San Francisco State University Michael BarECON 560 Fall 2013

Final Exam

Tuesday, December 17

2 hours

Name: ___________________________________ Instructions

1. This is closed book, closed notes exam.

2. No calculators of any kind are allowed.

3. Show all the calculations.

4. If you need more space, use the back of the page.

5. Fully label all graphs.

Good Luck

1

1. (10 points). The next figure shows the natural log of real GDP per capita in two countries, A and B, over the 40 year period 1971 – 2010.

a. Based on the figure (circle the correct answer), i. Country A is growing faster than country B before 1990

ii. Both countries grow at the same rate before 1990 iii. Both countries grow at the same rate after 1990 iv. Country A’s real GDP per capita is growing at accelerating rate

before 1990.

b. Based on the figure, the approximate annual growth rate of GDP per capita, in country A, over the entire period is (circle the correct answer),

i. 1% ii. 2.5%

iii. 4% iv. 5%

c. Based on your answer to part b, and given that population in country A

grows at 1% per year, find the approximate annual growth rate of real GDP in country A.

%5.3

%1%5.2

RGDP

RGDP

POPRGDPPOP

RGDP

2

2. (20 points). Consider the Solow model discussed in class, and described as

follows. Output is produced according to 10,1 tttt LKAY . Capital

evolves according to ttt IKK )1(1 , where is the depreciation rate and

tI is aggregate investment. People save a fraction s of their income. This fraction

is exogenous. Thus, the total saving and total investment in this economy are

ttt sYIS . The population of workers grows at a constant rate of n, which is

exogenous in this model. Thus, tt LnL )1(1 .

a. Solve for the steady state capital per worker, output per worker, and consumption per worker (i.e. derive the expressions for ssssss cyk ,, ).

Deriving the law of motion of capital per worker:

n

sAkk

nk

Ln

LsAK

Ln

K

L

K

ttt

t

tt

t

t

t

t

11

1

)1()1(

)1(

1

1

1

1

Using the definition of a steady state: sstt kkk 1

n

sAk

sAknk

sAkknk

n

sAkk

nk

ss

ssss

ssssss

ssssss

1

)(

)1()1(

11

1

ssss

ssss

ss

ysc

Aky

n

sAk

)1(

1

1

3

b. Derive the golden rule saving rate, GRs (the saving rate which maximizes

the steady state consumption per worker). We start by deriving the optimal capital per worker:

ssss

ssssssssk

sAkkn

ts

sAkAkAkscss

)(

..

1max

The constraint means that indeed the capital per worker is at its steady state level. Plugging the constraint into the objective, gives:

ssssssk

knAkcss

)(max

The first order condition is:

1

1

1

1

)(

0)(

n

Ak

knA

nAk

GR

GR

GR

Comparing this to the steady state capital per worker, from the previous section,

1

1

n

sAkss , implies that the golden rule saving rate is:

GRs

4

3. (10 points). Suppose that you want to measure the productivity level, A, in a given economy, under the assumption that aggregate output is Cobb-Douglas, i.e.

10,1 tttt LKAY .

a. Write the formula that you would use, and describe what data you would need for obtaining a time series on tA .

1tt

tt

LK

YA

I would need data on real GDP, tY , physical capital, tK , labor input (either number of

workers or total hours), tL , and capital share, .

b. Suppose that you plotted )ln( tA as a function of time, and the resulting

graph, with the fitted linear trend and its equation, are presented below.

y = 0.0121x - 24.201

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

1940 1950 1960 1970 1980 1990 2000 2010

ln(T

FP

)

Years

ln(TFP) in U.S.

ln(A) Linear (ln(A))

Based on the above graph, approximately, what is your estimated average growth rate of productivity (in %)? Round your answer to the nearest tenth of a percent.

Approximately, 1.2% per year.

5

4. (10 points). Suppose that the technology production function is

ALA Aˆ

where A is the level of technology, 1,0 , AL is the number of researchers in the economy and is the constant cost of technological improvement in terms of researchers.

a. According to this model, higher level of technology, all else being the same, leads to slower growth rate of technology. True/false, circle the correct answer, and briefly explain.

The term A is a decreasing function of A, which means that if the stock of technology already in existence is large, it is harder to develop new technology because scientists

need to cover a lot of material. Therefore, the term A is referred to as the "fishing-out effect" of technology production. The analogy to fishing is that after catching all the big and lazy fish in the lake, it is harder to catch new fish.

b. Suppose that A is growing at constant rate. Derive the approximate

relationship between the growth rate of productivity ( A ) and the growth

rate in the number of researchers ( AL ). Show your derivations.

