nominal gdp vs. real gdp - yuantian.site · econ 202 macroeconomics i problem set 1 due: tuesday,...

ECON 202 MACROECONOMICS I

PROBLEM SET 1

DUE: TUESDAY, JUNE 26, 2012, IN CLASS

1. Nominal GDP vs. Real GDP

In this question we ask you to compute the GDP for a hypothetical economy that produces two goods

(computers and gasoline). The prices and quantities produced in each year are provided below:

2006 2007 2008

PricesComputers $2,000 $1,900 $1,200Gasoline $50/gal. $100/gal. $400/gal.

QuantitiesComputers 10 16 40Gasoline 100 gal. 120 gal. 110 gal.

(1) Compute the nominal GDP for each of the three years.

(2) Compute the real GDP for each of the three years, taking 2006 as the fixed base year.

(3) How does the real GDP in 2006 compare to the nominal GDP in 2006 when 2006 is used as the

fixed base year?

(4) Based on your calculations, comment on the economic well-being of the above two-good economy.

Have things improved over time for this country (in real-GDP terms)?

(5) State two reasons why real GDP is not a perfect measure of economic well-being.

2. Crusoe’s Preferences

Suppose that our poor Crusoe who does not have access to credit has a utility function:

u(c, l) = c� l�

where we assume that � > 1, c and l are defined as in class.

(1) Derive the expressions for the marginal utilities of consumption (uc) and labor (ul). What are the

signs of the marginal utilities (positive or negative)?

(2) Derive an equation for the indi↵erence curve of this utility function for an arbitrarily specified level

of utility, u, as we did in class. Graph the indi↵erence curve putting c on the vertical and l on the

horizontal axis.1

2 ECON 202 PROBLEM SET 1

(3) How would the graph above change if you put c on the vertical and 1 � l on the horizontal axis?

Provide the new graph and explain.

(4) Finally, compute the slope of the indi↵erence curve (also called the marginal rate of substitution,

MRS, or TMS in France).

3. Crusoe’s Optimal Choice

Now assume a utility function of the following form:

u(c, l) = c�(1� l)1��

where 0 � 1. The production function is the same as in class, y = Al↵, where 0 < ↵ < 1.

(1) Set up the constrained maximization problem.

(2) Solve for the optimal consumption choice, c⇤, and the optimal labor choice, l⇤, using the method of

the Lagrangian Multipliers.

(3) Remember that the l⇤ in the example we did in class was equal to ↵/(1 + ↵). For what value of �

will the l⇤ here be equal to ↵/(1 + ↵)?

4. Playing with the Production Function

Suppose the production function of an economy has the form:

y = Apl +B

(1) Parallel shift: using a graph, describe the wealth and substitution e↵ects on consumption and labor

from an increase in B.

(2) Proportional shift: using a graph, describe the wealth and substitution e↵ects on consumption and

labor from an increase in A.

5. Intertemporal Consumption Choice

Consider the two-period intertemporal consumption decision model that we discussed in class, where the

household’s utility function is given by:

U(c1, c2) = u(c1) + �u(c2).

where c1 and c2 denote period one and two consumption, respectively, and � < 1 is the discount factor.

(1) State the IBCs of the household assuming that the household comes and dies with nothing.

(2) Derive the lifetime budget constraint of the household by combining the two IBCs.

ECON 202 PROBLEM SET 1 3

(3) Simplify the expression that you obtained in part (2) above by setting X ⌘ y1 + y2

1+R . Plot the

simplified expression that you obtained in the c1-c2 space (i.e. with c2 on the vertical and c1 on the

horizontal axis). Mark the slope of the line on the graph. What does the slope represent in this case

(economically speaking)?

(4) Now add an arbitrary indi↵erence curve on the same graph above and indicate the optimal choice.

Why are the indi↵erence curves downward sloping this time?

(5) To analyze the wealth e↵ect, consider an increase in X. Graphically show what happens to the

budget line and mark the new optimal choice. What happens to c1 and c2?

(6) To analyze the (intertemporal) substitution e↵ect, consider an increase in the slope without a change

in wealth (the line will rotate around the initial optimal point). Graphically show what happens to

the budget line and mark the new optimal choice. What happens to c1 and c2 in this case?

