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: The intertemporal budget constraint is: 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

» 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: The intertemporal budget constraint is: 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 define the long interest rate as the product of one period interest rates. In particular, define (1 + r2,t)2 = (1 + rt)(1 + rt+1) (the squared term on 1 + r2,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 r2,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 r2,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 r2,t > r1,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.

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