Intermediate Macroeconomics: Economic Growth and the Solow Model

ing from money, meaning everything is denominated in units of goods, e.g. fruit). Total cost is the wage bill plus the capital bill. Let wt be the real wage rate – it is the number of goods the firm must pay each unit of labor. Let Rt be the real “rental rate” – it is the number of goods the firm must give up to lease a unit of capital. The firm is a price-taker, so it takes these as given. Profit is therefore: Πt = AF (Kt, Nt)− wtNt −RtKt (3) The firm wants to pick capital and labor to maximize profit. The problem is therefore: max Kt,Nt AF (Kt, Nt)− wtNt −RtKt The solution is characterized by taking the partial derivatives of the production function with respect to each input and setting them equal to zero: ∂Πt ∂Kt = 0⇔ AFK(Kt, Nt) = Rt (4) ∂Πt ∂Nt = 0⇔ AFN (Kt, Nt) = wt (5) Because of the concavity assumption, these two conditions imply downward sloping demand curves for each factor input – the bigger the wage, for example, the less labor a firm will want, holding all factors constant. An increase in A will shift the factor demand curves out for both capital and labor, meaning that firms will want more of both inputs at given factor prices. Because of the constant returns to scale assumption, it turns out that the firm will earn no profits. This is easiest to see by using the Cobb-Douglas form of the production function, so that the optimality conditions are: αKα−1 t N 1−α t = Rt (1− α)K t N−α t = wt With these factor demands, we see that RtKt = αYt and wtNt = (1 − α)Yt. Therefore Πt = Yt − αYt − (1 − α)Yt = 0, so there are no profits. Also, with this functional form, α has the interpretation as the share of total income that gets paid out to capital, and 1− α as the share of total income paid out to labor. So α will sometimes be called “capital’s share.” You may take issue with the notion that the firm earns no profits, because firms in the real world do earn profits. It’s important to draw the distinction between accounting and economic profit. The way I’ve set the model up here there is no distinction between the two, and this is because of a particular way of modeling the ownership structure of capital. In the real world, firms typically own their own capital, and firms are owned by households via shares of common stock. The way