Question

AutoSave OFF OF HW2P2 - Compatibility Mode - Saved to my Mac a Home Insert Draw Design Layout References Mailings Review View Tell me Share Comments Times New... v v 12 A A Aa A. 215 AakbCelld Ankl DdE A RICcDdE ABCDde AaBbCcDc Aabb Dee Heading 1 Normal Na Spacing Hercing 2 Title Subtitle Pagic B ! | + | x x = = Dictate Sensitivity Sty es Pare 1. Country X and Country Y have the following production function: Y = F(K, L.) VK.L. a. Does this production function have constant returns to scale? Explain. b. What is the per-worker production function (i.c. y=fki from locture)? c. Assume that acither country experiences aby population growth or tochnological change, and that 5% of capital depreciates cach year. Assure further that Country X saves 10% of output each year and Country Y saves 20% of output each year. Using your answer from part(b) and the steady-state condition that investment equals clepreciation, find the steady state level of capital per wirker for each country. Then find the stendy-state levels of income per worker and consumption per worker. 2. "Devoting a larger share of national extput to investment would help restore rapid productivity growth and rising Living standards (ie rising consumption)." Do you ngroc with this claim? Explain using the Solow model from lecture 3. Draw a well-labeled graph that illustrates the steady-state of the Solow model from class and include population growth. Use the graph to find what happens to steady-state capital pier wirker (k) and income per worker (Y) in response to each of the following changes. a. A cbroge in consumer preferences increases the savings rate. b. A change in weather patterns incrosses the deprecintion rate. c. Better birth-control methods reduce the rate of population growth. <LA one-time, permanent improvement in technology increases the amount of'ulput that can be produced from any given amount of capital and labor. 4. Many clerigraphers predict that the U.S. will have zenci population growth in the 219 century, in contrast to average population growth of about 1% per year in the 20th century. Use the Solow model from class to forecast the effect of this slowlown in population growth ank Page 1 of 1 311 words DE English (United States Focus --+ 100%

Please complete question 1, parts A,B, and C. Use formulas and graphs to answer the problems! Please show all work, and use the formulas for the equations.

Answer 1

Ans.

y у 0.05k 0.05k 0.10y 4 0.20y 2 4 k For Country X 16 k For Country Y

a) Production function, Y = (K*L)^0.5

To check for returns to scale, let both the inputs be increased by factor t, then new output,

Y' = (tK * tL)^0.5 = t*(K*L)^0.5 = t*Y

As increase in both the inputs by t lead to an increase in output by the factor t, so, this production function exhibits constant returns to scale.

b) Production function, Y = (K*L)^0.5

=> Y/L = [(K*L)^0.5]/L

=> y = k^0.5

Here, output per worker = Y/L = y

and capital per worker = K/L = k

From the marginal product of capital per worker, MPk = dy/dk = 1/k^0.5,

we see that per worker prodction function is increasing function of k but increase will be at a decreasing rate, as seen from the slope of MPk = dMPk/dk = -0.5/k^1.5
Hence, per worker priduction function will be concave upward sloping curve.

c) The change in capital per worker,

dk = s*y - c*k

Here, s = saving rate

c = depriation

At steady state, dk = 0

=> y/k = c/s

Substituting y = k^0.5 in the above equation, we get,

k^0.5 / k = c/s

=> k = (s/c)^2 ---> Eq1

Thus, steady state level of capital per worker is (s/c)^2

Substituting Eq 1 in per worker production function,

y = s/c --> Eq2

Thus, steady state level of output per worker = s/c

Now, at steady state consumption, C = (1-s)*y = (1-s)*s/c ---->Eq3

For country X, s = 0.10 and c = 0.05

Steady state value of,

k = (0.10/0.05)^2 = 4 units (From Eq1)

y = 0.10/0.05 = 2 units (From Eq2)

C = (1-0.10)*0.10/0.05 = 1.8 units (From Eq3)

For country Y, s = 0.20 and c = 0.05

Steady state value of,

k = (0.20/0.05) = 16 units (From Eq1)

y = 0.20/0.05 = 4 units (From Eq2)

C = (1-0.20)*0.20/0.05 = 3.2 units (From Eq3)

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