Econ Exam Vault (UCSD)

8/18/2019 Econ Exam Vault (UCSD)

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ECON 110A Final Exam A Solutions Name________________________

Prof. Levkoff PID #_________________________

Fall Quarter 2014 Seat #________________________

Directions: You will have the full three hours to complete the exam after everyone has been

seated and the exam distributed. The exam is 12 pages (front and back). Additional scratch

paper is provided in the back of the exam. There are two parts- one short answer/free response

section and one problem solving/analysis section. To receive full credit, SHOW ALL WORK and

graphs where appropriate and CLEARLY BOX YOUR FINAL ANSWERS for any calculations.

Make sure your name, PID #, and seat # (for the seat you are physically sitting in) are on the

examination sheet before submitting your exam.


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c) (7 points) The bathtub model of unemployment is a model of structural unemployment

used to explain the steady state natural rate of unemployment and suggests that the

long run rate of unemployment is increasing in the rate of job separation and decreasing

in the rate of job finding.

ANSWER: The “bathtub” model of unemployment is a model of frictional (NOT

structural, which

recall

is

caused

by

wage

rigidity)

unemployment.

The

remainder

of

the statement regarding the steady state rate and the effects of an increase in the rate

of job separation and a decrease in the rate of job finding since the steady state

unemployment rate in the bathtub model is U/L=s/(s+f).

3 points for identifying that structural is wrong and that frictional is correct

4 points for either proving or identifying the s.s level of U/L for the bathtub model

that verifies the second half of the statement.

d)

(7 points)

The

key

assumption

driving

output

per

capita

towards

the

steady

state

in

the

Solow growth model with population growth is the increasing marginal product of

capital per capita.

ANSWER: False. The statement would be correct if instead it read “…decreasing

marginal product of capital per capita.”

7 points for either identifying in written argument (as above) what the correct version

of the statement would be OR showing on a diagram a concave vs. convex production

function and arguing that we used the concave one in the Solow model. If students

also argue that it is depreciation that is the key opposing force to savings/investment

that drives us to the steady state in the Solow model, (which it would regardless of

returns to

scale

for

many

specifications),

this

is

also

an

acceptable

answer

for

full

credit.

2 points if students try and suggest the statement would be correct if we either

switched Solow to Romer or to Combined Solow ‐Romer since the Romer model

doesn’t use capital and the combined model still utilizes diminishing marginal product

to capital (since the exponent on K is 1/3, capital’s share of income).


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Part II: Analysis / Problems

#1) (32 points) Consider the following variation of the Romer model where all exogenous

variables have a above the corresponding parameter: The production technology for aggregate output is defined by

/

,.

The law

of

motion

for

the

stock

of

ideas is

given

by

∆ ̅, .

The resource constraint for labor is given by , , . The fraction of total labor allocated directly to the production ideas is given by .̅ The economy is endowed with an initial stock of ideas given by .

a) (4 points) Use the law of motion for the stock of ideas to write an expression for as a function of the initial stock of ideas , time t , and the growth rate of the stock of ideas, . How does depend on only the exogenous model parameters? Explain.

ANSWER: Dividing the law of motion for the stock of ideas by yields the growth rate

,

̅

. So since we know that

grows geometrically at a rate of

, then given it’s initial stock of , we can compute the value of the stock of ideas at any given time by ̅ so that an increase in , ,̅ or would increase the growth rate of . Increases in the initial stock of knowledge increase the level of . 1 point for correct growth rate as a function of exogenous parameters

2 points for correct formula for given geometric growth rate 1 point for explaining at least 3 of the 4 exogenous effects.

b)

(6

points)

Use

your

answer

from

a)

to

write

an

expression

for

output

per

capita

as

a

function of all of the exogenous model parameters and t , time. Which exogenous

variables exhibit growth effects on output per capita? Which ones exhibit level

effects?

ANSWER: Substituting in for into the production technology using our answer from a) and the remaining exogenous model parameters recalling that , ̅, we find that output per capita is

̅ / ̅

, ,̅ and exhibit growth effects. only exhibits a level effect on . ̅also exhibits a level effect on

(so it exhibits both a growth and level effect).

3 points for correctly writing out expression for output per capita – deduct one

point if instead, student writes it as total output forgetting to divide by 1 point for correct growth effects

1 point for correct level effect

1 point for identifying that ̅exhibits both


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c) (4 points) What is the rate of growth of output per capita as a function of the

exogenous model parameters? Explain.

ANSWER: By taking the log‐time derivative of the production technology and

writing everything out in terms of growth rates noting that there is no growth in

the labor force , we find that %∆ %∆ %∆, %∆ % ̅ %∆

where the last equality follows from the fact that the labor force is fixed.

4 points for correct growth rate in output per capita (if they fail to distinguish, still

give them full credit since the answer is the same anyhow since is fixed). 2 points if the miss the ¼ but do attempt to do the growth accounting above

1 point if they miss the ¼ because they forget to do the growth accounting above

For the remainder of the problem, we make two minor adjustments to augment the Romer model with

some elements

of

the

Solow

model

with

capital

() accumulation. Namely, make the following two changes / additions:

The production technology is now changed to //,/

Add the law of motion for aggregate capital given by ∆ ̅ ̅ d)

(6 points) Use the law of motion for aggregate capital to find a relationship for the

capital‐to‐output ratio along a balanced growth path (BGP) as a function of only

exogenous model parameters by assuming that the growth rate of capital is equal to

̅. ANSWER: We are told to assume that ∗

∗ (by dividing through by K to

get the

growth

rate

on

the

left

hand

side

of

the

law

of

motion).

Note

that

the

left

hand side being constant along a BGP implies that the right hand side of the

equation is constant also, implying that the capital to income ratio ∗∗

must

also be constant.

No partial credit

e) (6 points) Use your answers from a)‐d) to find an expression for output per capita

as a function of all of the exogenous model parameters,

̅, and time, t , in the

combined modification

of

the

Romer

‐Solow

model.

