In: Economics
1. Consumption-Savings. The representative consumer’s utility
function is
u(c1,c2)=lnc1 + Blnc2 , 0<B<1
in which the parameter (the Greek lowercase letter “beta”) is a
fixed number between
zero and one.
The consumer begins period 1 with zero net wealth. The period-1 and
period-2 budget
constraints, stated in real units, are, respectively, c1+a1=y1 and
c2+a2 = y2+(1+r)a1 .
a. Based on the utility function above, construct the sequential
Lagrange function.
b. Based on the sequential Lagrange function from part a, obtain
the five first-order
conditions.
c. Using the first-order conditions from part b, construct the
consumption-savings
optimality condition. Provide algebraic steps as needed for
clarity.
d. Based on the consumption-savings optimality condition obtained
in part c above,
compute the point elasticity of perod-2 consumption (i.e., of the
optimal choice
c2*) with respect to the gross real interest rate (1+r). (Note on
terminology: r is
referred to as the net real interest rate, and 1+r is referred to
as the gross real interest