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Problem Set 3 - Solutions 1. Leo consumes only nuts and berries. Fortunately, he likes both goods. The consumption bundle where Leo consumes cj units of nuts per week and x2 units of berries per week is written as (21, $2). (a) The set of consumption bundles (r1, 72) such that Leo is indifferent between (21, r2) and (16, 4) is the set of bundles such that 21 2 0, x2 2 0, and x2 = 20-4 /r1. Plot several points that lie on the indifference curve that passes through the point (16, 4) and sketch this curve.2. Consider two countries, 1 and 2, that are described by the Solow model. Both countries are on their balanced growth paths, and the only difference between them is that s2 >s. Then: A. Output, consumption, and investment are all higher in country 2. B. Output is higher in country 2, but consumption and investment can both be either higher, lower, or the same as in country 1. C. Output and investment are higher in country 2, but consumption can be either higher, lower, or the same as in country 1. D. Investment is higher in country 2, but output and consumption can both be either higher, lower, or the same as in country 1. 3. Romer, Problem 1.9. 4. This question asks you to use a Solow-style model to investigate some ideas that have been discussed in the context of Thomas Piketty's recent work. Consider an economy described by the assumptions of the Solow model, except that factors are paid their marginal products (as in the previous problem), and all labor income is consumed and all other income is saved. Thus, C(t) = L(t) [aL(6)] a. Show that the properties of the production function imply that the capital-output ratio, K/Y, is rising if and only if the growth rate of K is greater than n + g - that is, if and only if & is rising. b. Assume that the initial conditions are such that dy/OK at r = 0 is strictly greater than n + g + 8. Describe the qualitative behavior of the capital-output ratio over time. (For example, does it grow or fall without bound? Gradually approach some constant level from above or below? Something else?) Explain your reasoning. c. Many popular summaries of Piketty's work describe his thesis as: Since the return to capital exceeds the growth rate of the economy, the capital-output ratio tends to grow without bound. By assumption, this economy starts in a situation where the return to capital exceeds the economy's growth rate. If you found in (b) that K/Y grows without bound, explain intuitively whether the driving force of this unbounded growth is that the return to capital exceeds the economy's growth rate. Alternatively, if you found in (b) that K/Y does not grow without bound, explain intuitively what is wrong with the statement that the return to capital exceeding the economy's growth rate tends to cause K/Y to grow without bound. d. Suppose F() is Cobb-Douglas. Describe the qualitative behavior over time of the share of net capital income (that is, K(t) and [excel - 5 ) in net output (that is, Y(t) - 6K (t)). Explain your reasoning. Is the common statement that an excess of the return to capital over the economy's growth rate causes capital's share to rise over time correct in this case? EXTRA PROBLEMS (NOT TO BE HANDED IN; ONLY SKETCHES OF ANSWERS WILL BE PROVIDED) 5. Describe how, if at all, each of the following developments affects the break-even and actual investment lines in our basic diagram for the Solow model: a. The rate of population growth falls. b. The rate of technological progress rises. c. The production function is Cobb-Douglas, F(K,AL) = K"(AL)"", and capital's share, a, rises. d. Workers exert more effort, so that output per unit of effective labor for a given value of capital per unit of effective labor is higher than before. 6. (Hicks meets Solow.) Consider the Solow model with one change: the production function is Y = AF(K,L). All other assumptions of the model are unchanged. (The type of technical change assumed in the Solow model is known as labor-augmenting or Harrod-neutral technical change; the form of technical change assumed in this problem is called Hicks-neutral.) a. Show what happens if we try to derive a balanced growth path like the one derived in class. b. What can you say in the special case F(K,L) = K"L ", 0 0 are factor shares. Assume a fixed share sk > 0 of output is invested in physical capital, a share s# > 0 of ouput is invested in human capital. Assume zero depreciation. Raw labor suppply grows at a fixed rate n and TFP grows at fixed rate g. For part (a), assume that S = $# = 0, as in the standard Solow model. a) Derive the steady state capital stock, steadty state human capital stock, and steady state ouput, all per efficiency unit of labor. b) Derive the quantitative impact on steady state output per efficiency unit of labor of a marginal increase in sk and sy. Describe under what conditions sy has a greater impaft on output than SH c) Consider the limiting case where a + 8 - 1. Explain why and in what sense the growth becomes endogenous in the limit. Derive the economy's growth rate for this case, assuming that g = 0. What are the determinants of the growth rate? Problem 2.6. Consider a Solow growth model with exogenous savings rate, s, population growth, n, and TFP growth, g;. Output is given by Y = Ko(AL)]-a with 0 ni. Determine how &* changes. Sketch the time path of k. b) Suppose n1 = 1%, n2 = 1.1%, s = 20%, g = 1%, 8 = 3%, and a = 1/3. Compute the percentage change in &*. c) Detemine how the growth rate of Y/L changes at time 1. Sketch the time path. d) Suppose productivity growth is endogenous and linked to population growth, g = An for some A > 0. If n increases at time t1, are there A values so that the growth of Y/L accelerates? Will Y/L grow faster or more slowly in the long run? e) Now suppose Y = F(K, AL) is a general CRS production function. Derive a formula for the percentage change in &* in response to a marginal increase in n (no change in g). Compute ther percentage change for the same numbers as in part b. Assume that capital share is again 1/3