Consider an economy with complete financial markets and a representative agent with CRRA utility, u(C) = C1−γ

1−γ , where γ > 0, and a time preference rate of δ. The aggregate consumption level C is assumed to follow the stochastic process dct = ct

(a1X2 t + a2Xt + a3

)

dt + σc dzt



, where z = (zt) is a standard one-dimensional Brownian motion under the real-life probability measure P and where a1, a2, a3, σc are constants with σc > 0. Furthermore, X = (Xt)

is a stochastic process with dynamics dXt = −κXt dt + dzt, where κ is a positive constant.

(a) Show that the short-term interest rate is of the form rt = d1X2 t + d2Xt + d3 and determine the constants d1, d2, d3.

(b) Find a parameter condition under which the short-term interest rate is always nonnegative.

(c) Write up the dynamics of rt.

(d) What is the market price of risk in this economy?

Suppose that the above applies to the real economy and that money has no effect on the real economy. The Consumer Price Index F˜t is supposed to have dynamics dF˜t = F˜t



μFt dt + ρCFσFt dzt +

'

1 − ρ2 CFσFt dzˆt



, where ρCF is a constant correlation coefficient and zˆ = (zˆt) is another standard Brownian motion independent of z. Assume that μFt and σFt are of the form

μFt = b1X2 t + b2Xt + b3, σFt = kXt.

(e) Write up an expression for the nominal short-term interest rate, r˜t.

Assume in the rest of the problem that γ a1 + b1 = k2.

(f) Show that the nominal short rate r˜t is affine in Xt and express Xt as an affine function of r˜t.

(g) Compute the nominal market price of risk λ˜t.

(h) Determine the dynamics of the nominal short rate. The drift and volatility should be expressed in terms of r˜t, not Xt.