)ˆ1()ˆ1(1

,

11, ALAL

ALA

ttA

ttA

Taking logs:

)ˆ1ln()ˆ1ln(0 ALA This approximately:

A

A

LA

AL

ˆˆ

ˆˆ0

6

5. (10 points). Suppose that the total output, as a function of labor, in industries 1

and 2 is given by 111 LAY and 2/1222 LAY , and the total labor available for

these two industries is 1021 LL . Also suppose that the productivities are given

by 4,1 21 AA . a. Find the efficient allocation of labor between the two industries.

Efficient allocation equalizes the marginal products in the two sectors:

6,4

2

2

1

*1

*2

2/12

2/1221

21

LL

L

LAA

MPMP LL

b. Suppose that the current allocation is 7,3 21 LL . Determine whether this allocation is (i) efficient, (ii) overallocation to sector 1, or (iii) overallocation to sector 2, and illustrate your answer graphically.

73

2LMP 1

LMP

Output loss due to overallocation to sector 2.

Efficient allocation:

4,6 *2

*1 LL

7

6. (20 points). John and Melissa work as waiters in a restaurant, and earn their income from tips. Suppose that each of them can choose one of three effort levels: {0, 1, 2}. The individual costs (in $) of these effort levels are {0, 49, 100}. The income from tips that each of them receives, depends on their individual effort levels as follows:

Individual effort level 0 1 2 Tips earned in $ (individual) 0 100 200

a. How much effort will each of them choose, and how much tips will each

of them earn, if tips earned are private (not shared with other waiters)? Explain briefly.

Individual effort level 0 1 2 Tips earned in $ (individual) 0 100 200Individual cost 0 49 100Individual profit (Revenue – Cost) 0 51 100

The highest individual profit earned is with effort level of 2, and each of them will earn $200 in tips.

b. Now suppose that the restaurant policy is that all waiters share the tips earned equally, regardless of how much tips each of them earned. How much effort will each of them choose, and how much tips will each of them earn, under this policy? To answer this question you are required to construct the payoff matrix and use the concept of Nash Equilibrium.

Melissa 0 1 2

John 0 0, 0 50, 1 100, 0 1 1, 50 51, 51 101, 50 2 0, 100 50, 101 100, 100

There is only one Nash Equilibrium in the above game: {1, 1}, which means that each of them will work at effort level 1. Notice that 1 is a dominant strategy for both players, and therefore it is always a best response. As a result, each of them will earn $100 worth of tips, instead of $200 under the policy where tips are not shared.

8

c. What do we learn from the above example about the inefficiencies embodied in communal (socialist) industry?

With shared tips, workers have incentives to shirk (free ride). If John works hard (at effort level 2), Melissa has an incentive to only work at effort level 1. And vise versa. Each worker in a communal industry (e.g. "The Great Leap Forward" plan by Mao), benefits from efforts of others, and therefore has a negative incentive to work hard.

9

7. (20 points). Suppose that the quantity of fish that grows each year in Lake

Superior depends on the existing stock of fish as follows:

1000

5.0500 ttt

SSG

.

a. Calculate the optimal stock of fish in the lake. The optimal stock maximizes growth, and thus allows for maximal harvest. To find it, we solve:

25.05005.0500max SSSSS

The first order condition is:

500

0500*

S

S

b. Calculate the maximum sustainable yield. The maximum sustainable yield:

125

1000

5005.0500500

1000

5.0500 ***

SSG

10

c. Suppose that the current stock is 200tS and the current harvest (fish

caught) is 100tH . Calculate the next period stock of fish in the lake,

1tS .

The growth in the current period is:

80

1000

2005.0500200

1000

5.0500

tt

tSS

G

The next period stock is therefore:

180100802001

tttt HGSS

Thus, the next period stock is smaller than this period.

d. Suppose that due to overfishing, the government wants to reduce the number of fishermen in the lake. Suggest a policy that will achieve this goal.

Fishing permit. It increases the cost of fishing, and reduces the number of people fishing in the lake.

e. What is the difficulty with applying similar policy you suggested in the last section to combat global warming?

Atmosphere, as opposed to most lakes, does not belong to a single country. Thus, a permit that reduces pollution and abuse of atmosphere requires coordination and agreement of many countries.