Answer Key to Problem 1, Econ 601 first half exam, Fall 2012
I. The log-linear consumption Euler Equation under CRRA utility
For this problem, you may take as given (i.e. you don’t need to derive) that the Euler Equation
for consumption under the benchmark life cycle model is given in general by
(*)
Et [U’(ct+1)/U’(ct)] = [β(1+rt+1)]-1
where c is consumption, β is a discount factor, and rt+1 is a potentially time-varying risk free
interest rate between periods t and t+1. Assume for this problem that utility is CRRA:
U(c) = (1/1-α) c1-α
where α > 0 is the coefficient of relative risk aversion. Your goal in this problem is to derive the
following log-linear version of the consumption Euler equation under CRRA utility:
(**)
Et ∆ln(ct+1) = A + B log(1+rt+1) + C σ2∆ln(ct+1)
where A, B and C are functions of model parameters that you will solve for, where ∆ln(ct+1) is
the change in the natural logarithm of consumption between t and t+1 (commonly known as “log
consumption growth”) and where σ2∆ln(ct+1) is the conditional variance (as of time t) of log
consumption growth.
(a) Starting from (*), prove that (**) holds exactly for CRRA utility, if we assume that ∆ln(ct+1)
is a normal random variable. Derive expressions for A, B and C in terms of model parameters.
[HINT: recall that if X is distributed as a normal random variable with mean μ and variance σ2,
then E[exp(X)] = exp(μ + σ2/2).]
ANSWER TO PART (a): (30 points)
First note that for CRRA utility, (*) can be written as
(a.1)
Et[(ct+1/ct)-α] = [β(1+rt+1)]-1
Next, note that I can rewrite the left hand side of (a.1) as Et[exp(-α * ∆ln(ct+1))]. From the
hint and the assumption that ∆ln(ct+1) is normal, we can then rewrite (a.1) as
(a.2)
exp(-α Et∆ln(ct+1) + (α2/2) * σ2∆ln(ct+1)) = [β(1+rt+1)]-1
Taking logs of both sides of (a.2) and rearranging, we get (**), with A = (log β)/α; B = (1/α);
and C = (α/2).
(b) Prove that (**) holds approximately for CRRA utility regardless of the distribution of
∆ln(ct+1). Show that A, B and C are the same functions of model parameters as in part (a).
[HINT: note that one can re-write (*) as [U’(ct+1)/U’(ct)] = [β(1+rt+1)]-1 * (1 + εt+1), where εt+1 is
a shock to the growth in the marginal utility of consumption at time t+1, satisfying Etεt+1 = 0.
Take a 2nd-order Taylor’s approximation to log(1 + εt+1) around the point εt+1 = 0. What is the
relationship between the variance of εt+1 and the variance of ∆ln(ct+1)?]
ANSWER TO PART (b) [30 points]
Plug in CRRA utility into the version of (*) given in the hint, take logs of both sides and
rearrange to get:
(b.1)
∆ln(ct+1) = (log β)/α + (1/α)*log(1+rt+1) – (1/α) * log(1 + εt+1)
A second-order Taylor’s expansion of log(1 + εt+1) around the point ε = 0 yields
(b.2)
log(1 + εt+1) ≈ log(1) + εt+1 - (1/2)* εt+12
Plugging in to (b.1) and taking expectations of both sides yields
(b.3)
Et ∆ln(ct+1) = (log β)/α + (1/α)*log(1+rt+1) – (1/2α) * σ2εt+1
Note that εt+1 is the innovation to -α ∆ln(ct+1), so that we can write σ2εt+1 = α2 * σ2∆ln(ct+1), so
that (b.3) is identical to the answer from part (a).
(c) Given (**) and your expression for C in terms of model parameters, what is the relationship
between risk aversion, uncertainty and average consumption growth? Does this relationship
make sense given our discussion of precautionary saving in class? Explain intuitively (no math
required for this part).
ANSWER TO PART (c) [20 points]: from parts (a) and (b), the impact of the variance of
consumption on expected consumption growth is given by (α/2). Thus, higher conditional
uncertainty increases expected consumption growth, and this effect is stronger the higher
the degree of risk aversion.
