Suppose a firm uses both labor L and capital K as inputs to production. Itsproduction function is of the Cobb-Douglas form, i.e.F(K, L) = K13 L23The firm charges a price P for every good it sells, pays a nominal rental rateR to every unit of capital it hires and pays a nominal wage W to every unit oflabor it hires.
1. Express the firm's real revenues, real costs and real profits in terms of thevariables defined above.
2. Assume the firm operates in goods markets, capital markets and labormarkets that are perfectly competitive - that is, it takes P , R and W asgiven. Derive two conditions that will need to hold in order for the firmto maximize its profits ( Set the partial derivative of profits with respectto L equal to zero and the partial derivative of profits with respect to Kequal to zero).
3. The first of these conditions should include the real wage rate WPas wellas K and L. This is the labor demand equation. The second conditionshould involve the real rental rate of capital RPas well as K and L. This isthe capital demand equation. Let us assume that labor is fixed L = 8 andfocus on the capital demand equation. Plot the capital demand functionwhen L = 8 (Show at least three points of the curve).
4. Suppose the supply of capital is inelastic and given by Ks = 27. Plot thecapital supply line on the same graph as the capital demand curve. Findthe real rental rate of capital that clears the market and show it on thegraph.5. How many goods will that firm produce?