(7) Finally, consider an increase in R. Graphically show what happens to the budget line and mark the

new optimal choice. Break down the change into substitution and wealth e↵ects. Can you tell what

will happen to c1 for sure? How about c2? Comment on your findings.

6. Habit Formation

Consider a two-period utility function of the form:

U(c1, c2) = ln(c1) + � ln(c2 � c1)

Notice that here, c1 enters into the second period utility function too. The utility of consumption in the

second period depends not simply on c2, but also on c1. The higher is first-period consumption, the higher

c2 must be to provide a given level of utility. Economists use utility functions of this form to capture habit

formation. When you consume more, you “get used to it”, and it creates a strong distaste for consuming

less in later periods.

(1) Set up the standard two-period intertemporal consumption optimization problem, and derive the

Euler equation for this utility function. Assume that b0 = 0, b2 = 0, and that the first- and second-

period incomes are y1 and y2, respectively.

(2) Assume that � = 1/(1 + R). What does the Euler equation say about the ratio of c2 to c1? What

is the ratio in the case we did in class, where U(c1, c2) = ln(c1) + � ln(c2)?

(3) Give the economic intuition behind why the c2 to c1 ratio is (higher/lower) in the habit-formation

case than in the regular case.


PROBLEM SET 2

DUE: TUESDAY, JULY 3, 2012, IN CLASS

1. Intertemporal Consumption with Heterogeneous Households

Suppose that an economy is made up of two types of households: poor and rich. There are NP poor

households with period two income yP2 , and NR rich households with period two income yR2 = kyP2 , where

k > 1. Both types of households have the same period one income, y1, and the same utility function:

U(c1, c2) = ln(c1) + � ln(c2)

where c1 and c2 denote period one and two consumption, respectively, and � 1 is the discount factor which

represents patience.

(1) State the poor household’s optimization problem and calculate the optimal choice of c1, c2 and b1.

Label the poor household’s choices as (c⇤1P , c⇤2P , b

⇤1P ).

(2) Repeat the above exercise for the rich household and label the optimal choices as (c⇤1R, c⇤2R, b

⇤1R).

(3) Give the definition of a competitive equilibrium for this economy. (Hint: Use the definition that

we talked about in class. State what prices are taken as given, what is chosen optimally and which

markets clear. Carefully write down the market clearing conditions).

(4) Using the market clearing condition for the credit market, solve for the equilibrium interest rate R⇤.

(5) Calculate @R⇤/@k and @R⇤/@(NRNP

). Are these partial derivatives positive or negative? What is the

economic intuition for the observed sign in each case? (Hint: Think about the role of the interest

rate in the credit market that we discussed in class).

(6) Use the equilibrium interest rate R⇤ you calculated in part(4) above to solve for b⇤1P and b⇤1R in terms

of the exogenous variables/parameters of the model. Is borrowing/lending possible in equilibrium?

(7) How do your results here compare to the model with identical households that we talked about in

class? Briefly discuss the di↵erences/similarities also giving the intuition.

1


2. The Infinite-Period Model

Before you attempt this question, please read your lecture notes on the infinite period model and then

close your notebook and attempt to do this exercise without looking at your notes. Make sure you can

replicate each step correctly.

Consider the infinite-period model of consumption choice, where each representative household solves:

max{ct,bt}1

t=1

1X

t=1

�t�1u(ct) s.t. pyt + (1 +R)bt�1 = pct + bt

for all t 2 {1, 2, . . .}. Again, � 1 is the discount factor which represents patience.

(1) Set up the Beast as we did in class.

(2) Find the period-t first order conditions with respect to c and b, and derive the Euler equation.

(3) Assume that the period utility function u(ct) = ln(ct) for all t, and � = 1/(1 + R). What do these

assumptions imply about the optimal intertemporal consumption choice?

3. General Equilibrium Model with a Labor Market

In an economy, there are N identical households and M identical firms. Each firm hires capital, kd, and

labor, ld, to produce a consumption good using the production function ys = 2pkdld. The price of the

consumption good, P , is one dollar.