This is consistent with what we learned in class from studying precautionary saving. We
found that, in general, the precautionary saving motive leads to more rapid expected
consumption growth. Intuitively, if we save more today for precautionary reasons, this
reduces today’s consumption and increases assets next period, which will increase
tomorrow’s consumption; for both reasons, expected consumption growth between today
and tomorrow rises. Moreover, from Zeldes’ and Deaton’s papers, we saw that the
precautionary saving motive increases in the degree of uncertainty (specifically uncertainty
about income in Zeldes’ and Deaton’s papers, but in general higher income uncertainty
should lead to higher consumption uncertainty), and in the degree of risk aversion.
(d) Suppose we had data for a group of households for years t and t+1 on each household’s actual
log consumption growth between t and t+1; each household’s real risk-free interest rate on
saving between t and t+1; a measure of each household’s conditional variance (as of time t) of
log consumption growth between t and t+1; and other household-level information available as
of time t on labor market variables such as income, occupation, and so on.
Write down a linear OLS regression that one could estimate to test whether the log-linear Euler
Equation (**) holds for this group of households. The dependent (left hand side) variable should
be log consumption growth for each household. What variables would you include on the right
hand side? What would be your null hypothesis, and what would this null hypothesis predict for
the coefficients of your regression? Under what conditions would your test of the null
hypothesis be powerful?
ANSWER TO PART (d) [20 points]
The null hypothesis should be that the log-linear life-cycle Euler Equation under CRRA
utility is valid. Given the data described, I would test this null using the following OLS
regression:
(d.1)
∆ln(cit+1) = A + B log(1+rit+1) + C σ2∆ln(cit+1) + D zit + eit+1
where ∆ln(cit+1) is household i’s observed consumption growth between t and t+1; where
the right hand side variables include household i’s risk-free interest rate and the proxy for
household i’s conditional consumption variance; where zit is a vector of variables in the
household’s information set at time t; and where eit+1 is a mean-zero error term.
Under the null, we would expect to find A, B and C to be positive and statistically
significant. Also, we’d expect to find D = 0; in other words, aside from interest rates and
conditional consumption variance, no other variables in the household’s information set at
time t should be useful for forecasting consumption growth at t+1. (In practical
applications, we might want to include some additional control variables xit+1 on the right
hand side of (d.1) to control for household preference shifters such as changes in family size
that could influence consumption even under the null; note that the utility function we
specify in the problem doesn’t include preference shifters like family size, but we could
easily add such preference shifters to our utility function, and changes in these preference
shifters would also affect consumption growth under the null).
What would make this test powerful? Recall that the power of a test depends on its ability
to reject the null hypothesis under plausible alternatives. The leading plausible
alternatives to the life cycle model include Keynesian/myopic behavior and liquidity
constraints, both of which predict that predictable changes in income between t and t+1
could influence consumption growth between t and t+1, contrary to the life cycle Euler
Equation. Thus, the ideal zit would include variables in the household’s information set at
time t that are useful for predicting household income growth between t and t+1.
Answer Key, Econ 601 First Half Exam Question 2, Fall 2012
II. Consumption and habit formation
A number of recent papers have explored the idea that consumption is affected by habit
formation. The idea of habit formation is that higher levels of consumption in the recent past
will increase the marginal utility of consumption today, because consumption is addictive or
habit forming.
Habit formation is often captured by assuming that utility takes the form u(ct – bct-1) or the form
u[ct/(ct-1p)], where u(.) is a concave function and where b > 0 and p > 0. The term bct-1 or ct-1p
represents the “habit” for the household; a higher level of past consumption reduces the level of
utility from consumption today, and/or increases the marginal utility of consumption today.
First, consider a generic version of a household’s problem with habit formation. The household
maximizes discounted expected lifetime utility:
T
Max E0 ∑ βt u(ct, ct-1)
t=0
subject to an initial condition on wealth, solvency, and the usual dynamic budget constraint
at+1 = (1+r)(at + yt – ct)
where at is financial wealth at the start of t and yt is household labor income in period t, which
may be stochastic. The utility function u(ct, ct-1) represents flows of utility experienced by the
household at time t. This function allows in a general way for both current and last-period
consumption to affect current utility; this form encompasses the aforementioned examples of
habit formation, as well as other possible formulations.
(a) Identify the state and control variables for this problem. Write down the Bellman equation,
and take the first order and envelope conditions.
