K and L by the factor z magnifies output produced by the factor z. Define

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1 Intermediate Macroeconomic Theory II, Fall 2014 Instructor: Dmytro Hryshko Solutions to Problem Set 1 1. (15 points) Let the economy s production function be Y = 5K 1/2 (EL) 1/2. Households save 40% of their income; population growth, n, is equal to 2%; the depreciation rate, δ, is equal to 1%; the growth rate in the efficiency of labor, g, is 2%. (a) (2 points) Show that the aggregate production function is constant returns to scale in K and L (1 point), and express the production function in per-effectiveworker terms (1 point). (b) (1 point) Is production function increasing/constant/decreasing returns to scale in 3 factors of production, K, E, and L? Show how you arrived at the conclusion. (c) (3 points) Calculate the steady state level of capital per effective worker (1 point) output per effective worker (1 point) and consumption per effective worker (1 point). (d) (2 points) Find the ratio of capital per worker to output per worker in the economy on a balanced growth path, and the real rental price of capital. (e) (1 point) What is the growth rate of the real wage in the economy on a balanced growth path? What is the growth rate of the total output? (f) (1 point) If you were a social planner who maximizes consumption per worker in the economy, what savings rate would you choose? (You need not show your calculations here if you see the answer.) (g) (3 points) Find the golden rule level of capital per effective worker (1 point) and the corresponding output per effective worker in the golden rule steady state (1 point), and consumption per effective worker (1 point). (h) (2 points) Assume the government can tax proportionally capital and wage incomes at the rate τ, and invests only the fraction s (equal to the private savings rate) of the tax proceedings, while wasting the rest. What is the level of consumption per effective worker in the very long run under this tax policy, assuming that s equals 40%? (Hint: it will be a function of the tax rate.) How does it compare to consumption per effective worker in (1c)? (a) F (zk, zl) = 5(zK) 1 2 (EzL) 1 2 = z 1 2 z 1 2 5K (EL) 2 = zy raising utilization of }{{} =Y K and L by the factor z magnifies output produced by the factor z. Define 1

2 z = 1. Then, Y = F ( K, ( EL EL EL EL EL) = 5 K ) 1 ( ) 1 2 EL 2. Thus, y EL EL pew = 5kpew, 1/2 where y pew = Y and k EL pew = K. EL (b) Production function is increasing returns to scale in K, E and L: F (zk, zezl) = 5 (zk) 1/2 (zezl) 1/2 = z 3/2 5K 1/2 (EL) 1/2 > zy. Thus, doubling the utilization of K, E and L more than doubles output, it magnifies output by the }{{} =Y factor 2 3/2. (c) In steady state, k pew = s5kpew 1/2 (n + δ + g)k pew = 0. It follows that ( ) 1 ( ) kpew 5s 1 1/2 2 ( 5s = = n+g+δ n+g+δ = ) = ypew = = 200. c pew = (1 s)ypew = = 120. (d) k pw y pw = k pew y pew = = 8. The rental price of capital, R = α Y K = α y pw k pw = α y pew k pew = = 1 16 = (e) The growth rate in the real wage will be equal to y pew + E = 0 + g = 0.02, ypew E or 2%. The balanced growth rate in total output is n + g = 4%. (f) You should set the savings rate equal to the golden rule savings rate of 50%. ( ) 2 (g) At the golden rule steady state, s = α, and so kgold = ( 5α n+g+δ = ) = ypew,gold = = 250; c gold = (1 α)y pew,gold = 125. (h) The effective savings rate in the economy will equal s(1 τ) + sτ = s. Since s = 40%, the steady-state level of capital and output per effective worker will be the same as in c). Consumption per effective worker, however, will equal (1 s)(1 τ)200 = 120(1 τ), and lower than in c) the higher is the proportional tax rate τ. 2. (15 points) Let us modify the Solow growth model without technological progress by including government spending as follows. The government purchases G units of consumption goods in the current period where G = gl, and g is a positive constant. The government finances its purchases through lump-sum taxes on consumers, where T denotes total taxes, and the government budget is balanced each period, so that G = T. Consumers consume a constant fraction of disposable income, that is, C = (1 s)(y T ), where s is the savings rate, saving the rest. Assume that capital depreciates at the rate δ, labor force grows at the rate n, and production function is constant-returns to scale in K and L, Y = F (K, L). (a) (5 points) Derive the fundamental equation for the law of motion of capital per worker. (b) (5 points) Show graphically that there can be two steady states, one with high capital per worker, k, and one with low k. 2

3 (c) (5 points) Ignore the steady state with low k. (It can be shown that this steady state is unstable. ) Determine the effects of an increase in g on capital per worker and output per worker in the steady state. What are the effects on the growth rates of aggregate output, aggregate consumption, and aggregate investment? Thoroughly explain your results. (a) The law of motion of aggregate capital is K = I δk = s(y T ) δk = s(y gl) δk. Since capital per worker, k = K/L, k k = K K L L = sy k sgl K δ = sy k sg k δ n. It follows that k = sy sg (n + δ)k = tion sf(k) [sg + (n + δ)k]. }{{}}{{} investment func- break-even investment line Steady state occurs when k = 0, at the value of capital per worker k when: sf(k ) = sg + (n + δ)k. Notice that relative to our previous analysis, the break-even investment line is augmented with an additional non-zero constant, sg, so that the break-even line has an intercept sg and a slope n + δ. Alternatively, you could have modified the investment function to s[f(k) g] in which case the investment }{{} y function would have shifted downwards with the intercept of sg at k = 0. In the latter case, you should have used the original break-even investment line (n + δ)k. (b) See the graphs below. Ignore z on the graphs (set it to 1). Vertical axis plots investment and break-even investment functions. k1 denotes the old steadystate, and k2 denotes the new steady state, when g = g 2 > g 1 (for question 2c)). 3

