Let M be an SDF process and Y a labor income process. Assume E



T 0

Mt|Yt|dt



< ∞

for each finite T. The intertemporal budget constraint is dW = rW dt + φ

(μ− rι)dt +Y dt − Cdt + φ

σ dB. (14.34)

(a) Suppose that (C,W,φ)satisfies the intertemporal budget constraint

(14.34), C ≥ 0, and the nonnegativity constraint (14.8) holds.

(i) Suppose the horizon is finite. Show that(C,W) satisfies the static budget constraint W0 +E



T 0

MtYt dt



≥ E



T 0

MtCt dt + MTWT



(14.35)

by showing that t

0 Ms(Cs − Ys)ds+ MtWt is a supermartingale.

Hint: Show that it is a local martingale and at least as large as the martingale −Xt, where Xt = Et



T 0

MsYs ds



.

This implies the supermartingale property (Appendix A.13.)

(ii) Suppose the horizon is infinite and limT→∞ E[MTWT] ≥ 0.

Assume Y ≥ 0. Show that the static budget constraint W0 +E



∞

0 MtYt dt



≥ E



∞

0 MtCt dt



holds.

(b) Suppose the horizon is finite, markets are complete, C ≥ 0, and (C,W)

satisfies the static budget constraint (14.35) as an equality. Show that there exists φ such that (C,W,φ)satisfies the intertemporal budget constraint (14.34).