Economics 5118 Assignment 3

Economics 5118 Assignment 3

Due: March 22, 2016

1. Say that the aggregate production function for some economy is Cobb-Douglas:

Yt = A KtaNt1-awith A>0 and 0<a<1

where: Y is output, K is capital, N is labour (number of workers) and ‘A’ and ‘a’ are constants.

(a) (i) Show that this production function has constant returns to scale.

(ii) Derive the per worker version of this production function (define y=Y/N and k=K/N).

(iii) Does this function have diminishing returns in k? Explain.

(b) In the Solow model total savings (S) is a constant share of output (call this share ‘s’ i.e. ‘s’ is the savings rate). So: St = s Yt

Savings funds investment:St = It

The capital accumulation equation is: Kt+1 = It-Kt where  is the depreciation rate.

Combine these three equations and the production function to derive an expression for the growth

rate in capital (Kt+1/Kt) as a function of y/k. Show your work.

(b), what condition must be satisfied if ‘k’ is to be growing over time? Explain the economic sense

behind thiscondition.

(d) This economy will be in a steady state if k is constant over time. State the condition that must be

satisfied for this to be true. Use the condition to solve for the steady state value of k(i.e. find the

solution where k is a function of exogenous parameters, show your work).

(e) Now find the steady state values of y and c (c is consumption per worker).

(f)Now do some comparative statics(take derivatives of your solutions for k, y and c from (d) and (e)).

(i) What happens to k, y and c if s falls? Show your work. Does this make economic sense? Explain.

(ii) What happens if n rises? Show your work. Provide a diagram of the effect of this change.

(g) Say that instead of a constant savings rate ‘s’ is sP for poor countries and sR for rich countries where

sRsP (assume s rises sharply from sP to sR once y is above some critical value). How might this change the steady state outcome in the model? Is there more than one steady state for an economy of this sort? Explain and provide a diagram.

2. Let’s think about some data on the United States, China and Saudi Arabia in terms of the model in question 1. The data comes from the “Penn World Tables” which generates national accounts data for a large sample of countries (see: The data is fromversion 8.0 of the tables and looks at 1980 and 2011 (the most recent year available).

China / United States / Saudia Arabia
1980 / 2011 / 1980 / 2011 / 1980 / 2011
N / 479.1169 / 784.4269 / 102.5131 / 141.807 / 2.496 / 10.529
Y / 1278194 / 10685676 / 5750466 / 13351904 / 241470 / 717691
K / 2274871 / 45263624 / 17983890 / 41501056 / 565021 / 2452500

N = is the number of people working (in millions), Y = real GDP (converted to 2005 US dollars), K = the real value of the capital stock (in 2005 US dollars).

(a) Calculate y=Y/N and k=K/N in for each country for both years. Also calculate the average annual growth rate for Nfor the 1980-2011 period (call it ‘n’).**

(**The average annual growth rate in some variable x over the period t to t+s can be obtained

from: xt+s = xt(1+g)s (solve for g). )

(b) What was the average annual growth rate in y and k in China for the 1980-2011 period? What were the average annual growth rates in y and k in the US and Saudi Arabia for this period?

(c) Say that the aggregate production function for each country is Cobb-Douglas (Yt = A KtaNt1-a) with a=.35 (this is a pretty typical value). Put the production function in per worker terms. Use the per worker production function and your figures for y and k in part (a) to calculate the value of ‘A’ for each country in each of the two years. How large is ‘A’ in China compared to the US in 1980? What might account for this difference? How has this difference changed 1980-2011? What is happening to A in Saudi Arabia? What might account for this trend in Saudi Arabia?

(d) Question 3 below gives the production function with exogenous technological change. Using this and your estimates of A from part (c) calculate the values of  for China, the US and Saudi Arabia for the 1980-2011 period.

(e) Based upon the figures above and your calculations sketch the per worker production functions of each country in 2011 and indicate the points at which the countries were located. Based on the diagram how can you account for the difference in y between China and the US in 2011? Why did China grow faster 1980-2011? How does Saudia Arabia compare to the US in 2011? What might account for this?

