Problem 1. Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, 6, and a population growth rate, n. The production function is given by Y = AK + BK³3/4L1/4 where A and B are positive constants. Note that this production is a mixture of Romer's AK model and the neoclassical Cobb- Douglas production function. • (1) Does this production function exhibit constant returns to scale? Explain why. • (i) Does it exhibit diminishing returns to physical capital? Explain why. • (i) Express output per person, y =, as a function of capital per person, k =. • (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 < 8 + 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? • (vii) Imagine that we have two countries with the same parameters (same A, B, s, 8, 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 > 8 + 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, 8 = .25, and n =0.10, the growth rate converges to some value as time goes to infinity. What is this value?
Problem 1. Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, 6, and a population growth rate, n. The production function is given by Y = AK + BK³3/4L1/4 where A and B are positive constants. Note that this production is a mixture of Romer's AK model and the neoclassical Cobb- Douglas production function. • (1) Does this production function exhibit constant returns to scale? Explain why. • (i) Does it exhibit diminishing returns to physical capital? Explain why. • (i) Express output per person, y =, as a function of capital per person, k =. • (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 < 8 + 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? • (vii) Imagine that we have two countries with the same parameters (same A, B, s, 8, 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 > 8 + 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, 8 = .25, and n =0.10, the growth rate converges to some value as time goes to infinity. What is this value?
Chapter1: Making Economics Decisions
Section: Chapter Questions
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Transcribed Image Text:Problem 1.
Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, 8, and a population growth rate, n. The
production function is given by
Y = AK + BK³3/4L1/4
where A and B are positive constants. Note that this production is a mixture of Romer's AK model and the neoclassical Cobb-
Douglas 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.
• (ii) Express output per person, y =-
-, as a function of capital per person, k =.
• (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 < 8 + 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, 8, 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 > 8 + 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, 8 = .25, and n =0.10, the growth rate converges to some value as time goes to infinity. What is this
value?
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