Department of Applied Economics
Johns Hopkins University
Economics 602
Macroeconomic Theory and Policy
Midterm Exam Suggested Solutions
Professor Sanjay Chugh
Summer 2011
NAME:
The Exam has a total of five (5) problems and pages numbered one (1) through fourteen (14).
Each problem’s total number of points is shown below. Your solutions should consist of some
appropriate combination of mathematical analysis, graphical analysis, logical analysis, and
economic intuition, but in no case do solutions need to be exceptionally long. Your solutions
should get straight to the point – solutions with irrelevant discussions and derivations will be
penalized. You are to answer all questions in the spaces provided.
You may use one page (double-sided) of notes. You may not use a calculator.
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
TOTAL
/ 25
/ 10
/ 15
/ 30
/ 20
/ 100
Problem 1: Two-Period Consumption-Savings Framework (25 points). Consider the twoperiod economy (with zero government spending and zero taxation), in which the representative
consumer has no control over his real income (y1 in period 1 and y2 in period 2). The lifetime
utility function of the representative consumer is
u ( c1 , c2 ) = ln c1 + ln c2 .
The lifetime budget constraint (in real terms) of the consumer is, as usual,
c1 +
c2
y
= y1 + 2 + (1 + r )a0 .
1+ r
1+ r
Suppose the consumer begins period 1 with zero net assets (a0 = 0), and as per the notation in
Chapters 3 and 4, r denotes the real interest rate.
For use below, it is convenient to define the gross real interest rate as R = 1+r (as a point of
terminology, “r” is the net real interest rate).
a. (5 points) Set up a lifetime Lagrangian formulation for the representative consumer’s
lifetime utility maximization problem. Define any new notation you introduce.
y
c ⎤
⎡
Solution: The lifetime Lagrangian is ln c1 + ln c2 + λ ⎢ y1 + 2 + (1 + r )a0 − c1 − 2 ⎥ , which
1+ r
1+ r ⎦
⎣
contains Lagrange multiplier λ.
1
Problem 1 continued
b. (10 points) Based on the Lagrangian from part a, compute the first-order conditions with
respect to c1 and c2. Then, use these first-order conditions to derive the consumption-savings
optimality condition for the given utility function. NOTE: Your final expression of the
c
consumption-savings optimality condition should be presented in terms of the ratio 2 .
c1
Furthermore, in obtaining the representation of the consumption-savings optimality
condition, you should express any (1+r) terms that appear as R instead (if you have not
already done so).
Thus, the final form of the condition to present is
c2
= ...
c1
in which the right hand side is for you to determine. Your final expression may NOT
include any Lagrange multipliers in it. Clearly present the important steps and logic of
your analysis.
Solution: The first-order conditions with respect to c1 and c2 are
1
−λ = 0
c1
1
λ
−
=0
c2 1 + r
The second expression tells us λ =
1+ r
. Inserting this expression for the multiplier into the first
c2
1 1+ r
=
. Replacing (1+r) by R, and multiplying both sides by c2, the
c1
c2
consumption-savings optimality condition in the requested form is
expression gives
c2
= R.
c1
2
Problem 1b continued (more work space)
c. (4 points) Starting from your expression in part b (that is, starting from the expression you
c
obtained that has the form 2 = ... ), construct the natural logarithm of the expression. In
c1
doing so, recall the following results regarding algebraic manipulation of natural logs: for
any two x > 0 and y > 0, ln(xy) = ln x + ln y, and ln( x y ) = y ln x . Your final expression here
⎛c ⎞
should be of the form ln ⎜ 2 ⎟ = ... . Clearly present the important steps and logic of your
⎝ c1 ⎠
analysis.
Solution: Applying the natural logarithm to both sides of the final expression in part b, we have
⎛c ⎞
ln ⎜ 2 ⎟ = ln R .
⎝ c1 ⎠
3
Problem 1 continued
Recall from basic microeconomics that the elasticity of a variable x with respect to another
variable y is defined as the percentage change in x induced by a one-percent change in y. As you
studied in basic microeconomics, elasticities are especially useful measures of the sensitivity of
one variable to another because they do not depend on the units of measurement of either
variable.