Each household chooses how much to consume and how much to work according to the utility function

U(c, l) = ln(c) + ln(24� l), where c denotes consumption and l denotes the number of hours spend at work,

with 0 l 24. The household has two sources of income: i) it owns a certain amount of capital and rents

ks units to firms at the ongoing rental rate r, ii) it supplies its labor, ls, to firms at wage rate w.

(1) State the maximization problem of the firm, calculate the FOCs, and find the equation of the firm’s

labor demand curve. Draw the labor demand curve with w on the horizontal axis. How does the

curve change as k changes? What is the economic intuition for this?

(2) State the maximization problem of the household, calculate the FOCs, and find the equation of the

household’s labor supply curve. Draw the labor supply curve on the same graph as labor demand.

How does an increase in k or r a↵ect the labor supply curve? What is the economic intuition?

(3) Formally define a competitive equilibrium for this economy. (Hint: Use the definition that we talked

about in class. State what prices are taken as given, what is chosen optimally and which markets

clear. Carefully write down the market clearing conditions).


(4) Use the labor market clearing condition that you stated in part(3) to solve for the equilibrium wage

rate, w⇤. How does w⇤ change when k increases? How about when (M/N) increases? Give the

economic intuition for the observed changes.

(5) Finally, assume that M = N , and that ks = kd = 1. Solve for the equilibrium wage rate, rental rate,

labor supply, consumption and profits.

4. General Equilibrium with Two Different Types of Households

Suppose that an economy is made up of two types of agents, A and B, with di↵erent intertemporal preferences:

Agent A: UA(cA1 , cA2 ) = ln(cA1 ) + � ln(cA2 )

Agent B: UB(cB1 , cB2 ) = cB1 + �cB2

Both types of agents live for two periods, 1 and 2, and are endowed with the same amount of income, y1

in period 1 and y2 in period 2. Note also that the agents share the same subjective discount factor �.

Each type of agent has a population of 1. Assume that income is not storable at the aggregate level, but

each type of agent can trade private bonds in period 1, which yields a net return of R in period 2.

Let the bond holdings of agents A and B be denoted by bA and bB , respectively. Assume that the price

of consumption goods in each period is 1 dollar.

(1) Which agent cares more about intertemporal consumption smoothing? Explain.

(2) State each agent’s intertemporal budget constraint for each period 1 and 2, assuming that the agents

come with nothing and die with nothing.

(3) Using the intertemporal budget constraints, derive each agent’s lifetime budget constraint.

(4) State the goods market clearing condition for this economy.

(5) Given your answers to parts 2 and 4 above, prove that the bond market clears.

(6) Carefully define a competitive equilibrium for this economy. State what prices are taken as

given, what is chosen optimally, and which markets clear.

(7) Now assume that

� = 0.5, y1 = 1, y2 = 5

Also assume that R > 1 and cA1 , cB1 , cA2 , cB2 = 0.

Compute the optimal cA1 , cB1 c

A2 , c

B2 , b

A, bB and the equilibrium interest rate R.


PROBLEM SET 2.5

DUE: ...NEVER...

1. A Simple Money Demand Model

Show graphically and mathematically the direction of the e↵ects of the following changes on the optimal

interval between fund withdrawals in the Baumol-Tobin model.

(1) A decrease in the real consumption c;

(2) A decrease in the nominal transaction cost �;

(3) A decrease in the nominal interest rate R.

2. Money Market Equilibrium

Suppose that our economy is in equilibrium so that real-money demand is equal to the real-money supply,

as given by the following equation:

�(R, Y, �real, . . .) =Ms

P

Consider the following scenarios and in each case explain what happens in the economy a) in the short-run,

b) in the long-run, using the above equation.

(1) The central bank gives money to people and people spend all the extra money to buy goods.

(2) The central bank gives money to the government and the government spends it all to buy goods.

(3) The central bank gives money to people and people put all the extra money under the mattress.

(4) The money supply stays the same, output increases due to an improvement in production technology.

(5) An adverse supply shock hits the economy and the central bank increases the money supply.

