Macroeconomics (Fourth Edition)
4th Edition
ISBN: 9780393603767
Author: Charles I. Jones
Publisher: W. W. Norton & Company
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Question
Chapter 5, Problem 12E
(a)
To determine
The Solow diagram in the economy.
(b)
To determine
The evolution of the economy over time in the Solow model.
(c)
To determine
The growth rate of per capita GDP over time.
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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 + BK¹/2 H¹/4L¹/4
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.
(a) Express output per person, y =Y/L, as a function of capital per person, k =K/L.
Assume a production function is Cobb - Douglas in capital and labour. Y = ZF(K, N) = zK\alpha N1-\alpha (a) Derive
the per worker production function, y = zf (k) where y = Y/N, k = K/N (b) Use the Solow Model to derive the steady
state level of capital per worker, for given s, d and n. (c) Show diagrammatically the impact on the steady state solution
of i) a rise in z; ii) a rise in s, using both the Solow Model diagram and time path diagrams of Iny and Inc. (d)
Showdiagrammatically (ideallysupplementedwithkeyequations)howtode rive the Golden Rule, and explain why this
matters for your answer to part c) ii) (e) Show diagrammatically the impact of a fall in n, in the short and long term
Consider the Solow model without technological progress and an economy with the following production function,
Y=A[Kα+Gα]1/α
where α<1, K is private capital and G is public capital that is used freely and provided by the government. The level of technology A is fixed and assumed to be equal to 1.
A. Does this production function feature constant returns to scale? Explain.
In order to finance public capital, the government taxes all investment on private capital at the rate 0<τ<1. So, the revenue raised by the government in each period is sKYt(1−τ) where sK is the private savings rate so that sKYt is pre-tax private savings. Public investment towards public capital is a constant fraction sG of total revenue. Then, the accumulation equations for private and public capital, respectively, are,
Kt+1−Kt=sKYt(1−τ)−δKt
Gt+1−Gt=sG(sKYtτ)−δGt
where δ is the common depreciation rate
B. Consider a balanced growth path where the growth rates of private capital is equal to the growth…
Chapter 5 Solutions
Macroeconomics (Fourth Edition)
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- 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 and depreciation curves, making sure to label the steady state level of capital(if it exists). Under these…arrow_forwardProblem 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 and depreciation curves, making sure to label the steady state level of capital(if it exists). Under these circumstances,…arrow_forwardProblem 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. (iii) 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 and depreciation curves, making sure to label the steady state level of capital(if it exists). Under these…arrow_forward
- Suppose in a Solow model, we have the following parameter values: n = 0, s = 0.2, a = 0.33. There is no growth in the total factor productivity so that A, = A = 1. Moreover, we know that at time 0, the economy is at a steady state so that k = k, =1. Now imagine that a deadly pandemic hits the economy at time t=1. As a result, the population at time t =1 is 10% lower than the population at time t=0. The pandemic is a one-time shock so that population growth rate remains the same, i.e., from t-2 onward, the population remains the same as the population at time t=1. The total capital stock, however, is unchanged so that K, Ko. What is the growth rate of per-capita capital in percentage (rounded to the 2 decimal places, e.g., answer 1.08 if your calculation shows the growth rate is 0.01079) at time t=3 from time t=2? %3!arrow_forwardConsider the Solow model with a production function Y(t) = A*K(t)αL(t)1-α, Where A is a fixed technological parameter. Explicitly solve for the steady-state value of the per capita capital stock and per capita income. How do these values change in response to a rise in (a) the technological parameter A, (b) the rate of saving s, (c) α , (d) δ, the depreciation rate, and the population growth rate n?arrow_forwardConsider the Solow model with a production function Y(t) = A*K(t)^α*L(t)^(1-α), Where A is a fixed technological parameter. Explicitly solve for the steady-state value of the per capita capital stock and per capita income. How do these values change in response to a rise in (a) the technological parameter A, (b) the rate of saving s, (c) α , (d) δ, the depreciation rate, and (e) the population growth rate n?arrow_forward
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