Consider an economy that exhibits both population growth (L grows at rate n) and technological progress (A grows at rate a) described by the production function, Y = F(K, AL) = Ka (AL)¹-α Here K is capital and Y is output. (a) Show that this production function exhibits constant returns to scale. [2 marks] (b) What is the per-effective-worker production function, y = f(k), (where y = Y/AL k K/AL)? (Show your working.) [2 marks] (c) Find expressions for the steady-state capital-output ratio, capital stock per effective worker, and output per effective worker, as a function of the saving rate (s), the depreciation rate (8), the population growth rate (n), the rate of technological progress (a), and the coefficient a. (You may assume the condition that capital per effective worker evolves according to Ak = sf (k) - (a+n+8)k.) [5 marks] (d) Show that at the Golden Rule steady state the saving rate for this economy is equal to the parameter a. [6 marks]
Consider an economy that exhibits both population growth (L grows at rate n) and technological progress (A grows at rate a) described by the production function, Y = F(K, AL) = Ka (AL)¹-α Here K is capital and Y is output. (a) Show that this production function exhibits constant returns to scale. [2 marks] (b) What is the per-effective-worker production function, y = f(k), (where y = Y/AL k K/AL)? (Show your working.) [2 marks] (c) Find expressions for the steady-state capital-output ratio, capital stock per effective worker, and output per effective worker, as a function of the saving rate (s), the depreciation rate (8), the population growth rate (n), the rate of technological progress (a), and the coefficient a. (You may assume the condition that capital per effective worker evolves according to Ak = sf (k) - (a+n+8)k.) [5 marks] (d) Show that at the Golden Rule steady state the saving rate for this economy is equal to the parameter a. [6 marks]
Chapter9: Production Functions
Section: Chapter Questions
Problem 9.8P
Related questions
Question
![Consider an economy that exhibits both population growth (L grows at rate n) and
technological progress (A grows at rate a) described by the production function,
Y = F(K, AL) = Ka (AL)¹-α
Here K is capital and Y is output.
(a) Show that this production function exhibits constant returns to scale. [2 marks]
(b) What is the per-effective-worker production function, y = f(k), (where y = Y/AL
k K/AL)? (Show your working.)
[2 marks]
(c) Find expressions for the steady-state capital-output ratio, capital stock per
effective worker, and output per effective worker, as a function of the saving rate
(s), the depreciation rate (8), the population growth rate (n), the rate of
technological progress (a), and the coefficient a. (You may assume the condition
that capital per effective worker evolves according to Ak = sf (k) - (a+n+8)k.)
[5 marks]
(d) Show that at the Golden Rule steady state the saving rate for this economy is
equal to the parameter a.
[6 marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe67b9373-07fc-4c99-ae45-ace7cb60cf24%2F8e4bc10f-e32b-4f48-839c-688898bb4d05%2Fpza89hp_processed.png&w=3840&q=75)
Transcribed Image Text:Consider an economy that exhibits both population growth (L grows at rate n) and
technological progress (A grows at rate a) described by the production function,
Y = F(K, AL) = Ka (AL)¹-α
Here K is capital and Y is output.
(a) Show that this production function exhibits constant returns to scale. [2 marks]
(b) What is the per-effective-worker production function, y = f(k), (where y = Y/AL
k K/AL)? (Show your working.)
[2 marks]
(c) Find expressions for the steady-state capital-output ratio, capital stock per
effective worker, and output per effective worker, as a function of the saving rate
(s), the depreciation rate (8), the population growth rate (n), the rate of
technological progress (a), and the coefficient a. (You may assume the condition
that capital per effective worker evolves according to Ak = sf (k) - (a+n+8)k.)
[5 marks]
(d) Show that at the Golden Rule steady state the saving rate for this economy is
equal to the parameter a.
[6 marks]
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