4 (c) An increase in g lowers capital per worker and output per worker in the (new) steady state. When the economy reaches its new steady state, the growth rates of aggregate output, investment and consumption will equal the growth rate in population n, the growth rate of the aggregates in the previous steady state. Since government in our example takes away from the public investable resources and doesn t invest itself, a higher level of government consumption will lead to lower values of capital per worker and output per worker in the steady state. 3. (15 points) Consider the Permanent Income Hypothesis we studied in class. Preferences are quadratic, u(c t ) = 1 2 (c t c) 2 ; planning horizon is infinite and consumer starts her life at time 0; β(1 + r) = 1; income stream is known as of time 0. Consider individual X whose income is 0 in even periods (periods 0, 2, 4, 6, etc.) and 190 in odd periods (periods 1, 3, 5, 7, etc.); and individual Y whose income equals 100 in all periods. The real interest rate, r, equals 1/9, and 1 1+r = 0.9. (a) (5 points) Find the present discounted value of incomes for individual X and Y. Who is lifetime richer? (b) (5 points) Find the optimal consumption plan for both individuals assuming they are not borrowing constrained. (c) (5 points) Qualitatively describe the optimal consumption plan for both individuals if they are not allowed to borrow at all (there s no need for calculations here but you may provide exact numbers if you choose so). (a) Individual X s PDV of incomes is = (1 + r) (1 + r) 3 (1 + r) r 1 1 = = 900. (1+r) 2 Individual Y s PDV of incomes is r (1 + r) (1 + r) (1 + r) = r = = Individual Y is lifetime richer and will be able to achieve a higher consumption level. (b) Consumption of individual X, in all periods, will equal r 1 + r 900 = = Consumption of individual Y, in all periods, will equal r 1 + r 1000 = =

5 (c) Since individual Y is neither a borrower nor a saver, she will eat all of her endowments in each period, and borrowing constraints will not affect her consumption plan. Individual X s consumption will be zero in period 0, after that, however, she will be able to smooth consumption perfectly, setting aside some income in odd periods (when she is earning 190) to support her consumption in even periods (when she is earning 0). 4. Briefly comment on the following statements, indicating whether they are true or false where appropriate, and show how you arrived at your conclusion. (a) (3 points) The real interest rate is constant in the steady-state of the Solow economy with population growth but no technological growth, while it is growing on a balanced growth path of the economy with positive population and technological growth. (b) (3 points) Ceteris paribus, we can predict that the price of capital and the real interest rate will go down if some part of the economy s capital stock gets destroyed, say, due to war. (c) (3 points) Ceteris paribus, we can predict that the price of capital and the real interest rate will go down if there is an inflow of immigrants into the economy. (d) (3 points) Consider the following situation. An individual receives some news that she will receive a large inheritance in 10 years from now with a very high likelihood. The Permanent Income Hypothesis predicts that the person will not act on this news now in terms of current consumption but will rather wait for 10 years, and only then change her consumption if that inheritance is indeed received. (e) (3 points) Consider the Solow economy without technological growth, in its steady state equilibrium. At some point in time, part of the economy s stock of capital is destroyed but otherwise the economy remains the same as before, that is the savings patterns are the same, population growth, depreciation rate, as well as the aggregate technology are the same. What will happen eventually, in the very long run, to this economy in terms of capital per worker and output per worker? What will happen to the economy during its transition to that long-run state? (a) False. The real interest rate can be written as r = α Y δ = α y pw K k pw δ = α ypew k pew δ. The last two equalities highlight the fact that the real interest rate is constant in the steady state of the economy with positive population but no technological growth (next-to-last equality) and on a balanced growth path of the economy with positive population and technological growth (last equality). (b) False. The real interest rate is related to the marginal productivity of capital, and it will be higher for lower levels of capital by the law of diminishing returns. Intuitively, when a factor of production becomes scarce, it commands a higher price. 5

6 (c) False. With an inflow of immigrants, the productivity of capital goes up, and one unit of capital will command a higher price (again, you can think in terms of relative scarcity of labor and capital). (d) False. The point of the PIH that one would want to smooth consumption and will react to the news about permanent income when the news is received, not when the actual change in resources happen. Thus, provided our consumer is not liquidity constrained, she will reset her consumption now to a higher value. (e) The economy will eventually return to exactly the same levels of capital per worker, output per worker and consumption per worker. The effects of the shock are not permanent. Of course, there will be transitional dynamics. On the impact of the shock, the capital-labor ratio will fall, the amount of output per worker produced will fall as well, the economy will start accumulating capital so that the capital-labor ratio as well as output per worker will eventually return to their pre-shock values. 6

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