Problem Set 6
Intermediate Macroeconomics, Fall 2015
The University of Notre Dame
Professor Sims
Instructions: You may work on this problem set in groups of up to four people. Should you
choose to do so, please make sure to legibly write each group member’s name on the first page of
your solutions. This problem set is due in class on Wednesday October 14.
(1) Endowment Economy Equilibrium, Graphical Analysis Suppose that the economy is
populated by many identical agents. These agents are price-takers, and also take their current and
future incomes as given. These agents live for two periods – t and t + 1. They solve a standard
consumption/saving problem, which yields a consumption function as the optimal decision rule:
Ct = C(Yt , Yt+1 , rt )
(a) What are the signs of the partial derivatives of the consumption function? Explain briefly.
(b) Suppose that there is an increase in Yt , holding Yt+1 fixed. How does a household want to
adjust its consumption and saving, holding the real interest rate fixed? Explain in words.
(c) Suppose that there is an increase in Yt+1 , holding Yt fixed. How does a household want to
adjust its consumption and saving, holding the real interest rate fixed? Explain in words.
(d) Write down the generic definition of a competitive equilibrium.
(e) Define the Y d curve and graphically derive it.
(f) Market-clearing in this context means that Yt = Ct , and production is taken to be exogenous,
so the Y s curve is vertical. Graphically show how you determine the equilibrium interest rate.
(g) Suppose that there is an increase in Yt . Graphically show how this affects the equilibrium real
interest rate.
(h) Suppose that there is an increase in Yt+1 . Graphically show how this affects the equilibrium
real interest rate.
(i) In this setup, what might the equilibrium real interest rate tell you about expectations of Yt+1
relative to Yt ? Explain the intuition for this result.
(2) A Three Period Model and the Yield Curve: Suppose that the economy is populated by
many identical agents. These agents live for three periods – t, t + 1, and t + 2. Lifetime utility is
given by:
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U = ln Ct + β ln Ct+1 + β 2 ln Ct+2
The household has access to two different bonds which differ by maturity. Type 1 bonds are savings
vehicles which pay out r1,t in the next period. Type 2 bonds are savings vehicles which pay out
r2,t in two periods. Formally, the three period budget constraints are:
Ct + S1,t + S2,t = Yt
C1,t+1 + S1,t+1 = Yt+1 + (1 + r1,t )S1,t
Ct+2 = Yt+2 + (1 + r1,t+1 )S1,t+1 + (1 + r2,t )S2,t
To be clear here, S1,t is the stock of one period bonds you take from t to t + 1, which pay out
r1,t in period t + 1. S1,t+1 is the stock of one period bonds you take from t + 1 to t + 2, which
pay out r1,t+1 in period t + 2. S2,t is the stock of two period bonds you take from period t,
which pay out r2,t in period t + 2. The subscripts on the interest rates denote (i) the type of
bond and (ii) the period’s saving to which the interest rate applies (the bonds pay out in a period
subsequent to this period); so r1,t is the interest rate on type 1 bonds which pay out in period
t + 1; r1,t+1 is the interest rate on type 1 bonds which are accumulated in t + 1 and pay out in
t+2; r2,t is the interest rate on type 2 bonds which are accumulated in period t and pay out in t+2.
The objective of the household is to maximize:
max
Ct ,Ct+1 ,Ct+2 ,S1,t ,S2,t ,S1,t+1
U = ln Ct + β ln Ct+1 + β 2 ln Ct+2
s.t.
Ct + S1,t + S2,t = Yt
C1,t+1 + S1,t+1 = Yt+1 + (1 + r1,t )S1,t
Ct+2 = Yt+2 + (1 + r1,t+1 )S1,t+1 + (1 + r2,t )S2,t
(a) Find the first order optimality conditions characterizing an optimal consumption plan. To do
this, the easiest way is to eliminate Ct , Ct+1 , and Ct+2 from the objective function, plugging in the
period budget constraints. Then the problem is one of choosing S1,t , S1,t+1 , and S2,t .
(b) Given your optimality conditions from part (a), what must be true about the relationship between r2,t and r1,t and r1,t+1 ? What is your intuition for this?
(c) This is an endowment economy, so Yt , Yt+1 , and Yt+2 are exogenous (and assume that they are
known). If Yt = Yt+1 = Yt+2 , solve for expressions for r1,t , r1,t+1 , and r2,t .
(d) A “Yield Curve” plots interest rates at date t as a function of their time to maturity. At time t,
there are two interest rates in this economy – r1,t , the interest rate on the bond with a one period
maturity, and r2,t , the interest rate on the bond with a two period maturity. The yield curve for
this model would be a plot of r1,t and r2,t against the time to maturity (1 and 2 periods). Suppose
that β = 0.95. For this value of β and for the the endowment pattern from part (c), plot the yield
curve here. Is it flat, upward-sloping, or downward-sloping?
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(e) Now suppose that the endowment pattern is Yt = 1, Yt+1 = 1, and Yt+2 = 0.92. Continue
to assume that β = 0.95. Plot the yield curve (r1,t and r2,t against time to maturity) for this
endowment pattern.
(f) It has been empirically documented than a so-called “inverted yield curve” (a yield curve which
is downward-sloping) is a predictor of a future recession (period where output is low). Does that
make sense theoretically given your work on this problem? Try to provide some intuition.
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