2-period Endowment Economy with Heterogeneous Consumers (10 pts)

Suppose we have an economy with two types of households: Minnesotans (M) and Wisconsinites (W). Both

types live for two periods and have the same utility function and discount rates (0, 1), but may differ in

their endowments and consumption allocation. There is an equal population of each type. The market clearing

condition for the first period is cM + cW = yM + yW , and likewise similar for the second period. There are no

firms or government. The utility maximization problem for a household of type i {M, W} is:

max

ci,c0

i

{log ci + log c

0

i}

subject to

ci +

c

0

i

1 + r

= yi +

y

0

i

1 + r

Problems: (a) Define the competitive equilibrium in this economy. Make sure to specify all the relevant problems

and conditions. For households, you may write out each type's problem seperately or write single

maximization problem indexed by i.

(b) Solve for the intertemporal Euler equation for a household of type i.

(Set up the Lagrangian for a household, find the FOCs, and then combine them to get an expression

about the Marginal Rate of Substitution between goods today and goods tomorrow.)

(c) Solve for consumption allocations in period 1 and 2 for household of type i as functions of their income

(yi

, y0

i

) and prices (r)

(d) Suppose that the endowments are given by: (yM, y0

M) = (2, 1) and (yW , y0

W ) = (1, 2), and that = 0.8

Use the households' Euler conditions and market clearing conditions to solve for the market-clearing

interest rate in this economy.

(e) Substitute the market clearing interest rate back into your consumption functions from part (c) and solve

for the equilibrium allocations. Which household is borrowing in the first period and which household is

saving? What is the economic intuition for this?

(f) Why does household M have higher consumption in each period? Explain.

(g) Now suppose the Midwest opens up to trade with the rest of the world, and acts as a small open economy.

This means two things:

Open: The Midwest can import and export goods if their consumption allocations don't match up

with their endowments.

Small: The Midwest can now borrow and lend at the exogenous world interest rate rW = 0.5.

How does the definition of competitive equilibrium change?

(h) Calculate the consumption allocations for consumers of each type in the small open economy.

(i) What are the Net Exports for the Midwest in each period? (Hint: Y=C+I+G+NX, and there is no

investment or government in this model, so Y = C + NX)