Assume a production function is Cobb - Douglas in capital and labour. Y = ZF(K, N) = zK\alpha N1-\alpha (a) Derivethe per worker production function, y = zf (k) where y = Y/N, k = K/N (b) Use the Solow Model to derive the steadystate level of capital per worker, for given s, d and n. (c) Show diagrammatically the impact on the steady state solutionof 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 thismatters for your answer to part c) ii) (e) Show diagrammatically the impact of a fall in n, in the short and long term

The objective of the question is to understand the Cobb-Douglas production function and its…

Answered: Assume a production function is Cobb -…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Consider 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?
Given the following aggregate production function: Y = K0.25 (AL) 0.75, where technology A grows at A a fixed rate: = g> 0 A (a) Obtain the marginal product of capital algebraically, also discussing the second derivative. (b) Transform the production function into efficiency-worker terms, showing how =ỹ depends Y AL K on = k. AL (c) Working with the Solow model, where capital evolution follows K = I - SK, investment is a fixed proportion of income I sy, production is given by the Cobb Douglas function above, and the workforce grows at a positive rate, algebraically derive the fundamental Solow equation in efficiency-worker terms. Show all workings. (d) Present and discuss the graphical counterpart to this equation, explaining the dynamics of capital per efficiency worker that result from the logic of the model. (e) Working with the fundamental equation you derived in part (c), find the equilibrium solution for k algebraically. Relate this to your diagram in part (d). (f) Find the…
Consider the Solow model with Cobb-Douglas production function with productivity parameter A, exponents on capital equal to 1/4, and exponent on labor equal to 3/4. Derive the MPL funcion as a function of K, L and constant A Note: xa is the same as x power a O (3/4) A(K/L) 0.25 O(1/4) A(L/KY0.76 O (1/4) A(KML)+0.75 O (3/4) A(UK) 0 25
Consider 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?
Assume the rate of technological progress is constant in a SSGM with Cobb- Douglas production function. Show that a steady-state cannot be identifiede if the technological progress is capital augmented, i.e. F(K, L) = [A(t)K(t)]*L(t)'_ª, where A(t) is the efficiency term growinge with a constant rate g.t
Consider the continuous-time Solow growth model as discussed in the lecture. The economy is on its balanced growth path with labor augmenting technological progress at rate g and population growth at rate n. The depreciation rate is 8. (a) Derive the dynamic equation of aggregate capital K. (b) Normalize the aggregate capital K by technology A and population L and denote the capital per unit of effective labor as k = K/(AL). Derive the dynamic equation of k .(c) Use the phase diagram to illustrate the steady state level of k and the dynamics of k if k starts from a level lower than the steady state level. (d) Suppose that the economy is on its balanced growth path at time to. Unex- pectedly, the population growth rate drops down to n' <n at to, stays at n' until t1, and resumes to n after t1. How does k respond, between to and t1, and afterwards?
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.
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. What is the per-effective-worker production function? (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.) (d) Show that at the Golden Rule steady state the saving rate for this economy is equal to the parameter a.
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…
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 circumstances,…
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. (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…
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…

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