1


PROBLEM SET 3

DUE: TUESDAY, JULY 17, 2012, IN CLASS

1. Adaptive Expectations

The table below gives the path of inflation observed in a particular economy:

Time 1 2 3 4 5 6 7 8 9 10

Inflation, % 2 2 2.5 3 3.2 4 4.5 5.6 7 8.6

Suppose that people form their expectations about inflation at time t according to the following model:

⇡et = ✓⇡t�1 + (1� ✓)⇡e

t�1

Recall that by definition ⇡t is not observed at time t. This formulation is called “adaptive expectations”

and it is a simple way of modeling how people form expectations about inflation. You can use Excel or a

similar program of your choice to help answer the questions below.

(1) Calculate the path of expected inflation assuming that ⇡e1 = 2 and ✓ = 0.2.

(2) The forecast error for inflation is defined as (⇡et � ⇡t). Calculate and plot the forecast error as a

function of time. How does the forecast error behave?

(3) Repeat the exercise for ⇡e1 = 2 and ✓ = 0.9. How does your answer to parts (1) and (2) di↵er in this

case? Can people be systematically fooled under adaptive expectations?

2. The Infinite Period B.C. with Inflation

Recall the period-t budget constraint of the representative consumer:

Ptyt + (1 +R)bt�1 = Ptct + bt

(1) The period-t budget constraint is a di↵erence equation in b. Recursively solve the di↵erence equation

forward and express bt�1 as a function of the history of nominal consumption expenditures, nominal1


income and savings in period T . You should obtain an expression that looks like the following:

bt�1 =1

1 +R

"TX

i=0

Pt+ict+i

(1 +R)i�

TX

i=0

Pt+iyt+i

(1 +R)i+

1

(1 +R)Tbt+T

#

(2) Assume that limT!1(1 + R)�T bt+T = 0. What is the economic intuition behind this limit? Use

this limit and the expression you derived above to get the lifetime PDV budget constraint for an

infinitely lived consumer (Hint: what happens as T ! 1?).

(3) Assume that the price level evolves according to Pt+1 = (1 + ⇡)Pt. How does the lifetime PDV

budget constraint change when you introduce inflation? What is the intuition behind this result and

how does it compare to the two-period case that we discussed in class?

3. Credibility of the Central Bank

Brazil is one of the countries that has su↵ered an extended period of chronic inflation. High money growth

and the continuously broken promises of the elected o�cials to decrease it was the underlying cause of

the observed inflation, which averaged around 45% a year during the 1980s. The episode ended with the

establishment of a new inflation fighting program and a new currency. In this question we will explore the

importance of credibility in fighting inflation, in the context of the model we developed in class.

(1) Assume that Brazil started with an annual money growth rate of 10% in 1970 and has had the same

money growth rate for the next 10 years. The annual real interest rate was 2% throughout the whole

period. Assuming no changes occurred in the goods market, what was the annual inflation rate in

Brazil during the 1970s? Graph the paths of the money stock and the price level on a logarithmic

scale, assuming that the money stock was e4 and the price level was e2 to start with in the beginning

of 1970. Calculate the level of the money stock and the price level at the beginning of 1980.

(2) Show the path of real money balances and inflation during the 1970s on two separate graphs. How

much would a house that cost 100,000 Brazilian Reals in 1970 cost in 1980? How much would it cost

if the same level of inflation was maintained for an extra 10 years (i.e until 1990)?

(3) In 1980, the government unexpectedly started a populist program in an e↵ort to secure a victory

in the upcoming elections. Flowers were put on the streets, the sidewalks and shabby buildings

were restored, and a lot of money was spent on the election campaign. All of these activities were

funded by freshly printed money and caused the annual money growth rate to jump to 45%. Even

worse, from that year on, the 45% annual money growth was the norm in Brazil for the next 10

years. Everything in the goods market stayed the same. How does this change a↵ect the paths of

the money stock and the price level on the logarithmic scale, assuming that the real money demand


function is given by �(Rt, Yt) = Yt/Rt? Calculate the level of the money stock and the price level

at the beginning of 1990.

(4) Show how the above increase in the money growth rate a↵ects the path of real money balances and

inflation. How much would the house that cost 100,000 Brazilian Reals in 1970 cost in 1990 now,

with the higher money growth rate?

