Write down the three first-order conditions for the problem

ECON 4021 Fall 2016
Christopher Gunn
Assignment 1
Due date: Tuesday, October 4, in class
Total Marks: 100
1. (10 marks) In your own words, (briefly) describe the “Lucas critique”, and explain
how the “modern micro-founded” method of macroeconomic modeling attempts to
address the issue that the Lucas critique raised about the “traditional” approach
to macroeconomics that was dominant up to the 1970s.
2. (10 marks) Imagine that we have a time series of real GDP per capita, Yt, where
Yis the level of the observation of the series for period t. Assume that through
some transformation we obtain a measure of a “trend” or growth component of Yt,
which we call g
. Defining dYt!gYtg
t
as the fractional deviation of the level of
real GDP per capital from its trend component, show that for small deviations,
we can approximate this deviation by computing the di↵erence between a series
constructed with the log of Yt, and the log of Ytg.
3. (10 marks) Robert Lucas said that “…business cycles are all alike”, yet from Chapter 3, we see that the times series of deviations from trend in real GDP is “choppy”,
and there is no regularity in either the amplitude or frequency of fluctuations in
real GDP about trend. What did Lucas mean by this statement?
4. (40 marks) Consider the consumer’s optimization problem from the Appendix to
chapter 4 in Williamson,
maxc,lU(c, l)
(1)
subject to w(l) + ⇡ T,
where U(·) is increasing in both arguments, strictly quasiconcave, and twice di↵erentiable.
(a) (3 marks) Write down the three first-order conditions for the problem, using
the Langranian method, and defining ” as the Lagrange multiplier.
(b) (10 marks) Now totally di↵erentiate these three first-order conditions, except
that instead of substituting out the Lagrange multipler ” as Williamson does,
1
ECON 4021 Fall 2016
Christopher Gunn
keep it in the problem, so that you obtain a system in the form
0 @d dc dl “1 A = D,
where and are comformable matrices, and the matrix contains the
di↵erentials of the exogenous variables as in Williamson.
(c) (2 marks) Do you notice anything special about the form of the matrix A?
(d) (5 marks) Calculate the determinant of A|A|.
(e) (3 marks) Calculate the determinant of the Bordered Hessian.
(f) (2 marks) What does the assumption that u(·) is strictly quasiconcave imply
about the sign of the Bordered Hessian?
(g) (10 marks) Determine expressions for the derivatives @ @⇡ C@⇡ @l
(h) (5 marks) What does the assumption that and are normal goods imply
about the sign of @ @⇡ C and @⇡ @l ?
5. (30 marks) Over the past century in the U.S., measures of labour input per person
have been remarkably constant, displaying little or no trend, which is potentially
surprising, since the real wage has displayed a distinct positive trend over this
period. In order to address this and other balanced growth facts, King, Plosser and
Rebelo (1988) developed a class of preferences of the general form
U(c, l) = 1
#
{[cv(l)]1!1}(2)
where v(l) is some twice di↵erentiable function of leisure with specific regularity
conditions that for the purpose of this problem we can assume that implies that
U(c, l) is strictly quasiconcave.
(a) (5 marks) Show that for these preferences,
lim
#!1
u(c, l) = ln + ln(v(l))(3)
Hint: Use L’hopital’s rule
(b) (15 marks) Derive the expression for the comparative statics derivative @ @w l for
the consumer’s optimization problem
maxc,lU(c, l) subject to c = w(h ! l) + ⇡ ! T, (4)whereU(c, l) = 11 ! #{[cv(l)]1!# ! 1}. (5)Note that you can work directly from the comparative statics expressionsobtained in the consumer’s optimization problem in the Appendix to Chapter2ECON 4021 Fall 2016Christopher Gunn4 in Williamson, since all we are doing are is placing some restrictions on thefunctional form of the utility function u(c, l) that Williamson uses (ie there isno need to re-do the total di↵erentiations etc).(c) (5 marks) Now assume that all variables in the expression above for @ @w l areequilibrium quantities and/or prices. Impose the following equilibrium condition on your expression for @ @w l ,C = w(h ! l), (6)(which corresponds to ⇡ ! T = 0, implying that in aggregate there is wageincome only) , and simplify.(d) (3 marks) Can you interpret the meaning your result? Hint: think about thethe income and substitution e↵ects.(e) (2 marks) Although this is a one-period model and so technically speakingthere is no “growth”, can you speculate (briefly) about how in a model ofmultiple periods (ie where you can loosely assume that your comparative staticexpression holds in each period), this analytical result could help a model beconsistent with the empirical observation that as the wage grows over time,hours-worked seems relatively constant? Note: no calculations are necessaryfor this answer.3

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