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Graduate Macroeconomics I: Midterm
Prof. Vollrath
October 5th, 2010
Instructions:
Write your answers on blank paper. Start each problem on a new sheet. Write your
name and the problem number of EVERY page. Number every page.
Read the whole problem before you start answering. Make sure you answer all the parts of each
problem. If the problem asks you to graph something, graph it and label the axes correctly.
You have 90 minutes for the exam. Make sure you at least right down some ideas about how
the problem should be answered if you cannot Fgure out a mathematical answer. You will need a
calculator.
There are a total of 100 points.
Problem 1 (30 points):
People live for two periods. They have utility of
U
= ln
c
1
+
β
ln
c
2
.
Wages are
w
1
and
w
2
, and are given exogenously. People face an interest rate of (1 +
r
), and can
borrow and lend freely.
There is an additional constraint on people: they must consume a minimum of
c
in the Frst
period, or
c
1
≥
c
.
(A) Write down the Lagrangian for this consumption problem, including both the regular budget

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