A convenient method for computing an elasticity (which we will not prove here) is that the
elasticity of one variable (say, x) with respect to another variable (say, y) is equal to the first
derivative of the natural log of x with respect to the natural log of y. (Read this statement
very carefully.)
d. (6 points) Starting from your expression in part c (that is, starting from the expression you
⎛c ⎞
c
obtained that has the form ln ⎜ 2 ⎟ = ... ), compute the elasticity of the ratio 2 with respect
c1
⎝ c1 ⎠
to the gross real interest rate R. The resulting expression is the elasticity of consumption
growth (between period one and period two) with respect to the (gross) real interest
rate for the given utility function. Clearly present the important steps and logic of your
analysis.
Solution: Being careful about applying the method of computation of an elasticity described
above, the elasticity of consumption growth with respect to the (gross) real interest rate is
∂ ln(c2 / c1 )
=1.
∂ ln R
This follows immediately from the final expression in part c.
4
Problem 2: Government Budgets and Government Asset Positions (10 points). Just as we
can analyze the economic behavior of consumers over many time periods, we can analyze the
economic behavior of the government over many time periods. Suppose that at the beginning
of period t, the government has zero net assets. Also assume that the real interest rate is
always r = 0. The following table describes the real quantities of government spending and real
tax revenue the government collects starting in period t and for several periods thereafter.
Period
t
t+1
t+2
t+3
t+4
Real government
expenditure (g)
during the period
10
8
15
10
8
Real tax collections
during the period
Quantity of net government
assets at the END of the period
12
2
14
8
10
3
10
3
12
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a. (6 points) Complete the last column of the table based on the information given. Briefly
explain the logic behind how you calculate these values.
Solution: The algebra to compute the list of numerical values given in the last column above is
to repeatedly apply the government flow budget constraint bt = tt − gt + (1 + r )bt −1 , being careful
to update the time period and the new bond position at the end of every time period.
b. (4 points) Suppose instead the government ran a balanced budget every period (i.e., every
period it collected in taxes exactly the amount of its expenditures that period). In this
balanced-budget scenario, what would be the government’s net assets at the end of period
t+4? Briefly explain/justify.
Solution: Again using the government flow budget constraint bt = tt − gt + (1 + r )bt −1 , a balanced
budget every period would mean tt – gt = 0 in every period. Doing the same procedure as in part
a would clearly give that the government’s net asset position at the end of every time period,
including at the end of period t+4, is exactly zero (given that the government begins period t with
zero net assets).
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Problem 3: A Contraction in Credit Availability (15 points). The graph below (on the next
page) shows the usual two-period indifference-curve/budget constraint diagram, with period-1
consumption plotted on the horizontal axis, period-2 consumption plotted on the vertical axis,
and the downward-sloping line representing, as usual, the consumer’s LBC. Throughout all of
the analysis here, assume that r = 0 always. Furthermore, there is no government, hence never
any taxes.
Suppose that the representative consumer has lifetime utility function u (c1 , c2 ) = ln c1 + ln c2 , and
that the real income of the consumer in period 1 and period 2 is y1 = 12 and y2 = 8. Finally,
suppose that the initial quantity of net assets the consumer has is a0 = 0. EVERY consumer in
the economy is described by this utility function and these values of y1, y2, and a0.
a. (6 points) If there are no problems in credit markets whatsoever (so that consumers can
borrow or save as much or as little as they want), compute the numerical value of the
optimal quantity of period-1 consumption. (Note: if you can solve this problem without
setting up a Lagrangian, you are free to do so as long as you explain your logic.)
Solution: The consumption-savings optimality condition (given the natural-log utility function)
is given by c2/c1 = 1+r = 1 (the second equality follows because r = 0 here). Thus, at the optimal
choice, it is the case that c1 = c2. Using this relationship (and again using the fact that r = 0
here), we can express the consumer’s LBC as c1 + c1 = y1 + y2 = 20 , which obviously implies the
optimal choice of period-1 consumption is c1 = 10.