ANSWER: The answer to the previous part implies the following relationship

between capital and output along a BGP:

∗ ∗ We can substitute this into the new production technology along with our

expression for and , just like we did in part b to find that


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∗ ̅ / ∗/

̅/ ∗/ ̅ /

/ ̅/

∗ ̅ / ∗/ ̅ Dividing through by yields

∗ ∗

̅ / ∗

/ ̅

3 points for correctly substituting into the production function for capital along

the BGP

3 points given an error in the previous part, correctly find output per capita –

deduct one point again if they forget to divide through by .

f) (6 points) What is the growth rate of output per capita along the BGP in the

combined modification of the Solow‐Romer model expressed as functions of only

exogenous parameters? By what factor does this growth rate exceed the growth

rate in the Romer model without the two minor adjustments? Explain.

ANSWER: Again by growth accounting we can find that, using the new production

technology,

%∆ %∆ %∆

%∆,

̅

% %∆

Where again, the last equality follows from the fact that there is no growth in . Recalling

that

along

a BGP,

the

ratio

of

Y/K

was

constant,

so

that

Y and

K

must

also

be growing at the same rates. This implies that %∆ ≡ so that the equation above becomes

̅

So that

̅ ∗

̅

̅

So that the growth rate of output per capita in the combined model exceeds that

of the Romer model without capital accumulation by a factor of (3/8)/(1/4)=3/2,

which is a 50% increase in the growth rate.

3 points for computing the correct growth rate using the growth accounting

3 points for identifying that it exceeds the original value by a factor of (3/2) or an

increase in 50% of the original model’s growth rate (either is fine)


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#2) (20 points) Consider an individual that lives for only two periods, t and t+1. The consumer,

as a rational, forward looking individual, wants to maximize the subjectively discounted lifetime

utility given by the expression ̅, where ̅ is an exogenous parameter.

The instantaneous utility in a given period is given by the function /

The consumer is endowed with income

and

in periods t and t+1,

respectively.

The rate of return on the amount of income saved, s, from period t to t+1 is

given by ̅. a) (4 points) Write out the resource (budget) constraints for the individual for each

period.

ANSWER:

Period t: Period t+1:

2 points for each correct constraint

Also give

full

credit

if

someone

uses

the

lifetime

budget

constraint

by

eliminating

s

b)

(4 points) Write out the individual’s inter temporal optimization problem as a

function of the amount of income saved, s, by using your answers from a).

ANSWER: The agent solves

, subject to the constraints in a) Substituting the constraints directly into the objective function, we find that the

agent’s lifetime utility maximization problem can be re‐written as

/ / where the second equation follows from directly substituting the constraints from

a) into the objective function.

4 points for correctly writing out the maximization problem as a function of the

savings rate s (even if they didn’t use the functional form of the utility function

provided)

2 points if they only write the optimization problem out as function of

consumption in each period.

c) (6 points) Derive the first order condition with respect to savings, s, and interpret

your results.

Explain

the

intuition.

(hint:

it

may

be

useful

to

consider

the

roles

of

and in the explanation) ANSWER: The first order condition with respect to s, the quantity saved in the

present, is given by

/


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Following the hint, we can substitute back in for the consumption values using the

constraints in the reverse direction and note that the first order condition is also

equivalent to

This is the Euler ‐consumption equation that explains the optimal savings behavior.

It suggests

that

the

individual

optimally

allocates

his

consumption

until

the

marginal utility of what is given up today (left hand side) is just offset by the

subjectively discounted future return on the marginal utility gained in the future

(right hand side). That is, the individual is optimally allocating his consumption

across periods if he is just indifferent between saving the last unit for the future or

consuming it today.

3 points for the correct FOC

3 points for the correctly identifying and interpreting the Euler equation (even if

they didn’t use the hint, they should have been able to explain this idea)

d)

(6 points) Compute the optimal value of savings, ∗. How does this value depend on ̅, , and ? Explain the intuition. ANSWER: By simplifying the first order condition (the Euler equation), and using

the constraints, we can find that

∗

Note that as increases (as the agent becomes more patient), the first fraction gets bigger while the second fraction (what is being subtracted) gets smaller,

resulting in

the

entire

savings

getting

larger.

This

should

make

sense

–

as

the

agent becomes more patient and places more subjective weight on the future,

he/she will save more. The same effect is true for present income. The more the agent is endowed with in the present, the more he will be able to save for the

future. The opposite is true for future income. If the agent expects to receive a

higher future income endowment of , then he will not need to save as much to consume the same amount in the future.

3 points for the correct savings function (doesn’t have to be identically simplified)

1 point for the correct effect of each of the 3 exogenous parameters (3 points

total)


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#3) (20 points) Using the terms velocity , inflation, output and money supply , explain what

theory explains the phenomena observable in the data below. In what time frame does the

theory apply? Explain.

ANSWER: The phenomena observable in the data above can be explained by the quantity

theory of money described by the quantity equation

since velocity remains fairly constant. Taking the log‐derivative and re‐writing in terms of

growth rates, this implies that ≡ %∆ %∆ %∆ so that any growth in the money supply in excess of the growth in output will translate into long run inflation. (same

argument can be made if we assume money is neutral in the long run an so that

%∆

resulting from

a change

in

the

money

supply

due

to

the

classical

dichotomy)

2 points for mentioning each of those keywords (8 points total)

4 points for writing out the correct quantity theory relationship MV=PY (in level of

percentage change form regardless of what is assumed to be constant provided it is

explained)

4 points for identifying the relationship between growth rate in the money supply and the

inflation rate

4 points for mentioning that the theory only applies in explaining long run phenomena


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SCRATCH PAPER

(DO NOT DETATCH)


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SCRATCH PAPER

(DO NOT DETATCH)


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Part II Total___________________/72

#1) Total______________________/32

a)___________________________/4

b)___________________________/6

c)___________________________/4

d)___________________________/6

e)___________________________/6

f)___________________________/6

#2) Total_____________________/20

a)___________________________/4

b)___________________________/4

c)___________________________/6

d)___________________________/6

#3) Total_____________________/20

SCORING RUBRIC

(FOR INSTRUCTOR/TA USE ONLY)

Exam Score____________________/100

Part I Total____________________/28

a)___________________________/7

b)___________________________/7

c)___________________________/7

d)___________________________/7


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ECON 110A Final Exam B Solutions Name________________________

Prof. Levkoff PID #_________________________


Directions: You will have the full three hours to complete the exam after everyone has been

seated and the exam distributed. The exam is 12 pages (front and back). Additional scratch







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Part I: Evaluate the validity of each of the following statements using graphs, equations, and written

arguments where appropriate. (28 points)

a) (7 points) The key assumption driving output per capita towards the steady state in the

Solow growth model with population growth is the increasing marginal product of

capital per capita.