Endogenous state variables: ct-1, at
Exogenous state variables: yt
Control variables: ct, at+1
Bellman Equation:
(*)
V(ct-1, at) = Max(ct) u(ct, ct-1) + β Et V(ct, at+1)
at+1 = (1+r)(at + yt – ct)
s.t.
The first order and envelope conditions:
(1)
ct: u1(ct, ct-1) + βEtV1(ct, at+1) =β(1+r)EtV2(ct, at+1)
(2)
ct-1: V1 (ct-1, at) = u2(ct, ct-1)
(3)
at:
V2 (ct-1, at) = β(1+r)Et V2(ct, at+1)
Updating (2) as V1 (ct, at+1) = u2(ct+1, ct) , plugging this and (3) in (1) we have
(4)
V2 (ct-1, at) = u1(ct, ct-1) + βEt u2(ct+1, ct)
Updating (4) as V2 (ct, at+1) = u1(ct+1, ct) + βEt+1 u2(ct+2, ct+1) , plugging this and (4) in (3) we
compute the Euler equation
(5)
u1(ct, ct-1) + βEt u2(ct+1, ct) = β(1+r)Et [u1(ct+1, ct) + βu2(ct+2, ct+1)]
Note that we use the Law of Iterated Expectations for Et [Et+1 u2(ct+2, ct+1)]= Et u2(ct+2, ct+1).
(b) Now specialize to the log utility function: u(ct, ct-1) = ln(ct/ct-1p), where 0 < p < 1. Derive the
consumption Euler equation. How does this Euler equation compare to the Euler equation for
log utility without habit formation? (In answering this question, you can use the conditions you
derived in part (a); you don’t need to start from scratch).
Given u(ct, ct-1) = ln(ct /ct-1p) = ln ct – p ct-1, we compute the derivatives
u1(ct, ct-1) = 1/ct
u2(ct+1, ct) = -p/ct
Plugging these derivatives (and one period updated versions) in (5) we compute
(5’)
1/ct + βEt (-p/ct) = β(1+r)Et [1/ct+1 + β (-p/ct+1)]
Rearranging we have the Euler equation
[1-βp] 1/ct = β(1+r) [1-βp] Et 1/ct+1
1/ct = β(1+r) Et 1/ct+1
In the usual case without habit formation with u(ct) = ln ct we obtain exactly the same Euler
Equation. Due to additive separability of the utility function in ct and ct-1, habits decrease current
utility but do not affect optimal choices (marginal utility) compared to the case without habits.
(c) Now specialize to the quadratic utility function: u(ct, ct-1) = a*[ct – ct-1] - (b/2)*[ct – ct-1]2,
where a > 0 and b > 0. Derive the consumption Euler equation and compare it to the Euler
equation for quadratic utility without habit formation. For simplicity, you may assume for this
part only that β(1+r) = 1. (Again, you can use the conditions you derived in part (a); you don’t
need to start from scratch).
Given u(ct, ct-1) = a*[ct – ct-1] - (b/2)*[ct – ct-1]2, we compute the derivatives
u1(ct, ct-1) = a - b[ct – ct-1] = a – b Δct
u2(ct+1, ct) = -a +b [ct+1 – ct] = - [a – b Δct+1]
Plugging these derivatives (and one period updated versions) in (5), using β(1+r)=1 we compute
(5’’)
a – b Δct - β Et [a - b Δct+1] = Et [ a – b Δct+1 - β [a - b Δct+2]]
Canceling out a and b from both sides and rearranging we have the Euler equation
Δct - β Et Δct+1 = Et [Δct+1 - β Et +1 Δct+2]
Below are two possible solutions to this difference equation:
(A)
(B)
Δct = β Et Δct+1
Δct = Et Δct+1
Note that (A) and (B) are also difference equations. Opening these up, we have
(A’)
(B’)
ct - ct-1 = β Et ct+1 - βct
ct - ct-1 = Et ct+1 - ct
A solution to both (A’) and (B’) is Et-1 ct = ct-1, which coincides with the solution to the Euler
equation without habit formation with quadratic utility function. However, another solution to
(A’) is Et-1 ct = 1/β ct-1, which is different from the usual consumption Euler equation. Thus, habit
formation may lead to a consumption path, which is expected to grow over time.