Q1. Consider the two-period Real Business Cycle (RBC) model without uncertainty
presented in the lecture slides (also Romer, 2019, ch.5), but with one modification. Now
assume that the instantaneous utility function for households takes the form:
๐ข๐ก =
๐๐ก
1โ๐
1 โ ๐
+ ๐(1 โ โ๐ก)
where ct is consumption at time t and (1โโt) is leisure time at time t. Given that the time
endowment is normalised to 1, it follows that โt is hours worked at time t. Finally, ฮธ>0
and b>0 are parameters.
All households in the economy are assumed to be identical. We can therefore consider a
โrepresentative householdโ (henceforth โthe householdโ). Set t=1 for the present period
and set t=2 for the next period. For example, c1 is consumption in the present period and
c2 is consumption in the next period. Remember, this is a two-period model so there are
no time periods prior to t=1 and there are no time periods after t=2. Assume that the
household begins and ends life with no accumulated wealth and that the real interest rate
is r (where r>0).
Answer the following questions:
a) Present the Lagrangian (constrained maximisation) problem for the household
under this modified specification.
b) Derive the first order conditions for the household in this case. [Hint: the
household chooses c1, c2, โ1 and โ2].
c) Use the first order conditions for c1 and c2 to derive an expression for the
relative amount of consumption chosen by the household over the two time
periods. Express your final answer with c2/c1 on the left-hand side.
d) Provide an economic interpretation for your result in part (c).
e) Explain why the magnitude of the parameter ฮธ is an important element of the
model when comparing the predictions of the theory to relevant empirical
evidence for the US or the UK.
For parts (a)โ(c), ensure that you explain your mathematical procedure.
There is no word limit for Q1.
LH A
