# Intermediate macroeconomic

1. (6 points each) Consider our basic two-period model of consumption/saving decisions, with household lifetime
utility given by
U = u(c) + u(c′
)
The household earns exogenous income y in the first period and y′ in the second period, and can borrow
and save at the real interest rate r. Savings/borrowing is denoted by s. Assume that there are no taxes.
(a) Obtain the intertemporal budget constraint.
(b) Draw the diagram that shows how the optimal c and c′ are determined. Be sure to label the diagram
thoroughly.
(c) Mathematically, set up the optimization problem faced by households.
(d) Derive the Euler Equation. Describe what it is.
2
(e) Assume that preferences are logarithmic, i.e. u(c) = A(ln c + 2), where A is any real positive number.
Solve for the optimal c and c′
. (Hint: you need to obtain the consumption function by using the
intertemporal budget constraint.)
(f) How does an increase in future income a↵ect current consumption? Why? What is the assumption
regarding the utility function that gives rise to this result?
(g) Suppose that = 0.8, r = 0.1, y = 11 and y′ = 4. What is saving?
(h) Assuming the same parameter values, suppose that borrowing is completely forbidden. What is the
optimal c and c′ now? What is the marginal propensity to consume out of current income in this case?
3
(35 Points)
2. Consider an individual who lives for T periods, the first R of which he spends working. Income in those first
R periods is given by yt, and income in periods R+1 through T is zero. The individual has a discount factor
of and can save and borrow at the real interest rate r. The individual’s optimal behavior is dictated by
the standard Euler equation (you don’t need to derive it).
(a) (8 points) Assume that = 1, r = 0, R = 30, T = 45, and yt = 25 for 1 ≤ t ≤ R. Calculate the individual’s
consumption in each period.
(b) (6 points) Based on your findings in part (a), draw a diagram that shows the paths for consumption,
income, and assets over the individual’s lifetime. Be sure to label it carefully.
(c) (5 points) How would this individual change his consumption following a one-time increase in income?
What would his response be if the increase in income was permanent? Explain the similarities and/or
di↵erences between the two cases.
4
(d) (6 points) Would your answer in part 2c change if individuals were borrowing constrained?
(e) (10 points) Assume that (1 + r) = 1, r = 0.1, R = T = 55, and yt = 10 for 1 ≤ t ≤ R. Calculate the
individual’s consumption in each period. Remember that the formula for the sum a geometric sequence
is 1 + s + s2 + … + sT−1 = ∑T
t=1 st−1 = 1−sT
1−s .

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