Question

2. A firm produces (= units of a commodity when labour input is L units. The price obtained per unit of output is P, and the price per unit of labour is w, both positive. (a) Write down the profit function w. What choice of labour input L=L* maximizes profits? (b) Consider L* as a function L (P, w) of the two prices, and define the value function (P, w) = (L* (P, w), P, w) Verify that ax/aP = (L*, P, w) and asfaw = x (L.P,w), thus confirming the enve- lope theorem.

Answer 1

2. (a) The production function is given as $\dpi{80} Q = L^{0.5}$ and the cost of production is $\dpi{80} C = wL$ . The profit would be $\dpi{80} \pi = TR - C$ or $\dpi{80} \pi = PQ - C$ or $\dpi{80} \pi = P L^{0.5} - wL$ .

The profit would be maximum for the FOC as below.

$\dpi{80} \frac{\mathrm{d} \pi}{\mathrm{d} L} = 0$

or $\dpi{80} \frac{\mathrm{d} }{\mathrm{d} L}(P L^{0.5} - wL) = 0$

or $\dpi{80} 0.5 P L^{- 0.5} - w = 0$

or $\dpi{80} 0.5 P L^{- 0.5} = w$

or $\dpi{80} L^{- 0.5} = \frac{w}{0.5 P}$

or $\dpi{80} L^{0.5} = \frac{0.5 P}{w}$

or $\dpi{80} L^* = \frac{P^2}{4 w^2}$ . This the the labor demand, and this maximizes the profit.

(b) Putting this in the profit function, we have $\dpi{80} \pi^* = P (\frac{P^2}{4 w^2})^{0.5} - w (\frac{P^2}{4 w^2})$ or $\dpi{80} \pi^* = \frac{P^2}{2 w} - \frac{P^2}{4 w}$ or $\dpi{80} \pi^* = \frac{P^2}{4 w}$ .

We have $\dpi{80} \frac{\partial \pi^*}{\partial P} = \frac{\partial }{\partial P}(\frac{P^2}{4 w})$ or $\dpi{80} \frac{\partial \pi^*}{\partial P} = \frac{2 P}{4 w}$ or $\dpi{80} \frac{\partial \pi^*}{\partial P} = \frac{P}{2 w}$ ,

and $\dpi{80} \frac{\partial \pi}{\partial P} = \frac{\partial }{\partial P}(P L^{0.5} - wL)$ or $\dpi{80} \frac{\partial \pi}{\partial P} = L^{0.5}$ and putting L*, we have $\dpi{80} \frac{\partial \pi}{\partial P}_{L^*} = (\frac{P^2}{4 w^2})^{0.5}$ or $\dpi{80} \frac{\partial \pi}{\partial P}_{L^*} = \frac{P}{2 w}$ .

Hence, we have $\dpi{80} \frac{\partial \pi^*}{\partial P} = \frac{\partial \pi}{\partial P}_{L^*}$ .

We have $\dpi{80} \frac{\partial \pi^*}{\partial w} = \frac{\partial }{\partial w}(\frac{P^2}{4 w})$ or $\dpi{80} \frac{\partial \pi^*}{\partial w} = - \frac{P^2}{4 w^2}$ ,

and $\dpi{80} \frac{\partial \pi}{\partial w} = \frac{\partial }{\partial w}(P L^{0.5} - wL)$ or $\dpi{80} \frac{\partial \pi}{\partial w} = - L$ and putting L*, we have $\dpi{80} \frac{\partial \pi}{\partial w}_{L^*} = - \frac{P^2}{4 w^2}$ .

Hence, we have $\dpi{80} \frac{\partial \pi^*}{\partial w} = \frac{\partial \pi}{\partial w}_{L^*}$