ANSWER: False. The statement would be correct if instead it read “…decreasing

marginal product of capital per capita.”

7 points for either identifying in written argument (as above) what the correct version

of the statement would be OR showing on a diagram a concave vs. convex production

function and arguing that we used the concave one in the Solow model. If students

also argue that it is depreciation that is the key opposing force to savings/investment

that drives us to the steady state in the Solow model, (which it would regardless of

returns to scale for many specifications), this is also an acceptable answer for full

credit.

2 points

if

students

try

and

suggest

the

statement

would

be

correct

if

we

either

switched Solow to Romer or to Combined Solow ‐Romer since the Romer model

doesn’t use capital and the combined model still utilizes diminishing marginal product

to capital (since the exponent on K is 1/3, capital’s share of income).

b)

(7 points) Computing the growth rate of the consumer price index involves holding

prices fixed

and

allowing

the

quantities

purchased

to

fluctuate

between

years.

ANSWER: This is not the case. Calculating the growth rate in the CPI involves holding

the quantities fixed in a base year basket and allowing the prices to fluctuate. If

instead, the statement had replaced the CPI with real GDP , then the statement would

be valid.

7 points for both explanations

5 points for only one of the following explanations


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c) (7 points) The bathtub model of unemployment is a model of structural unemployment

used to explain the steady state natural rate of unemployment and suggests that the

long run rate of unemployment is increasing in the rate of job separation and decreasing

in the rate of job finding.

ANSWER: The “bathtub” model of unemployment is a model of frictional (NOT

structural, which

recall

is

caused

by

wage

rigidity)

unemployment.

The

remainder

of

the statement regarding the steady state rate and the effects of an increase in the rate

of job separation and a decrease in the rate of job finding since the steady state

unemployment rate in the bathtub model is U/L=s/(s+f).

3 points for identifying that structural is wrong and that frictional is correct

4 points for either proving or identifying the s.s level of U/L for the bathtub model

that verifies the second half of the statement.

d) (7 points) The loanable funds model predicts that a reduced budget deficit will increase

investment in equilibrium.

ANSWER: This is fairly accurate. Recall that in the loanable funds model, the supply

curve is national savings. Reducing the budget deficit means increasing national

savings (through the increase in government savings, the deficit reduction) which puts

downward pressure

on

the

real

interest

rate,

increasing

the

equilibrium

quantity

of

investment.

7 points for correct answer (either written / diagrammatic, or some combination of)

6 points for the correct answer but fail to mention the interest rate effect

3 points if they get the direction of the savings shift or the interest rate movement (or

both) wrong, but otherwise answer is consistent


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Part II: Analysis / Problems

#1) (20 points) Using the terms nominal interest rate, real interest rate, inflation rate and the

money supply , explain the causal effect observable in the data below. In what time frame does

the

theory

apply?

Explain.

ANSWER: The phenomena observable in the data above can be explained by the Fisher

effect :

For a particular real interest rate (which is determined by savings and investment in the

capital / loanable funds market), the Fisher equation implies that nominal interest rates

and inflation rates should move very closely together in the long run, when the inflation

rate

is

determined

by

the

growth

rate

of

the

money

supply.

2 points for mentioning each of those keywords (8 points total)

4 points for writing out the correct Fisher equation (if students provide the exact formula

including the extra → term, that is fine)

4 points for identifying the relationship between inflation rate and nominal interest rate

4 points for mentioning that the theory applies to the long run trend of these variables


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#2) (20 points) Consider an individual that lives for only two periods, t and t+1. The consumer,

as a rational, forward looking individual, wants to maximize the subjectively discounted lifetime

utility given by the expression ̅, where ̅ is an exogenous parameter.

The instantaneous utility in a given period is given by the function /

The consumer is endowed with income

and

in periods t and t+1,

respectively.

The rate of return on the amount of income saved, s, from period t to t+1 is

given by ̅. a) (4 points) Write out the resource (budget) constraints for the individual for each

period.

ANSWER:

Period t: Period t+1:

2 points for each correct constraint

Also give

full

credit

if

someone

uses

the

lifetime

budget

constraint

by

eliminating

s

b) (4 points) Write out the individual’s inter temporal optimization problem as a

function of the amount of income saved, s, by using your answers from a).

ANSWER: The agent solves

, subject to the constraints in a) Substituting the constraints directly into the objective function, we find that the

agent’s

lifetime

utility

maximization

problem

can

be

re‐

written

as

/ / where the second equation follows from directly substituting the constraints from

a) into the objective function.

4 points for correctly writing out the maximization problem as a function of the

savings rate s (even if they didn’t use the functional form of the utility function

provided)

2 points if they only write the optimization problem out as function of

consumption in each period.


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c) (6 points) Derive the first order condition with respect to savings, s, and interpret

your results. Explain the intuition. (hint: it may be useful to consider the roles of and in the explanation) ANSWER: The first order condition with respect to s, the quantity saved in the

present, is given by

/ Following the hint, we can substitute back in for the consumption values using the

constraints in the reverse direction and note that the first order condition is also

equivalent to

This is the Euler ‐consumption equation that explains the optimal savings behavior.

It suggests that the individual optimally allocates his consumption until the

marginal utility of what is given up today (left hand side) is just offset by the

subjectively discounted future return on the marginal utility gained in the future

(right hand side). That is, the individual is optimally allocating his consumption

across periods

if

he

is

just

indifferent

between

saving

the

last

unit

for

the

future

or

consuming it today.

