## (solution) please do question 1 and 3 ONLY. Slight change in question 1,

please do question 1 and 3 ONLY. Slight change in question 1, please use the production function of A x KL/K+L

ECON 100C: Midterm exam #1

October 15, 2015, 7pm-8:20pm

Show all work to receive full credit 1 Problem 1: Income di¤erences between US

and South Africa

Consider the simple production model where the aggregate production function takes the form:

p

(1)

F (K; L) = A KL with A &gt; 0;

where K denotes the capital stock rented by ?rms and L denotes the number

of workers hired by ?rms.

(i) Does this production function exhibit constant returns to scale? Justify

your answer. (ii) Write down the expressions for the wage and the rental price of capital

as a function of the capital stock per worker. (Make sure to explain where

these expressions come from.)

What is the value for the share of GDP paid to labor, also called &quot;labor

share&quot;? Is it consistent with the evidence for the US economy? 2 (iii) Express per capita GDP as a function of capital per person.

Draw a graphical representation of this relationship with capital per person on the horizontal axis and per capita GDP on the vertical axis. 3 (iv) According to Table 4.3 in your textbook the observed capital per person

in South Africa is equal to 16% the level in the US. The observed per capita

GDP is 18% the one in the US. Assuming South Africa and the US have

access to the same technology represented by (1), with the same TFP A, can

the production model account for the di¤erence of income between South

Africa and the US? Explain. (v) Normalize A = 1 for the US. What is the implied Total Factor Productivity for South Africa such that the observed per capita GDP and capital

per person are consistent with the production model? 4 Problem 2: Why do some countries produce

so much more output per worker than others?

In their 1999 article Robert Hall and Charles Jones use a version of the

production model where the production function takes the following form:

Y = K 1=3 (AhL)2=3 ; (2) where K denotes the stock of physical capital, h represents human capital

per worker (which increases with years of schooling), L is labor, and A is a

labor-augmenting measure of productivity.

(i) Explain the choice of the exponents, 1/3 and 2/3, in the production

function (2). (ii) Is it possible to rewrite the production function, (2), to make it similar

to the one used in the lecture, namely Y = AK 1=3 L2=3 ? 5 (iii) Show that output per worker, y = Y =L, can be expressed as:

Y

y=

=

L K

Y 1

2 where K=Y is the capital/output ratio. (HINT: Show ?rst that

K

AhL 1

3 and then compute Y =AhL.) 6 (3) Ah;

K

Y 1

2 = (iv) Robert Hall and Charles Jones report the following observations for the

year 1988.

country

U.S.

France

Mexico

China y

1.000

0.818

0.433

0.060 K 1=2

Y 1.000

1.091

0.868

0.891 h

A

1.000 1.000

0.666

0.538

0.632 Use Equation (3) to compute the implied productivity for the countries

in the Table above.

Based on this table how do you explain that China is so much poorer

than other countries in 1988? 7 Problem 3: Solow growth model

Consider a version of the Solow growth model where the aggregate production

function is:

p

Y = A KL:

We normalize the labor force to L = 1 and we assume that aggregate consumption is:

C = cY;

where c 2 (0; 1) the propensity to consume out of income. The depreciation

rate of capital is d.

(i) Write down the law of motion of the capital stock.

(You must obtain an equation with Kt+1 and Kt as the only endogenous

variables). 8 (ii) Give the expression for the steady-state capital stock.

Provide a graphical representation of the determination of the steadystate capital stock.

What is the e¤ect of an increase in c on capital and output? Explain. 9 (iii) Write down the closed-form solutions for output and consumption at the

steady state.

What is the value for c that maximizes steady-state consumption? 10

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