(This follows Rebel, 1991.) Assume that there are two sectors, one producing consumption goods and one producing capital goods, and two factors of production: capital and land. Capital is used in both sectors, but land is used only in producing consumption goods. Specifically, the production functions are C(t) = KC (t) αT1−α and K˙(t) = BKK (t), where KC and KK are the amounts of capital used in the two sectors (so KC(t) + KK (t) = K(t)) and T is the amount of land, and 0 < α="">< 1="" and="" b=""> 0. Factors are paid their marginal products, and capital can move freely between the two sectors. T is normalized to 1 for simplicity.

(a) Let PK (t) denote the price of capital goods relative to consumption goods at time t. Use the fact that the earnings of capital in units of consumption goods in the two sectors must be equal to derive a condition relating PK (t), KC(t), and the parameters α and B. If KC is growing at rate gK (t), at what rate must PK be growing (or falling)? Let gP (t) denote this growth rate.