3. 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 = 1110: #51110\"; #531110\": The interteniporal budget constraint is: 233 l:IH-l + Ctr! 'Y + Kel + Kw: 1+1\": [l+r,][l+r'm] ' 1+3\": {l+r=]{l+r,,1} (1+ 1': is the interest rate on saving ,n" borrowing between t and t+ 1. while 1'\". is the interest rate on saving ,f borrowing between t+ 1 and {+2. {a} Solve for C\": in the intertemporal budget constraint. and plug this into lifetime utility. This transforms the problem into one of choosing U: and CHI. Use calculus to derive two Euler equations one relating E." to CM, and the other relating CH1 to CH2. [h} In equilibrium, we must have C: = H, Cm = 1"\"1, and C\": = Hg. Derive expressions for r, and rm in terms of the exogenous endowment path and Ii. {c} One could dene the \"long\" interest rate as the product of one period interest rates. In particular, dene (1+ rm]2 = {I + r':}{1+ ml] {the squared term on 1 erg, reflects the fact that if you save for tam periods you get some compounding]. If there were a savings vehicle with a two period maturity, this condition would have to be satised (intuitively, because a household would be indierent between saving twice in one period bonds or once in a two period bond}. Derive an expression for my. {d} The yield curve plots interest rates as a function of time maturity. In this simple problem, one would plot 1', against 1 [there is a one period maturity} and r1: against 2 {there is a two period maturity}. If K = Y\". = 111-21 what is the sign of slope of the yield curve [i.e. if r1, >r._,, then the yield curve is upward-sloping}. {e} It is often claimed that an \"inverted yield curve\" is a predictor of a recession. if 1"\": is suciently low relative to Y: and Y\