each of the following changes affects the = 0 and = 0 lines and the position of the economy in ( . ) space at the moment of the change:

(a) An increase in n.

(b) An increase in fljf.

(c) An increase in .

3.5. Consider the economy described in Section 3.3, and assume p + < I and n > 0. Suppose the economy is initially on its balanced growth path, and that there is a permanent increase in s.

(a) How, if at all, does the change affect the = 0 and 0 loci? How, if at all, does it affect the location of the economy in [ . ) space at the time of the change?

(b) What are the dynamics of and after the increase in s? Sketch the path of log output per worker.

(c) Intuitively, how does the effect of the increase in s compare with its effect in the Solow model?

3.6. Consider the model of Section 3.3 with + = 1 and n = 0.

(a) Using (3.14) and (3.15), find the value that A/K must have for and to be equal.

(b) Using your result in part (a), find the growth rate of A and when gf; = -

(c) How does an increase in s affect the long-run growth rate of the economy?

id) What value of maximizes the long-run growth rate of the economy? Intuitively, why is this value not increasing in p, the importance of capital in the R&D sector?

3.7. Learning-by-doing. Suppose that output is given by equation (3.22), Y{t) = K(t)[A{t)Ut)], that L is constant and equal to 1, that (f) = sY{t), and that knowledge accumulation occurs as a side effect of goods production: A(r) = BY(t).

(a) Find expressions for (f) and (t) in terms of A(f), K{t), and the parameters.

(b) Sketch the = 0 and =0 lines in ( , ) space.

(c) Does the economy converge to a balanced growth path? If so, what are the growth rates of K,A, and Y on the balanced growth path?

(d) How does an increase in s affect long-run growth?

3.8. Suppose that output at firm / is given by , = lf,"Lj~" [If where K, and L, are the amounts of capital and labor used by the firm, and L are the aggregate amounts of capital and labor, and a > 0, > 0, and 0 < a + < 1. Assume that factors are paid their private marginal products; thus r = dY, / ,. Assume that the dynamics of and L are given by = sY and L = nL, and that K, /L, is the same for all firms.

ia) What is r as a function of K/L?

To see tliis, note that capital in the investment sector produces new capital at rate and changes in value relative to the consumption good at rate gp. (Because the return to capital is the same in the two sectors, the same must be true of capital in the consumption sector.)

(b) What is IL on the balanced growth path? What is r on the balanced growth path?

(c) "If an increase in domestic saving raises domestic investment, positive externalities from capital would mitigate the decline in the private marginal product of capital. Thus the combination of positive externalities from capital and moderate barriers to capital mobility may be the source of Feldstein and Horiokas findings about saving and investment described in Chapter 1." Does your analysis in parts (a) and (b) support this claim? Explain intuitively.

3.9. (This follows Rebelo, L991.) Assume that there are two factors of production, capital and land. Capital is used in both sectors, whereas land is used only in producing consumption goods. Specifically, the production functions are C(f) = Kc(tYR-" and (f) = (1), where Kc and are the amounts of capita] used in the two sectors (so Kc(t) + ( ) = K(t)) and R is the amount of land, and 0 < a < 1 and > 0. Factors are paid their marginal products, and capital can move freely between the two sectors. R is normalized to 1 for simplicity.

(a) Let ( ) denote the price of capital goods relative to consumption goods at time t. Use the fact that the eamings of capital in units of consumption goods in the two sectors must be equal to derive a condirion relating Pj? (t), Kcit), and the parameters a and B. If Kc is growing at rate guit), at what rate must be growing (or falling)? Let gp{t) denote this growth rate.

(b) The real interest rate in terms of consumption \sB+ gp(t). Thus, assuming that households have our standard utility function, (3.28), the growth rate of consumption must be (B + gp- p)/a = gc- Assume p < B.

(i) Use your results in part (a) to express gcit) in terms of gU) rather than gp(t).

(ii) Given the production function for consumption goods, at what rate must Kc be growing for to be growing at rate gc(t)7

(ih) Combine your answers to (i) and (ii) to solve for and gc(t) in terms of the underlying parameters.

(c) Suppose that investment income is taxed at rate , so that the real interest rate households face is (1 - )( -i- gp). How, if at all, does affect the equilibrium growth rate of consumption?

3.10. (This follows Krugman, 1979; see also Grossman and Helpman, 1991b.) Suppose the world consists of two regions, the "North" and the "South." Output and capital accumulation in region i (/ = N,S) are given by y,(f) = K,{t)"[A,{t){l - «1,)!,] k,{t) = s,Y,{t). New technologies are developed in the North. Specifically, Ajv(f) = BaLLNANit). Improvements in Southern technology, on the other hand, are made by learning from Northern technology: Asit) = p.aLsLs[AN(t) - As(t)] if AnU) > Asit); otherwise As(t) = 0.

is the fraction of the Northern labor force engaged in R&D, and ais is the fraction of the Southern labor force engaged in learning Northern technology; the rest of the notation is standard. Note that Ln and Ls are assumed constant.

(a) What is the long-run growth rate of Northern output per worlcer?

(b) Define Z(t) = As(t)/AN(t). Find an expression for Z as a function of Z and the parameters of the model. Is Z stable? If so, what value does it converge to? What is the long-run growth rate of Southern output per worker?

(c) Assume = flis and = ss- What is the ratio of output per worker in the South to output per worker in the North when both economies have converged to their balanced growth paths?

3.11. Delays in the transmission of knowledge to poor countries.

(a) Assume that the world consists of two regions, the North and the South. The North is described by Yit) = Ajv(t)(l - fliliw and A,v(f) = . The South does not do R&D but simply uses the technology developed in the North; however, the technology used in the South lags the Norths by T years. Thus Ys(t) = As{t)Ls and Asit) = A,v(f - ). If the growth rate of output per worker in the North is 3 percent per year, and if ai is close to 0, what must be for output per worker in the North to exceed that in the South by a factor of 10?

(b) Suppose instead that both the North and the South are described by the Solow model: y, ( ) = f(k, ( )), where y,(t) = Y, (t)/A, (t)L, (f) and k,(t) = (t)/A, (t)L,(f) (J =N,S). As in the Solow model, assume K, (t) = sY, (r) -SKiit) and L, (r) = nL,(f); the two countries are assumed to have the same saving rates and rates of population growth. Finally, ( ) = gANit) and Asit) = AN(t ~ t).

(i) Show that the value of on the balanced growth path, *, is the same for the two cottntries.

(h) Does introducing capital change the answer to part ( )? Explain. (Continue to assume g = 3%.)

3.12. Consider an economy described by the model of Part of this chapter that is on its balanced growth path. Suppose there is a permanent increase in the rate of population growth. How does this affect output per worker over time?

3.13. Con.sider the model of Part of the chapter.

(a) What is consumption per unit of effective labor on the balanced growth path?

ib) What values of Sk and Sh maximize this value?

3.14. Suppose that, despite the political obstacles, the United States permanently reduces its budget deficit from 3 percent of GDP to zero. Suppose that the economy is described by the model of Part of the chapter, and that a = 0.3 5 and p = 0.4. Suppose that initially = Sh = 0.15, and that Sk rises by the full amount of the fall in the deficit.