Note: although you were not asked to compute it, you could have computed the implied value of
the consumer’s asset position at the end of period one. Because a0 = 0, y1 = 12, and we just
computed c1 = 10, the asset position at the end of period one is a1 = y1 – c1 = 2 (i.e., positive 2).
b. (9 points) Now suppose that because of problems in the financial sector, no consumers
are allowed to be in debt at the end of period 1. With this financial restriction in place,
compute the numerical value of the optimal quantity of period-1 consumption. ALSO,
on the diagram on the next page, qualitatively and clearly sketch the optimal choice with
this financial restriction in place (qualitatively sketched already for you is the optimal
choice if there are no problems in financial markets). Your sketch should indicate both
the new optimal choice and an appropriately-drawn and labeled indifference curve that
contains the new optimal choice. (Note: if you can solve this problem without setting up
a Lagrangian, you are free to do so as long as you explain your logic.)
Solution: Because in part a (ie, without any credit restrictions), the representative consumer was
choosing to NOT be in debt at the end of period 1 (i.e., a1 > 0 under the optimal choice in part a),
the imposition of the credit restriction, nothing changes compared to part a. That is, the
optimal choice of period-1 consumption is still 10. Hence, in the diagram below, the optimal
choice in the presence of credit constraints is exactly the same as the optimal choice without
credit constraints. The general lesson to draw from this example and our analysis in class is
that it is not necessarily the case that financial market problems must and always spill over into
real economic activity (i.e., consumption in this case).
6
Problem 3b continued
c2
Optimal choice if no
credit-market problems
Consumer LBC
c1
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Problem 4: The Consumption-Leisure Framework (30 points). In this question, you will use
the basic (one period) consumption-leisure framework to consider some labor market issues.
Suppose the representative consumer has the following utility function over consumption and
labor,
u (c, l ) = ln c −
A 1+φ
n ,
1+ φ
where, as usual, c denotes consumption and n denotes the number of hours of labor the
consumer chooses to work. The constants A and φ are outside the control of the individual, but
each is strictly positive. (As usual, ln(⋅) is the natural log function.)
Suppose the budget constraint (expressed in real, rather than in nominal, terms) the individual
faces is c = (1 − t ) ⋅ w ⋅ n , where t is the labor tax rate, w is the real hourly wage rate, and n is
the number of hours the individual works.
Recall that in one week there are 168 hours, hence n + l = 168 must always be true.
The Lagrangian for this problem is
ln c −
A 1+φ
n + λ [ (1 − t ) wn − c ] ,
1+ φ
in which λ denotes the Lagrange multiplier on the budget constraint.
a.
(6 points) Based on the given Lagrangian, compute the representative consumer’s firstorder conditions with respect to consumption and with respect to labor. Clearly present
the important steps and logic of your analysis.
Solution: The first-order conditions with respect to consumption and labor are
1
−λ = 0
c
− Anφ + λ (1 − t ) w = 0
8
Problem 4 continued
b.
(7 points) Based on ONLY the first-order condition with respect to labor computed in part
a, qualitatively sketch two things in the diagram below. First, the general shape of the
relationship between w and n (perfectly vertical, perfectly horizontal, upward-sloping,
downward-sloping, or impossible to tell). Second, how changes in t affect the relationship
(shift it outwards, shift it in inwards, or impossible to determine). Briefly describe the
economics of how you obtained your conclusions. (IMPORTANT NOTE: In this
question, you are not to use the first-order condition with respect to consumption nor any
other conditions.)
real wage
Solution: The first-order condition with respect to labor can be rearranged to (if we want to put
1 Anφ
it in vertical axis/horizontal axis form) w =
. Given that φ > 0 , there is clearly an
λ (1 − t )
upward sloping relationship between w and n.
Plotting this below (and ignoring
convexity/concavity issues, which are governed by the particular magnitude of φ ) gives an
upward sloping relationship holding A, t, and λ constant. This is the labor supply function.
Then, starting from this upward-sloping relationship, a rise in the tax rate t (holding A, λ, and n
constant) causes the entire function to shift inwards. The latter effect is due to individuals
working fewer hours when the tax rate rises, all else equal, due to the decrease in their after-tax
real wage.
labor
9
Problem 4 continued
c.
(4 points) Now based on both of the two first-order conditions computed in part a,
construct the consumption-leisure optimality condition. Clearly present the important steps
and logic of your analysis.
Solution: As usual, this requires eliminating the Lagrange multiplier across the two expressions.
The first-order condition on consumption gives lambda = 1/c. Inserting this into the first-order
(1− t)w
. Or, multiplying through by c, the consumption-leisure
condition on labor gives Anφ =
c
optimality condition can be expressed as
Anφ
= (1 − t)w .
