Impatient consumers in the two-period world. Consider the 2-period environment from class. The economy is populated by

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Impatient consumers in the two-period world.

Consider the 2-period environment from class. The economy is populated by two types of agents A and B. There is an equal number (N) of each type. The agents have preferences which incorporate impatience: they value consumption in period 2 less than consumption in period 1. Specifically, the preferences are described by the utility function

U(c ₁, c ₂) = ln(c ₁) + βln(c ₂)

where β is positive but less than 1 (reflecting their impatience).

Type A's endowment pattern is (y ^A ₁, y ^A ₂) = (x, 0) where x > 0. Type B's endowment pattern is (y ^B ₁, y ^B ₂) = (0, z) where z > 0. Let 1 + r ₁ be the real rate of return.

(a) What is the marginal utility of consumption in period 1 (i.e., what is MU (c ₁))? What is MU (c ₂)?

(b) Write down the tangency condition for a type A and B agents.

(c) Find the equilibrium real rate of return. How does 1 +r ₁ depend on β? How does 1 + r ₁ depend on x and z? What is the intuition for these effects?