3 points for the correct FOC

3 points for the correctly identifying and interpreting the Euler equation (even if

they didn’t use the hint, they should have been able to explain this idea)

d) (6 points) Compute the optimal values of value of savings, ∗. How does this value depend on ̅, , and ? Explain the intuition. ANSWER: By simplifying the first order condition (the Euler equation), and using

the

constraints,

we

can

find

that

∗

Note that as increases (as the agent becomes more patient), the first fraction gets bigger while the second fraction (what is being subtracted) gets smaller,

resulting in the entire savings getting larger. This should make sense – as the

agent becomes more patient and places more subjective weight on the future,

he/she will save more. The same effect is true for present income. The more the agent is endowed with in the present, the more he will be able to save for the

future. The opposite is true for future income. If the agent expects to receive a

higher future income endowment of , then he will not need to save as much to consume the same amount in the future.

3 points for the correct savings function (doesn’t have to be identically simplified)

1 point for the correct effect of each of the 3 exogenous parameters (3 points

total)


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#3) (32 points) Consider the following variation of the Romer model where all exogenous

variables have a above the corresponding parameter: The production technology for aggregate output is defined by /, . The law of motion for the stock of ideas is given by ∆ ̅, .

The resource

constraint

for labor

is

given

by

, , . The fraction of total labor allocated directly to the production ideas is given by .̅ The economy is endowed with an initial stock of ideas given by .

a) (4 points) Use the law of motion for the stock of ideas to write an expression for as a function of the initial stock of ideas , time t , and the growth rate of the stock of ideas, . How does depend on only the exogenous model parameters? Explain.

ANSWER: Dividing the law of motion for the stock of ideas by yields the growth rate , ̅. So since we know that grows geometrically at a rate of , then given it’s initial stock of , we can compute the value of the stock of ideas at any given time by

̅ so that an increase

in , ̅, or would increase the growth rate of . Increases in the initial stock of knowledge increase the level of . 1 point for correct growth rate as a function of exogenous parameters

2 points for correct formula for given geometric growth rate 1 point for explaining at least 3 of the 4 exogenous effects.

b) (6 points) Use your answer from a) to write an expression for output per capita as a

function of all of the exogenous model parameters and t , time. Which exogenous

variables exhibit

growth

effects

on

output

per

capita?

Which

ones

exhibit

level

effects?

ANSWER: Substituting in for into the production technology using our answer from a) and the remaining exogenous model parameters recalling that , ̅, we find that output per capita is

̅ / ̅

, ,̅ and exhibit growth effects. only exhibits a level effect on . ̅also exhibits a level effect on (so it exhibits both a growth and level effect). 3 points for correctly writing out expression for output per capita – deduct one

point if instead, student writes it as total output forgetting to divide by 1 point for correct growth effects

1 point for correct level effect

1 point for identifying that ̅exhibits both


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c) (4 points) What is the rate of growth of output per capita as a function of the

exogenous model parameters? Explain.

ANSWER: By taking the log‐time derivative of the production technology and

writing everything out in terms of growth rates noting that there is no growth in

the labor force , we find that %∆ %∆ %∆, %∆ % ̅ %∆

where the last equality follows from the fact that the labor force is fixed.

4 points for correct growth rate in output per capita (if they fail to distinguish, still

give them full credit since the answer is the same anyhow since is fixed). 2 points if the miss the 1/5 but do attempt to do the growth accounting above

1 point if they miss the 1/5 because they forget to do the growth accounting

above

For the remainder of the problem, we make two minor adjustments to augment the Romer model with

some elements of the Solow model with capital () accumulation. Namely, make the following two changes / additions:

The production technology is now changed to //,/ Add the law of motion for aggregate capital given by ∆ ̅ ̅

d) (6 points) Use the law of motion for aggregate capital to find a relationship for the

capital‐to‐output ratio along a balanced growth path (BGP) as a function of only

exogenous model parameters by assuming that the growth rate of capital is equal to

̅. ANSWER: We are told to assume that ∗∗ (by dividing through by K to get the growth rate on the left hand side of the law of motion). Note that the left

hand side being constant along a BGP implies that the right hand side of the

equation is constant also, implying that the capital to income ratio ∗∗

must

also be constant.

No partial credit

e) (6 points) Use your answers from a)‐d) to find an expression for output per capita as

a function of all of the exogenous model parameters, ̅, and time, t , in the combined modification of the Romer‐Solow model.

ANSWER: The answer to the previous part implies the following relationship

between capital and output along a BGP:

∗ ∗


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We can substitute this into the new production technology along with our

expression for and , just like we did in part b to find that ∗ ̅ / ∗

/ ̅/

∗/ ̅ / /

̅/ ∗ ̅ / ∗

/ ̅

Dividing through by yields ∗

∗ ̅

/ ∗/

̅

3 points for correctly substituting into the production function for capital along

the BGP

3 points

given

an

error

in

the

previous

part,

correctly

find

output

per

capita

–

deduct one point again if they forget to divide through by .

f) (6 points) What is the growth rate of output per capita along the BGP in the

combined modification of the Solow‐Romer model expressed as functions of only

exogenous parameters? By what factor does this growth rate exceed the growth

rate in the Romer model without the two minor adjustments? Explain.

ANSWER: Again by growth accounting we can find that, using the new production

technology,

%∆ %∆ %∆ %∆, ̅ % %∆

Where again, the last equality follows from the fact that there is no growth in . Recalling that along a BGP, the ratio of Y/K was constant, so that Y and K must also

be growing at the same rates. This implies that %∆ ≡ so that the equation above becomes

̅

So that

̅ ∗

̅

̅

So that the growth rate of output per capita in the combined model exceeds that

of the

Romer

model

without

capital

accumulation

by

a factor

of

(3/10)/(1/5)=3/2,

which is a 50% increase in the growth rate.