1/ c
d.
(7 points) Based on both the consumption-leisure optimality condition obtained in part c
and on the budget constraint, qualitatively sketch two things in the diagram below. First,
the general shape of the relationship between w and n (perfectly vertical, perfectly
horizontal, upward-sloping, downward-sloping, or impossible to tell). Second, how
changes in t affect the relationship (shift it outwards, shift it in inwards, or impossible to
determine). Briefly describe the economics of how you obtained your conclusions.
labor
10
Problem 4d continued (more work space)
Solution: The budget constraint says that c = (1− t)wn . Substituing this into the consumptionleisure optimality condition from part c, we have (1− t)wn ⋅ Anφ = (1− t)w . The (1-t)w terms on
the left-hand and right-hand sides obviously cancel, leaving n ⋅ Anφ = 1 , or, combing the powers
in n, An1+φ = 1 . Plotting this in the space above, we will have
1
⎛ 1 ⎞ 1+φ
n=⎜ ⎟
⎝ A⎠
which clearly does not depend on the real wage w at all. Hence, this is a vertical line at the value
1
⎛ 1 ⎞ 1+φ
⎜⎝ ⎟⎠ .
A
e.
(6 points) How do the conclusions in part d compare with those in part b? Are they
broadly similar? Are they very different? Is it impossible to compare them? Describe as
much as you can about the economics when comparing the pair of diagrams.
Solution: Broadly, the difference between part b and part d is that part b is a “microeconomic”
analysis, while part d is a “macroeconomic” analysis. More precisely, part b is, intuitively, a
purely “slope” argument, rather than both a “slope” and a “level” argument in part d. The
analysis in part b is tantamount to analyzing the effects of policy on just the labor market (why?
– because the analysis there treats consumption as a constant). The analysis in part d instead is
tantamount to analyzing jointly the effects of policy on labor markets and goods markets. To
the extent that there are feedback effects between the two markets, there is no reason to think the
answers from the analyses must be the same.
The latter is the basis for thinking of the analysis in part b as a “microeconomic” analysis and the
analysis in part d as a “macroeconomic” analysis. What this implies is that one way (perhaps the
most important way) to understand the difference between “microeconomic” analysis and
“macroeconomic” analysis is that the latter routinely considers feedback effects across markets,
whereas the former usually does not.
11
Problem 5: Two Types of Stock (20 points). Consider a variation of the Chapter 8 infiniteperiod “stock-pricing” model. The variation here is that there are two “types” of stock that the
the
representative consumer can buy: “Dow” stock and “S&P” stock. Denote by atDOW
−1
representative consumer’s holdings of Dow stock at the beginning of period t and by atSP
−1 the
representative consumer’s holdings of S&P stock at the beginning of period t. Likewise, let
Pt DOW and Pt SP denote, respectively, the nominal price of Dow and S&P stock in period t, and
DtDOW and DtSP denote, respectively, the per-share nominal dividend that Dow and S&P stock
pay in period t. The period-t budget constraint of the representative consumer is thus
SP SP
DOW DOW
,
Pc
at
= Yt + ( StSP + DtSP )atSP−1 + ( StDOW + DtDOW )atDOW
−1
t t + St at + St
in which all of the other notation is standard: Yt denotes nominal income (over which the
consumer has no control) in period t, ct is real units of consumption, and Pt is the nominal price
of each unit of consumption. Also as usual, the lifetime utility of the consumer starting from
period t onwards is u (ct ) + β u (ct +1 ) + β 2u (ct + 2 ) + β 3u (ct +3 ) + ... , where β ∈ (0,1] is the usual
measure of consumer impatience.
The Lagrangian for the consumer lifetime utility maximization problem, starting from the
perspective of the beginning of period t, is
u (ct ) + β u (ct +1 ) + β 2u (ct + 2 ) + ...
SP SP
DOW DOW
at ⎤⎦
+ λt ⎡⎣Yt + StSP atSP−1 + DtSP atSP−1 + StDOW atDOW
+ DtDOW atDOW
− Pc
−1
−1
t t − St at − St
SP
SP SP
DOW DOW
DOW
SP
DOW DOW
⎤
+ βλt +1 ⎡⎣Yt +1 + StSP
+ DtDOW
− Pt +1ct +1 − StSP
+1at + Dt +1at + St +1 at
+1 at
+1at +1 − St +1 at +1 ⎦
+ ....
in which λt denotes the Lagrange multiplier on the budget constraint of period t, λt+1 denotes the
Lagrange multiplier on the budget constraint of period t+1, and so on.
a. (3 points) Based on the given Lagrangian, compute first-order conditions with respect to
atDOW and atSP . NOTE: You do not need to compute any other first-order conditions.
