assume that all firms can hire labor at wage wA and rent capital al cost r, and that all firms have the same value of A. (a) Consider the problem of a firm trying to produce Y units of output at minimum cost. Show that the costminimizing level of is uniquely defined and is independent of Y, and that all firms therefore choose the same value of k. (b) Show that the total output of the N costminimizing firms equals the output that a single firm with the same production function has if it uses all of the labor and capital used by the N firms. 2.2. The elasticity of substitution with constantrelativeriskaversion utility. Consider an individual who lives lor two periods and whose utility is given by equation (2.46). Let P] and P2 denote the prices of consumption in the two periods, and let W denote the value of the mdividuals lifetime income; thus the budget constramt is Pi Ci + 2 = W. (a) What are the individuals utihtymaximizing choices of Ci and C2 given Pi, P2, and ? ib) The elasticity of substitution between consumption in the two periods is [(P, / 2) /(Ci/G)][<9(C, /C2)/(Pi/ 2)]. or  /Q)/ln(P] / 2). Show that with the utihty function (2.46), the elasticity of substitution between Cj and C2 is 1/e. 2.3. Assume that the instantaneous utility function u{C) m equation (2.1) is InC. Consider the problem of a household maximizing (2.1) subject to (2.5). Find an expression for at each time as a function of initial wealth plus the present value of labor mcome, the path of r(t), and the parameters of the utility function.
2.4. Consider a iiouseiiold witii utility given by (2.1)(2.2). Assume that the real interest rate is constant, and let W denote the households initial wealth plus the present value of its lifetime labor income (the righthand side of [2.5]). Find the utilitymaximizing path of given r, W, and the parameters of the utility function. 2.5. The productivity slowdown and saving. Consider a RamseyCassKoopmans economy that is on its balanced growth path, and suppose there is a permanent fall ing. (a) How, if at all, does this affect the = 0 curve? (b) How, if at all, does this affect the = 0 curve? (c) What happens to at the time of the change? (d) Find an expression for the impact of a marginal change in g on the fraction of output that is saved on the balanced growth path. Can one tell whether this expression is positive or negative? (e) For the case where the production function is CobbDouglas, f{k) = k", rewrite your answer to part (d) in terms of p, n, g, , and a. (Hint: use the fact that fik*) = + .) 2.6. Describe how each of the following affect the = 0 and = 0 curves in Figure 2.5, and thus how they affect the balancedgrowthpath values of and k: ia) A rise in . (b) A downward shift of the production function. (c) A change in the rate of depreciation from the value of zero assumed in the text to some positive level. 2.7. Derive an expression analogous to (2.37) for the case of a positive depreciation rate. 2.8. Capital taxation in the RamseyCassKoopmans model. Consider a RamseyCassKoopmans economy that is on its balanced growth path. Suppose that at some time, which we will call time 0, the govemment switches to a policy of taxing investment income al rale . Thus the real interest rate that households face is now given by r{t) = (1  T)f{k(t)). Assume that the government returns the revenue it collects from this lax through lumpsum transfers. Finally, assume that this change in lax policy is unanticipated. (a) How does the tax affect the = 0 locus? The = 0 locus? ib) How does the economy respond to the adoption of the tax at time 0? What are the dynamics after time 0? (c) How do the values of and on the new balanced growth path compare with their values on the old balanced growth path? (d) (This is based on Barro, Mankiw, and SalaiMartin, 1995.) Suppose there are many economies like this one. Workers preferences are the same tn each country, but the tax rates on investment income may vary across countries. Assume that each country is on its balanced growth path.
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