3 points for computing the correct growth rate using the growth accounting

3 points for identifying that it exceeds the original value by a factor of (3/2) or an

increase in 50% of the original model’s growth rate (either is fine)


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SCRATCH PAPER

(DO NOT DETATCH)


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SCRATCH PAPER

(DO NOT DETATCH)


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Part II Total___________________/72

#1) Total______________________/20

#2) Total_____________________/20

a)___________________________/4

b)___________________________/4

c)___________________________/6

d)___________________________/6

#3) Total______________________/32

a)___________________________/4

b)___________________________/6

c)___________________________/4

d)___________________________/6

e)___________________________/6

f)___________________________/6

SCORING RUBRIC


Exam Score____________________/100

Part I Total____________________/28

a)___________________________/7

b)___________________________/7

c)___________________________/7

d)___________________________/7


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ECON 110A Midterm Exam 2A Solutions Name________________________

Prof. Levkoff PID #_________________________


Directions: You will have an hour and twenty minutes to complete the exam after everyone has

been seated and the exam distributed. The exam is 10 pages (front and back). Additional scratch







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a) (10 points) The Solow model predicts that poorer countries should grow relatively faster than

wealthier

countries

(in

terms

of

per

capita

income).

However,

since

we

still

observe

some

poor

economies that grow slowly and rich economies that grow quickly, the Solow growth model

must be making an incorrect prediction.

ANSWER: This is not the case. The Solow model predicts conditional convergence – that each

country will grow towards its own unique steady state. The statement would be correct if it

also made the qualification, ”…all other things equal…” (or ceteris paribus). If countries don’t

have the same underlying economic climate (described by the exogenous factors in our

model), then their economies will grow to different steady states at different speeds.

3 points for mentioning conditional convergence

7 points for mentioning that Solow makes a ceteris paribus assumption (or the equivalent)

b) (10 points) In the Solow model with population growth, the consumption function exhibits

diminishing marginal propensity to consume.

ANSWER: False. The consumption function for the Solow model is just . So the consumption function is linear in income implying that the MPC is a constant and equal to 1‐s.

3 points for correctly writing out the consumption function (or drawing it)

6 points for correctly arguing that it does not exhibit diminishing MPC in income

1 point

if

they

mention

that

it

exhibits

constant

MPC


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3

c) (10 points) The Solow model with population growth and technological progress predicts that,

in the long run steady state, an economy’s ratio of capital to income will be approximately

constant.

ANSWER: This is true. Y/L grows at a rate of g in the model, the rate of exogenous

technological progress.

So

does

K/L.

Thus,

the

ratio

of

Y/L

to

K/L,

or

Y/K

must

not

be

growing

at all.

5 points for correctly identifying that Y/L and K/L grow at the same rate

5 points for arguing that this implies that Y/K must be constant in the model.

Part II: Problems & Analysis (70 points)

#1) (21 points) Consider the following data on the growth rate of worldwide income (GDP) per capita:

Worldwide Income Per Capita

Years Annualized Growth Rate

0‐1700 0.0%

1700‐1820 0.1%

1820‐1913 0.9%

1913‐1950 0.9%

1950‐1970 2.8%

1970‐1990 1.3%

1990‐2014 1.9%

2014‐2050 2.5%

2050‐2070 1.5%

2070‐2100 1.2%

a) (7 points) If the current worldwide average income per capita is approximately $14,000

(measured in 2014 base year US dollars), compute the approximate level of the worldwide

average income in year zero (in 2014 base year US dollars) as precisely as possible.

ANSWER: Since there is 0% growth from 0‐1700, all we really need to do is find the level of

income per person in 1700 by applying the growth rates backwards:

/ ≅ / $,..... $,. (in 2014 dollars).

Deduct 2 points if the methodology looks correct, but there is a small discrepancy counting

the years.

Deduct 2 more points if they use the wrong growth rates.

No partial credit otherwise if both of these mistakes are made


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4

b) (7 points) If the future forecasts provided are accurate, compute a projection for the worldwide

average income per capita in 2100 (measured in 2014 base year dollars) as precisely as possible.

ANSWER: Now we just project the current $14,000 income forward to the year 2100:

/ $,... $,. (in 2014 dollars).


the years.



c) (7 points) If the growth projections are instead, expected to remain at 2.5% through 2100, by

what factor will the projected worldwide average income in 2100 (measured in current 2014

dollars) increase relative to your answer in b)?

ANSWER: Now we just project the current $14,000 income forward to the year 2100 assuming

a constant rate of 2.5% and compare the resulting figure with our answer from b):

/ $,. $,. (in 2014 dollars). This is precisely $, . / $ , . . times as great as when we projected that growth would slow down. That is, if income growth remains at around 2.5%, the standard of living

would be almost double (1.78) what it would be with the predicted slowdown.


the years.


No partial

credit

otherwise

if

both

of

these

mistakes

are

made

AND

the

students

get

the

factor

incorrect (if the factor is wrong because of a carry through error but otherwise, would have

been correct, do not deduct only 1 more point from the previous 4 that were deducted for

making multiple math errors).


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#2) (49 points) Consider the baseline Solow growth model that also incorporates population

growth.

The

production

technology

is

given

by , . The population grows at an

exogenous rate of 1%. The depreciation rate on capital is exogenous and equal to 9%. Assume

consumers save a constant exogenous fraction of their income in any given period.

a)

(7

points)

Derive

the

per

capita

production

function where / and /. What property of the production technology guarantees that you can do this? Explain.

ANSWER: , , /. We could only do this because the

technology exhibits constant returns to scale.

7 points for correct production function and mentioning CRS

4 points for only providing one of these

b) (7 points) Derive the law of motion for the capital stock per capita, ∆, as a function of only capital per capita, . If we were to give this economy some amount of starting capital, , and then let it grow towards its steady state through the capital accumulation process, what

value of would induce the highest growth rate as the country converges towards its steady state? (hint: use your work from b), to find the value as a function of the savings rate,

).

ANSWER:

∆

.. If we were to start the economy off with

some initial amount of capital, , on a trajectory that will provide the highest initial growth in k (the correspondingly largest / maximal value for ∆), then we just need to find the level of k that maximizes ∆ by solving:

∆ .

The corresponding first order condition for the maximization problem yields

∆

. Which can be solved for /.

4

points

for

the

law

of

motion

3 points for correctly optimizing it to find k_0

c) (7 points) Compute the steady state values ∗, ∗, and ∗ using your answer from b).


