PROFESSOR XAVIER PRESENTS...PROBLEM SET 2 Í THIS PROBLEM SET IS DUE ON WEDNESDAY, OCTOBER 20 BY 11:59 PM. YOU SHOULD SUBMIT YOUR ANSWERS ON CANVAS BY THE DEADLINE. YOU CAN EITHER TYPE YOUR ANSWERS OR SCAN THEM. MAKE SURE TO UPLOAD A UNIQUE FILE. LATE PROBLEMS WILL NOT BE ACCEPTED. Problem 1. Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, δ, and a population growth rate, n. The production function is given by 𝑌 = 𝐴𝐾 + 𝐵𝐾3/4𝐿1/4 where A and B are positive constants. Note that this production is a mixture of Romer’s AK model and the neoclassical CobbDouglas production function. • (i) Does this production function exhibit constant returns to scale? Explain why. • (ii) Does it exhibit diminishing returns to physical capital? Explain why. • (iii) Express output per person, 𝑦 = 𝑌 𝐿 𝐾 , as a function of capital per person, 𝑘 = . 𝐿 • (iv) Write down an expression for y/k as a function of k and graph. (Hint: as k goes to infinity, does the ratio y/k approach zero?) • (v) Use the production function in per capita terms to write the fundamental equation of the Solow-Swan model. • (vi) Suppose first that sA < δ + n. Draw the savings curve and the depreciation curve, making sure to label the steady state level of capital(if it exists). What number does the savings curve approach as k goes to zero? As k goes to infinity, the savings curve approaches a number: what number is that? Is it zero? • (vii) Under these parameters, will there be positive growth in the long run? (Remember that A and B are constants). Why? • (viii) Imagine that we have two countries with the same parameters (same A, B, s , δ, and n). One of them is rich and the other is poor. Which one of the two will grow faster? Why? Will those two countries eventually catch up? • (ix) Suppose now that sA > δ + n. Draw the savings and depreciation curves, making sure to label the steady state level of capital(if it exists). Under these circumstances, will there be positive growth in the long run? Why? • (x) If s =0.4, A =2, B =1, δ = .25, and n =0.10, the growth rate converges to some value as time goes to infinity. What is this value? 1 Problem 2. a. In the neoclassical growth model, there are diminishing returns to physical capital and diminishing returns to human capital. But there are constant returns to scale. Hence, the economy can grow forever by accumulating both kinds of capital without ever facing diminishing returns. Comment the various aspects of this statement. b. Discuss the “rivalry” and “excludability” properties of each of the following goods. • a. An aspirin • b. The formula to produce aspirin • c. Cable TV • d. Lobsters in the Atlantic Ocean • e. A cow in a farm • f. A zoom meeting • g. A grilled cheese sandwich • h. The Pythagoras theorem • i. The Solow model Problem 3. In class we argued that if people could accumulate human as well as physical capital, the production function would look like the “AK” production function. • (a) If the production function is AK and the savings rate is constant at rate “s”, and the rates of depreciation and population growth are δ and n respectively, what would the growth rate of the economy be? • (b) What would be the macroeconomic consequences of decreasing the savings rate in this economy? • (c) What would be the consequences of an increase in fertility in this economy? • (d) Would the consequences of decreasing fertility be UNAMBIGUOUSLY GOOD? • (e) Can human capital grow without bounds? Explain why or why not (make sure you discuss the physical nature of human capital). • (f) What is the growth rate of the economy (in the absence of technological progress) if human capital cannot grow without bounds? Problem 4 Consider the following production function: 𝑌 = 𝐴𝐾α𝐻1−α𝐿𝛽 𝑡 𝑡 𝑡 𝑡 where Kt is capital, Ht is human capital, Lt is the amount of workers and A is the (constant) level of technology. • (i) Does this production function satisfy all the neoclassical properties. Discuss the meaning of each property INTUITIVELY. Imagine that parents invest in the human capital of their children up to the point where the MARGINAL PRODUCT OF PHYSICAL CAPITAL, Kt, is equal to the MARGINAL PRODUCT OF HUMAN CAPITAL, Ht. 2 • (ii) What is the relation between Kt and Ht? Use this relation to write down total output as a function of Kt only. Imagine that the number of people in this economy, Nt, is different from the number of workers because some people do not work. Let 𝑙 = 𝐿𝑡 be the number of workers per capita (the fraction of the population that works). Let 𝑦 = 𝑌𝑡 be output per capita and 𝑘= 𝑡 𝑡 𝐾𝑡 𝑡 𝑁𝑡 𝑁𝑡 be capital per capita. Finally, let n be the rate of population growth and γ be the growth rate of labor. 𝐿 𝑁𝑡 • (iii) Using the “effective production function” you derived in (b), write down output per capita, yt , as a function of capital per capita, kt, labor per capita, lt, the level of population Nt, and the level of technology, A. Following Solow and Swan, assume there is no government and no net exports, that the depreciation rate of capital is the constant δ > 0 and the savings rate is constant 0 < s < 1. • (iv) DERIVE the fundamental equation of Solow-Swan. How does the growth rate of capital depend on employment per person, lt? Explain intuitively. • (v) Does the equation of the growth rate of capital depend on the growth rate of population, n, or the growth rate of employment? Explain intuitively. Problem 5 • a. Discuss the evolution of the world distribution of income over the last four decades. Is the distribution becoming unimodal? If so, how? If not, how? • b. Do you think that, without technological progress, the average level of human capital of an economy can increase without bounds? Why or why not? 3
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