(f) In the Solow model steady state differences in y, k and c reflect differences in the labour force growth rate (n), differences in the savings rate (s) and differences in the position of the production function (A). Differences in depreciation rates () could also have an effect however these seem to be about the same in the two countries (around .03). You have calculated ‘n’ and ‘A’ for Saudi Arabia, China and the US in (a). The PennWorld tables suggestthat ‘s’ in both China and Saudi Arabia averaged about .30 over the entire 1980-2011 period while ‘s’ in the US averaged about .20. Say depreciation rates were about .03 in all three countries. Based on these figures do you think the differences in y and k in 2011 are mainly due to differences in the steady states of the US and China or due to transitional differences? Explain.

3. Solow model with exogenous technological change. Let the production function be:

Yt = (1+)t KtNt1-

where is the per period rate of technological change.

(a) Give the expression for the growth rate in y (y=Y/N) as a function of technological progress and growth in k (do not assume a steady state). Go back to your data on China in question 2. Setting =.35 again and using your estimate of  from question 2 confirm that your growth rate equation is correct.

(b) As in question 1 assume that the labour grows at rate 'n' and capital growth reflects the effect of savings and investment (s) -- just like in question 1. Derive the expression for the growth rate in k (=K/N) assuming the production function is the Cobb-Douglas function above.

(c) Richer countries typically have high levels of capital per worker. If this (higher k) is the only difference between richer and poorer countries will output per worker (y) in richer countries grow faster or slower than y in poor countries? Show your work.

4. (a) In the centralized (planner’s) version of the Ramsey model the decision-maker must satisfy the following constraints (variables are defined as in the lecture notes):

yt = ct + it

kt+1 = it -kt (so it=kt+1 +kt )

yt = F(kt)

Combine these equations to derive the dynamic resource constraint of the economy.

(b) Look at your constraint from (a). What would the level of investment have to be in order to keep the

size of the capital stock constant over time? Explain. What would the level of consumption be?

(ii) Say that the production function F(kt) takes the form:

yt = Akta where 0<a<1

Substitute this into your dynamic resource constraint and then derive an expression for the

Golden Rule capital stock and consumption level (notice that since you have an explicit production function you can solve for the values of c and k as functions of the exogenous parameters).

(iii) Represent your outcome in a diagram.

(d) Say that the inter-temporal utility function has the form: (so U(c)=ln(c) )

Combine this utility function with the dynamic resource constraint from (a) assuming that F(kt)=Akta. Set up the Lagrangian and derive the first order conditions for period t+s.

(e) Use your first-order conditions to find the expression for the Euler condition.

(f) Use your results from (e) to find expressions for the steady state values of c and k as functions of the

exogenous parameters. Explain how you obtained them.

(g) How does the solution compare to the Golden rule outcome? Explain why the difference between the

outcomes makes sense.

(h) How does k and c change if A falls? Show your work.

5. A model with optimization, labour force growth but no technological progress.

Say that the labour force (N) grows at a constant rate ‘n’. The per worker production function is:

yt = A kt (A>0 is a constant, y is output per worker and k is capital per worker). The national income constraint in per worker terms is: yt=ct+itwhere it is investment per worker. The capital accumulation equation in per worker form is: (1+n)kt+1 = it+(1-kt (this is obtained by taking Kt+1 = It-Kt, dividing by Nt and using Nt+1=Nt(1+n) and k=K/N).

(a) Derive the dynamic resource constraint for this model.

(b) The utility function in this model must recognize that the number of people in this economy grows over time. One way to do this is to make utility a function of aggregate consumption (see Wickens). Let us assume instead that the utility function to be maximized is the sum of utilities of everyone in the economy. If everyone is identical this will just be:

Nt+sU(ct+s) in period t+s

Since labour force grows at the rate n: Nt+s= (1+n)sNtso you can write the intertemporal utility function:

Set up the Lagrangian for this model. Derive the first order conditions and Euler equation.

(c) Use the Euler equation to find the condition that determines the size of the steady state value of k. Solve for k. Now find the value of y and c in the steady state.

(d) In the Solow model the exogenous savings rate is a key determinant of the steady-state. In this model savings are determined by optimization and will reflect the effect on savings of exogenous variables like preferences and technology. How is the steady state outcome for k, y and c affected by an increase in ? Explain why this result makes economic sense.