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6

ANSWER: Using our law of motion for capital and setting it to zero (imposing the steady

state condition), we find that the steady state level of capital ∗ must satisfy ∆ . ∗ /

Substituting this into the production function yields

∗ ∗ / Substituting this into the consumption function yields

∗ ∗ ∗ /

3 points for correct k* (if done correctly with a carry through error from the law of

motion, award credit)

2 points for correct y* (if done correctly with a carry through error from calculating k*,

award credit)

2 points for correct c* (if done correctly with a carry through error from calculating y*,

award credit)

d)

(7

points)

Compute

the

value

of

the

savings

rate, ∗, that maximizes the steady state

consumption profile ∗ by solving the problem, ∗. Show your work!

From the previous part, we found that

∗ ∗ ∗ / Now all we need to do is maximize this expression with respect to s by solving:

∗

Where the second equality above comes from noting that we can pull out the constant 10

and it will not affect the optimization problem (if you keep it, you will see that it will

cancel out anyway once you take the first order condition).

∗

5 points for writing out the optimization problem and taking the first order conditions

correctly

2 points for finding the correct value (I proved in general, that this value will be equal to

capital’s share

of

income

and

some

students

may

try

and

circumvent

actually

solving

the

optimization problem to prove this – in this event, award these 3 points if they adequately

explain that we proved that capital’s share will equal the optimal savings rate in the Cobb‐

Douglas case, but DO NOT award the 4 points since they did not follow the directions)

e)

(7

points)

What

other

condition

must ∗∗ satisfy if consumption is maximized in the

steady

state?

Explain

using

a

diagram.


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7

ANSWER: If we are using the golden rule, then ∗∗ must also satisfy (no need to verify that it actually does – we know it must).

Note that the “depreciation line” is just our break even investment function which also

accounts for population growth in the case where it is not zero.

7 points for explicitly mentioning the condition and drawing the diagram

5 points for only doing one of these

f) (7 points) If instead, the aggregate production function is given by , , for what values of the savings rate will the economy grow forever? At what rate will capital and

output (per capita) grow? Explain. (hint: a diagram may help you here)

ANSWER: With the new production function, our law of motion for par capital per capita

changes slightly since we now are using . The new law of motion for the capital stock is given by

∆ . Dividing through by k and recognizing that y=k implies %∆ %∆ lets us re‐write the law of motion in terms of growth rates:

%∆ ∆

. %∆ Thus, the economy will exhibit positive perpetual growth if and only if %. The rate of growth above is determined by the amount which the savings rate exceeds the net

depreciation and

population

growth

rates.

g)

(7

points)

Explain

why

the

adjustment

to

the

model

in

f)

provides

a

more

accurate

prediction

in

terms

of

what

we

would

expect

to

happen

to

income

and

the

real

wage

in

the


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8

long run in comparison to the production model in the original question, which faced

diminishing marginal returns to capital? Assume that the input markets are in equilibrium.

ANSWER: The model in f) incorporates endogenous growth, whereby the economy never

reaches a steady state provided savings is sufficiently high. This is a much more realistic

description of

actual

phenomena

compared

to

the

earlier

parts

of

the

question

that

has

modeled production in a way such that the economy slows down to a steady state where

K/L doesn’t grow (due to diminishing returns to k).

If the real wage tracks average labor productivity (Y/L) and Y is growing forever, then we

would expect this model also predicts that the real wage should also grow. In the

previous parts of the problem where we use the Solow model without endogenous

growth (assuming diminishing returns to k), the steady state growth rate of Y/L is zero,

predicting absolutely zero growth in the real wage (unrealistic).

SCRATCH PAPER

(DO NOT DETATCH)


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9

SCORING RUBRIC


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10

Part II Total___________________/70

#1) Total______________________/21

a)___________________________/7

b)___________________________/7

c)___________________________/7

#2) Total_____________________/49

a)___________________________/7

b)___________________________/7

c)___________________________/7

d)___________________________/7

e)___________________________/7

f)___________________________/7

g)___________________________/7


Exam Score____________________/100

Part I Total____________________/30

a)___________________________/10

b)___________________________/10

c)___________________________/10


35/85

1

ECON 110A Midterm Exam 2B Solutions Name________________________

Prof. Levkoff PID #_________________________










36/85

2



a) (10 points) The Solow model predicts that poorer countries should grow relatively faster than

wealthier

countries

(in

terms

of

per

capita

income).

However,

since

we

still

observe

some

poor

economies that grow slowly and rich economies that grow quickly, the Solow growth model

must be making an incorrect prediction.

ANSWER: This is not the case. The Solow model predicts conditional convergence – that each

country will grow towards its own unique steady state. The statement would be correct if it

also made the qualification, ”…all other things equal…” (or ceteris paribus). If countries don’t

have the same underlying economic climate (described by the exogenous factors in our

model), then their economies will grow to different steady states at different speeds.

3 points for mentioning conditional convergence

7 points for mentioning that Solow makes a ceteris paribus assumption (or the equivalent)

b) (10 points) The Solow model with population growth and technological progress predicts that,

in the long run steady state, an economy’s ratio of capital to income will be approximately

constant.

ANSWER: This is true. Y/L grows at a rate of g in the model, the rate of exogenous

technological progress. So does K/L. Thus, the ratio of Y/L to K/L, or Y/K must not be growing

at all.

5 points

for

correctly

identifying

that

Y/L

and

K/L

grow

at

the

same

rate

5 points for arguing that this implies that Y/K must be constant in the model.


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3

c) (10 points) In the Solow model with population growth, the consumption function exhibits

diminishing marginal propensity to consume.

ANSWER: False. The consumption function for the Solow model is just . So the consumption function is linear in income implying that the MPC is a constant and equal to 1‐s.

3 points for correctly writing out the consumption function (or drawing it)

6 points for correctly arguing that it does not exhibit diminishing MPC in income

1 point if they mention that it exhibits constant MPC


#1) (49 points) Consider the baseline Solow growth model that also incorporates population

growth. The production technology is given by , . The population grows at an exogenous rate of 1%. The depreciation rate on capital is exogenous and equal to 4%. Assume

consumers save a constant exogenous fraction of their income in any given period. a) (7 points) Derive the per capita production function where / and /

. What property of the production technology guarantees that you can do this? Explain.

ANSWER: , , /. We could only do this because the

technology exhibits constant returns to scale.

