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:

c_t = y_t + (1 + r_t)a_t-1,

where a_t-1 is real assets in period t-1, y_t is real income, r_t is the real interest rate, and c_t is real consumption.

The period t budget constraint is: c_t + a_t - a_t-1 = y_t + r_ta_t-1.

Consumers have general utility function over consumption, u(c_t), and discount the future at rate β ∈ (0, 1). We will use λ_t to denote the Lagrange multiplier on the budget constraint and φ_t to denote the Lagrange multiplier on the credit constraint. We thus have sequential Lagrangian of:

u(c_t) + βu(c_t+1) + β²u(c_t+2) + ...

+λ_t[y_t + r_ta_t-1 - (c_t + a_t - a_t-1)]

+φ_t[y_t + (1 + r_t)a_t-1 - c_t]

βλ_t+1[y_t+1 + r_t+1a_t - (c_t+1 + a_t+1 - a_t)]

+ βφ_t+1[y_t+1 + (1 + r_t+1)a_t - c_t+1]

+...

4(a): Construct the first-order conditions with respect to c_t and a_t.

4(b): Write out the steady-state versions of the first-order condition with respect to a_t and c_t.

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?