Analysis of Borrower and Lender Households
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October 27, 2022Economic Models for Scarcity of Resources
October 27, 2022Analysis of Borrower and Lender Households
Macro Comp Question 1 August 2015
Suppose there are two types of households in the economy. “Borrower” households borrow funds from “lender” households to purchase housing. Borrower and lender households differ in two ways: first, only borrowers value housing, while both types value non-durable consumption. Second, borrowers discount future utility flows more heavily than lenders: 0 < βB < βL < 1. Both borrowing and lending are subject to constraints.
First, consider the problem of borrowers. Borrowers maximize the expected discounted value of period utility flows with an infinite horizon, where the period utility flow in period t depends on both housing consumption H and non-housing consumption C, as follows:
∞
(1)Max E0 ∑ (βB)t [u(CBt) + w(Ht)]
t=0
where u(C) and w(H) are increasing flow utility functions. Housing depreciates at the rate δ, between 0 and 1. There are no costs of adjustment for housing. Households enter period t with Ht units of housing, and have (1-δ)Ht units of housing available to retain or sell at the end of the period. In period t they choose their housing for next period Ht+1. The house price in period t is Pt, which may be time varying and stochastic. Non-housing consumption is the numeraire, with constant price 1.
In period t, borrowers may borrow Dt+1 ≥ 0 in funds from lender households, at a risk-free interest rate Rt+1, which may be time varying but is known at t. Borrowing is limited to a fraction of the value of next period’s housing stock by the following inequality constraint:
(2)Dt+1≤ θ Pt Ht+1
Borrower households enter period t with a stock of debt RtDt which they must repay to lenders, and receive an endowment of non-durable consumption goods Yt. The household’s period budget constraint is given by
(3) CBt + Pt Ht+1 = Yt + (1 – δ) Pt Ht – RtDt + Dt+1
(a) [5 points]: Identify the minimum set of exogenous and endogenous state variables and the minimum set of control variables, and write down the Bellman Equation for the borrower’s problem. Don’t forget the inequality constraint (2); let μt denote the multiplier on this constraint.
(b) [20 points]: Take first order and envelope conditions, and state the complementary slackness condition. Derive two Euler equations: one involving only μt and values of u'(C) at t and t+1, and the other involving μt, values of u'(C) at t and t+1, and w'(H) at t+1.