7 points for correct production function and mentioning CRS

4 points for only providing one of these


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4

b) (7 points) Derive the law of motion for the capital stock per capita, ∆, as a function of only capital per capita, . If we were to give this economy some amount of starting capital, , and

then

let

it

grow

towards

its

steady

state

through

the

capital

accumulation

process,

what

value

of would induce the highest growth rate as the country converges towards its

steady state? (hint: use your work from b), to find the value as a function of the savings rate,

). ANSWER: ∆ .. If we were to start the economy off with some initial amount of capital, , on a trajectory that will provide the highest initial growth in k (the correspondingly largest / maximal value for ∆), then we just need to find the level of k that maximizes ∆ by solving:

∆ .

The corresponding first order condition for the maximization problem yields

∆

. Which

can

be

solved

for

/.

4 points for the law of motion

3 points for correctly optimizing it to find k_0

c)

(7

points)

Compute

the

steady

state

values ∗, ∗, and ∗ using your answer from b).

ANSWER: Using our law of motion for capital and setting it to zero (imposing the steady

state condition), we find that the steady state level of capital ∗ must satisfy ∆ . ∗ /

Substituting this into the production function yields

∗ ∗ / Substituting this into the consumption function yields

∗ ∗ ∗ /

3 points for correct k* (if done correctly with a carry through error from the law of

motion, award credit)

2 points for correct y* (if done correctly with a carry through error from calculating k*,

award credit)

2 points for correct c* (if done correctly with a carry through error from calculating y*,

award credit)


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5

d) (7 points) Compute the value of the savings rate, ∗, that maximizes the steady state consumption

profile ∗ by solving the problem, ∗. Show your work!

From the previous part, we found that

∗ ∗ ∗ / Now

all

we

need

to

do

is

maximize

this

expression

with

respect

to

s by

solving:

∗

Where the second equality above comes from noting that we can pull out the constant 20

and it will not affect the optimization problem (if you keep it, you will see that it will

cancel out anyway once you take the first order condition).

∗

5 points for writing out the optimization problem and taking the first order conditions

correctly

2 points

for

finding

the

correct

value

(I

proved

in

general,

that

this

value

will

be

equal

to

capital’s share of income and some students may try and circumvent actually solving the

optimization problem to prove this – in this event, award these 3 points if they adequately

explain that we proved that capital’s share will equal the optimal savings rate in the Cobb‐

Douglas case, but DO NOT award the 4 points since they did not follow the directions)

e) (7 points) What other condition must ∗∗ satisfy if consumption is maximized in the steady state? Explain using a diagram.

ANSWER: If we are using the golden rule, then ∗∗ must also satisfy (no need to verify that it actually does – we know it must).

Note that the “depreciation line” is just our break even investment function which also

accounts for population growth in the case where it is not zero.

7 points for explicitly mentioning the condition and drawing the diagram

5 points for only doing one of these


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6

f) (7 points) If instead, the aggregate production function is given by , , for what values of the savings rate will the economy grow forever? At what rate will capital and

output

(per

capita)

grow?

Explain.

(hint:

a

diagram

may

help

you

here)

ANSWER: With the new production function, our law of motion for par capital per capita

changes slightly

since

we

now

are

using

. The new law of motion for the capital stock is given by ∆ .

Dividing through by k and recognizing that y=k implies %∆ %∆ lets us re‐write the law of motion in terms of growth rates:

%∆ ∆

. %∆ Thus, the economy will exhibit positive perpetual growth if and only if %. The rate of growth above is determined by the amount which the savings rate exceeds the net

depreciation and population growth rates.

g) (7 points) Explain why the adjustment to the model in f) provides a more accurate

prediction

in

terms

of

what

we

would

expect

to

happen

to

income

and

the

real

wage

in

the

long

run

in

comparison

to

the

production

model

in

the

original

question,

which

faced

diminishing marginal returns to capital? Assume that the input markets are in equilibrium.

ANSWER: The model in f) incorporates endogenous growth, whereby the economy never

reaches a steady state provided savings is sufficiently high. This is a much more realistic

description of actual phenomena compared to the earlier parts of the question that has

modeled production in a way such that the economy slows down to a steady state where

K/L doesn’t grow (due to diminishing returns to k).

If the real wage tracks average labor productivity (Y/L) and Y is growing forever, then we

would expect this model also predicts that the real wage should also grow. In the

previous parts of the problem where we use the Solow model without endogenous

growth (assuming diminishing returns to k), the steady state growth rate of Y/L is zero,

predicting absolutely

zero

growth

in

the

real

wage

(unrealistic).


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7

#2) (21 points) Consider the following data on the growth rate of worldwide income (GDP) per capita:

Worldwide Income Per Capita

Years Annualized Growth Rate

0‐1700 0.0%

1700‐1820 0.1%

1820‐1913 0.9%

1913‐1950 0.9%

1950‐1970 2.8%

1970‐1990 1.3%

1990‐2014 1.9%

2014‐2050 2.5%

2050‐2070 1.5%

2070‐2100 1.2%

a) (7 points) If the current worldwide average income per capita is approximately $13,000

(measured

in

2014

base

year

US

dollars),

compute

the

approximate

level

of

the

worldwide average income in year zero (in 2014 base year US dollars) as precisely as possible.

ANSWER: Since there is 0% growth from 0‐1700, all we really need to do is find the level of

income per person in 1700 by applying the growth rates backwards:

/ ≅ / $,..... $,. (in 2014 dollars).


the years.

Deduct 2 more

points

if

they

use

the

wrong

growth

rates.



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b) (7 points) If the future forecasts provided are accurate, compute a projection for the worldwide

average income per capita in 2100 (measured in 2014 base year dollars) as precisely as possible.

ANSWER: Now we just project the current $14,000 income forward to the year 2100:

/ $,... $,. (in 2014 dollars).


the years.



c)

(7

points)

If

the

growth

projections

are

instead,

expected

to

remain

at

2.5%

through

2100,

by

what

factor

will

the

projected

worldwide

average

income

in

2100

(measured

in

current

2014

dollars) increase relative to your answer in b)?

