# The Yield Curve Suppose you have an economy with one type of agent, but that time lasts for three…

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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