(5) Finally, at the end of 1990, a trustworthy central banker takes over the central bank and is given full

power and independence to coordinate monetary policy. She announces that starting in 1991, the

growth rate of the money supply is going to be reduced to 3% a year forever to curb inflation. The

public believes her announcement. Describe what happens to the paths of the money stock and the

price level on a logarithmic scale, and to inflation and real money balances, after the announcement

is made. Explain your reasoning.

4. Cash-in-Advance and Optimal Monetary Policy

Suppose that an economy is made up of N infinitely-lived identical households. The total utility for the

representative household is given by:

1X

t=1

�t�1

1

3ln(ct) +

2

3ln(1� lt)

�.

where 0 < � < 1 is the discount factor, c denotes consumption, and l represents the fraction of time spent

working during the day. The period-t intertemporal budget constraint of the household is given by:

Ptlt + (1 +Rt)bt�1 + vt +mt�1 = Ptct +mt + bt

where P denotes the price level, R is the nominal interest rate, b denotes borrowing/lending, v represents

the nominal transfer to/from the government, and m denotes money holdings. The household has to put

aside the money that will be spend on consumption one period in advance. Therefore, it is also bound by

the following cash-in-advance constraint:

Ptct mt�1.

(1) Give the intuition as to why the cash-in-advance constraint will hold with exact equality as long as

R > 0.

(2) Assuming that money grows at a constant rate µ, the price level grows at a constant rate ⇡, and that

the interest rate R, consumption c, and leisure l are constant over time, show that the cash-in-advance

constraint implies that µ = ⇡.


(3) Calculate the equilibrium consumption c⇤ as a function of the model’s parameters. What happens

to consumption as inflation goes up?

(4) Assume that this economy is governed by a benevolent central bank whose goal is to maximize social

welfare (which is represented by the total utility of the representative household). Using the same

assumptions as in part (2) above, calculate the level of optimal consumption, c⇤, that this central

bank should target. (Hint: what does c and l constant imply for the total utility function?)

(5) What would be the money growth rate µ that the central bank chooses to deliver the welfare maxi-

mizing consumption? What would this policy imply about R and ⇡?

(6) Would a real-life central bank be able to implement such an optimal policy? Why or why not?

Briefly discuss.


PROBLEM SET 3.5

DUE: ...NEVER...

1. The Solow Model

Consider an economy which has the production technology:

Yt = K↵t (AtLt)

1�↵

1.1. Labor and Capital Share. For simplicity, assume that there is one big competitive

firm that operates this production function and that hires workers and capital for wage wt

and rental rate rt. Formulate the profit maximization problem of this firm and find the

equilibrium wage and rental rate. Verify that the labor share (wtLt/Yt) and capital share

(rtKt/Yt) are constant in this economy. (Note: This is the primary reason why the Cobb-

Douglas production function is so popular: the fraction of output paid to workers, i.e. the

labor share, and the capital share have historically been pretty constant in the U.S.)

1.2. The Law of Motion of Capital. Now assume that productivity and population are

constant (At = A and Lt = L), that capital depreciates at rate �, where 0 < � < 1, and

fraction s of output is invested in new capital each period. The law of motion for capital,

K, is given by:

Kt+1 = (1� �)Kt + sYt.

Find an expression for the initial growth rate of output:

Y1 � Y0

Y0

as a function of the initial capital K0. Verify that the growth rate is decreasing in K0.1

2 ECON 202 PROBLEM SET 3.5

1.3. Steady State K⇤. Derive the steady-state level of capital K⇤ as a function of the

parameters. Use your solution for K⇤ to derive expressions for output Y ⇤ and consumption

C⇤ in the steady state. (Hint: recall that Yt = Ct + It, where It = sYt).

1.4. Solow Model with Population Growth. Contrary to the initial assumption, assume

that population grows at the constant rate n: Lt+1 = (1 + n)Lt, while At continues to be

constant. Rewrite the law of motion in terms of capital per worker kt ⌘ Kt/Lt, and find the

steady state level of k⇤, y⇤, c⇤ and i⇤ as a function of the model’s parameters. What is the

growth rate of output Yt at steady state? What happens when n increases? Use a Solow

model graph to illustrate your answer.

nominal gdp vs. real gdp - yuantian.site · econ 202 macroeconomics i problem set 1 due: tuesday,...

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