ANSWER: Now we just project the current $13,000 income forward to the year 2100 assuming

a constant

rate

of

2.5%

and

compare

the

resulting

figure

with

our

answer

from

b):

/ $,. $,. (in 2014 dollars). This is precisely $, . / $ , . . times as great as when we projected that growth would slow down. That is, if income growth remains at around 2.5%, the standard of living

would be almost double (1.78) what it would be with the predicted slowdown.


the years.


No partial credit otherwise if both of these mistakes are made AND the students get the factor

incorrect (if the factor is wrong because of a carry through error but otherwise, would have

been correct,

do

not

deduct

only

1 more

point

from

the

previous

4 that

were

deducted

for

making multiple math errors).


43/85

9

SCRATCH PAPER

(DO NOT DETATCH)


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Part II Total___________________/70

#1) Total_____________________/49

a)___________________________/7

b)___________________________/7

c)___________________________/7

d)___________________________/7

e)___________________________/7

f)___________________________/7

g)___________________________/7

#2) Total______________________/21

a)___________________________/7

b)___________________________/7

c)___________________________/7

SCORING RUBRIC


Exam Score____________________/100

Part I Total____________________/30

a)___________________________/10

b)___________________________/10

c)___________________________/10


45/85

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ECON 110A Midterm Exam 1A Solutions Name________________________

Prof. Levkoff PID #_________________________










46/85

2



a) (10 points) Consider a country that produces output using only capital and labor with a Cobb‐

Douglas technology. If the marginal product of capital is 3 and the average product of capital is

10, then labor’s share makes up 70% of the total national income. Assume that the labor and

capital markets are in equilibrium.

ANSWER: Recall that for a Cobb‐Douglas technology, , , , capital’s share of income. Substituting the information given, it must be the case that

%. This implies that labor’s share of national income is equal to 70% as stated accurately above.

5 points for recalling that the ratio of the marginal product to the average product is capital’s

share of income

5

points

for

identifying

the

remaining

share

as

labor’s

share

b) (10 points) Because of substitution bias and the introduction of new and improved goods and

services, the GDP Deflator will tend to understate the cost of living.

ANSWER: False. If the statement instead read “…the CPI will tend to overstate the cost…”, it

would be valid.

5 points for correctly identifying that this is a problem with the CPI basket not the deflator

5 points for identifying that because of these problems, the CPI overstates the cost of living


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3

c) (10 points) If nominal GDP increases by 4%, and during the same period, real GDP increases by

1.5%, then inflation must have been approximately 2.5%.

ANSWER: The statement is fairly accurate. Recall that NGDP=PY where P is the price level and

Y is

output

(real

GDP).

Using

the

log

‐time

derivative

trick,

we

can

rearrange

this

identity

in

percentage terms to find that

%∆ %∆ %∆. Thus, 4%‐1.5%=2.5%, the change in the price level (inflation). 5 points for correct relationship between nominal GDP, real GDP, and the price level in either

level or % change

5 points for showing that the statement is correct


#1) (40 points) Consider a country with an aggregate production technology given by , 5 2 where Y, K and L are aggregate output, capital and labor, respectively.

a)

(15 points)

According

to

the

neoclassical

theory

of

distribution

of

income,

what

are

the

equilibrium nominal wages and nominal rental rates for labor and capital, respectively, if the

price of output is equal to $1 and the endowments (supply) of capital and labor are fixed?

Explain.

ANSWER: Using the labor and capital market equilibrium conditions,

For the linear production technology above, MPK=5 and MPL=2, so the real rental rate

must be $5 and the wage rate $2. The equilibrium in the labor markets can be described

with a perfectly elastic (horizontal) demand for inputs (a constant marginal product) and a

completely inelastic

(vertical)

supply

of

these

inputs

–

the

flat

demand

curve

determines

the equilibrium price.

5 points for each correct marginal product (10 points total)

5 points for identifying the correct wage and rental rate through the equilibrium

conditions


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4

b) (10 points) Assuming that the labor force is equal to the population, compute the per capita

production function, where y and k are per capita output and capital, respectively. Graph the resulting technology with per capita output on the vertical axis and per capita

capital stock on the horizontal axis. Be precise.

ANSWER: Dividing

both

sides

by

L,

we

can

find

that

,

,

Plot should consist of an upward sloping line with a slope of 5 and vertical intercept of 2.

5 points for correct per capita production function

5 points for correct diagram – deduct points for each missing detail (slope and intercept)

c)

(15 points)

If

instead,

the

per

capita production

technology

is

given

by

√ , what rate of growth in the equilibrium real wage is implied by the neoclassical theory if the

aggregate capital stock, K, grows at a rate of 4% and the population, L, grows at a rate of

2%? Explain.

ANSWER: Recalling that the labor market equilibrium implies that, for a Cobb‐Douglas

technology,

. Since alpha is a constant, any growth in the

real wage will be equal to the growth in per capital output y (recall that y=Y/L is also

average labor productivity or equivalently, the average product of labor). Thus, we only

need to calculate the growth rate of y through the per capita production function given

our knowledge

of

the

aggregate

capital

stock’s

growth

and

the

population

growth

rates.

Taking the natural log of the production function yields

Differentiating the above with respect to time yields the growth rate relationship

%∆ %∆%∆ %% %

This is also the real wage growth rate as argued above earlier.

10 points for recognizing that the growth in the equilibrium real wage is equal to the

growth in the average product of labor.

5 points

for

correctly

computing

the

growth

rate

in

output

per

capita


49/85


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6

c) (10 points) Argue that if the quantity of savings exceeds the quantity of investment at the

current real interest rate, then it must also be the case that there is a surplus in the market

for goods and services and we should expect the real interest rate to fall in order to restore

equilibrium. (hint: a diagram of the loanable funds model along with the market clearing

condition should help)

ANSWER: if S>I, then it must be the case that r>r* so that r needs to fall to restore

equilibrium. Note also, that S>I implies Y‐C‐G>I which implies Y>C+G+I, so that supply

exceeds demand – that is, when there is a surplus in the market for loans, there is a

surplus in the market for goods and services.

5 points that r must fall to restore equilibrium

5 points for identifying that S>I implies Y>C+I+G or stating the equivalent with written

argument


51/85

7

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8/18/2019

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