Solution: Taking FOCs with respect to atSP and atDOW :
SP
−λt StSP + βλt +1 ( StSP
+1 + Dt +1 ) = 0
−λt StDOW + βλt +1 ( StDOW
+ DtDOW
)=0
+1
+1
12
Problem 5 continued
b. (3 points) Based on the first-order conditions obtained in part a, re-express them as period-t
stock-pricing expressions for both Dow stock and S&P stock. Your final expressions should
be of the form StDOW = ... and StSP = ... .
Solution: Rearranging the two expressions obtained in part a, we have
StSP =
StDOW
βλt +1 SP
St +1 + DtSP
(
+1 )
λt
βλ
= t +1 ( StDOW
+ DtDOW
)
+1
+1
λt
c. (6 points) Based on the stock-price expressions obtained in part b, is it the case that
StDOW = StSP ? If so, briefly explain why; if not, briefly explain why not; if it is impossible to
tell, explain why.
Solution: No, given the information at hand thus far, there is no way to know if the two stock
prices are equal to each other. In particular, you are thus far given no information on the
dividends that each of these two different assets pay.
13
Problem 5 continued
d. (8 points – Harder) Assume in this sub-question that β = 1 . Suppose the economy
eventually reaches a steady state. In this steady state, Dow stock pay zero dividends but S&P
stock pay a positive nominal dividend that is always one-tenth the nominal price of a share
of S&P stock. That is, in the steady state, DtSP = 0.1StSP in every period t. Further suppose
that in the steady state, the inflation rate of consumer goods prices between one period and
the next is always 10 percent (i.e., π = 0.10 ). Compute numerically the steady-state rate at
which the nominal price of each type of stock grows every period (i.e., what you are
being asked to compute is the “inflation” or “appreciation” rates of each of the two types of
stock). Justify your solution with any appropriate combination of mathematical, graphical, or
qualitative arguments. Also provide brief economic rationale/intuition for your findings.
Solution: The first point to observe is that we now need the first-order condition on ct (and its
period t+1 analog): u '(ct ) − λt Pt = 0 (the period t+1 analog is u '(ct +1 ) − λt +1 Pt +1 = 0 ). Solving
these expressions for the multipliers λt and λt+1, and inserting them in the stock price expressions
obtained in part b, gives
β u '(ct +1 ) SP
Pt
StSP =
St +1 + DtSP
(
+1 )
u '(ct )
Pt +1
β u '(ct +1 )
Pt
u '(ct )
Pt +1
= ct for all t, by definition. You are also told that β = 1 and
StDOW =
(S
DOW
t +1
W
+ DtDO
)
+1
Next, in the steady state, ct+1
Pt
1
1
=
=
(be careful with numerators and denominators). Imposing these three
Pt +1 1 + π t +1 1.1
conditions gives
1
SP
StSP = ( StSP
+1 + Dt +1 )
1.1
1
StDOW = ( StDOW
+ DtDOW
) 1.1
+1
+1
Finally, imposing the condition DtSP = 0.1StSP in every period t gives 1.1StSP = 1.1StSP
+1 and
1.1StDOW = StDOW
in the steady state. From these two steady-state expressions, it is clear how
+1
Dow prices and S&P prices are changing over time: from the latter expression, clearly S&P
prices are not changing over time, while from the former expression, Dow prices are rising at a
rate of 10 percent, the same as the rate of consumer price inflation.
The intuition behind these results is as follows. No matter which we way we measure the “real
interest rate” (whether using Dow returns or S&P returns), they must both must be equal to the
consumer’s MRS. The Dow stock pays no dividend, hence its entire return must come through
changes in the price of the stock itself – i.e., there are capital gains on the Dow stock. In
contrast, because S&P stocks do pay a dividend, the required capital gains on S&P stock are
lower. With the particular numerical values given, the required capital gain on S&P stock turn
out to be zero.
END OF EXAM
14