The Yield Curve Suppose you have an economy with one type of agent, but that time lasts for three periods instead of two. Lifetime utility for the household is:

U = lnCt + lnCt+1 + (^2)lnCt+2

The intertemporal budget constraint is:

Ct+ (Ct+1)/(1 + rt)+(Ct+2)/((1 + rt)(1 + (rt+1)))= Yt + (Yt+1)/(1 + rt) + (Yt+2)/((1 + rt)(1+(rt+1)))

rt is the interest rate on saving / borrowing between t and t + 1, while rt+1 is the interest rate on saving / borrowing between t + 1 and t + 2.

(a) Solve for Ct+2 in the intertemporal budget constraint, and plug this into lifetime utility. This transforms the problem into one of choosing Ct and Ct+1. Use calculus to derive two Euler equations: one relating Ct to Ct+1, and the other relating Ct+1 to Ct+2.

(b) In equilibrium, we must have Ct = Yt, Ct+1 = Yt+1, and Ct+2 = Yt+2. Derive

expressions for rt and rt+1 in terms of the exogenous endowment path

and .

(c) One could dene the \long" interest rate as the product of one period interest rates. In particular, dene (1 + r(2,t))^2 = (1 + rt)(1 + rt+1) (the squared term on 1 + r(2,t) reflects the fact that if you save for two periods you get some compounding). If there were a savings vehicle with a two period maturity, this condition would have to be satisfied (intuitively, because a household would be indifferent between saving twice in one period bonds or once in a two period bond). Derive an expression for r(2,t).

(d) The yield curve plots interest rates as a function of time maturity. In this simple problem, one would plot rt against 1 (there is a one period maturity) and r(2,t) against 2 (there is a two period maturity).

If Yt = Yt+1 = Yt+2, what is the sign of slope of the yield curve (i.e. if r(2,t) > r(1,t), then the yield curve is upward-sloping).

(e) It is often claimed that an "inverted yield curve" is a predictor of a recession. If Yt+2 is sufficiently low relative to Yt and Yt+1, could the yield curve in this simple model be "inverted" (i.e. opposite sign) from what you found in the above part? Explain.