Question - The Long-run Real Rate with a Credit Restriction
Consider the infinite-period consumer framework with a credit restriction in each period. This period-t credit restriction is:
ct = yt + (1 + rt)at-1,
where at-1 is real assets in period t-1, yt is real income, rt is the real interest rate, and ct is real consumption.
The period t budget constraint is: ct + at - at-1 = yt + rtat-1.
Consumers have general utility function over consumption, u(ct), and discount the future at rate β ∈ (0, 1). We will use λt to denote the Lagrange multiplier on the budget constraint and φt to denote the Lagrange multiplier on the credit constraint. We thus have sequential Lagrangian of:
u(ct) + βu(ct+1) + β2u(ct+2) + ...
+λt[yt + rtat-1 - (ct + at - at-1)]
+φt[yt + (1 + rt)at-1 - ct]
βλt+1[yt+1 + rt+1at - (ct+1 + at+1 - at)]
+ βφt+1[yt+1 + (1 + rt+1)at - ct+1]
+...
4(a): Construct the first-order conditions with respect to ct and at.
4(b): Write out the steady-state versions of the first-order condition with respect to at and ct.
4(c): Suppose the credit constraint does not bind in steady state such that φ = 0, where (as is the standard notation) φ¯ is the steady state value of the Lagrange multiplier on the credit constraint. Combine the two steady state equations from part b to recover the solution to r¯ (the state value of the long-run real rate) in terms of β.
4(d): Suppose the credit constraint binds in steady state such that φ¯ > 0. If c¯ is steady state consumption, write out r¯ as a function of u′(c¯), φ¯, and β, reducing as far as possible to look closest to your answer in part c.
4(e): Suppose φ¯/u′(c¯) < 2, is r¯ higher or lower than its steady state value without credit constraints?