Derivations: LR and SR Profit Maximization

Transcription

1 Derivations: LR and SR rofit Maximization Econ 50 - Lecture 5 February 5, 06 Consider the production function f(l, K) = L 4 K 4 This firm can purchase labor and capital at prices and r per unit; it can sell its output at the price per unit. Let s derive the folloing for the short run and the long run: The total cost function The supply function The profit-maximizing demand for labor The firm s maximized profits The firm s producer surplus (and confirm Hotelling s lemma) Short-Run rofit Maximization In the short run, suppose this firm s capital is fixed at K = 4.. Short run total cost In the short run, K = 4, so: Total cost is therefore q = L L = 4 q4 T C(q) = rk + L(q) = 4r + 4 q4 and marginal cost is the derivative of this ith respect to q:. Short-run supply function MC(q) = q In the short run, the firm ill set = MC(q). Using our expression for MC(q) this means = q q = ( ) q ( ) =

2 . Short-run profit-maximizing demand for labor We kno from before that the firm s required labor, as a function of q, is L(q) = 4 q4 lugging in our supply function q ( ) = ( ) in for q, this becomes [ ( L(q ( )) = ) ] 4 = ( ) Notice that the conditional labor demand L(q) as perfectly inelastic it didn t depend at all on the age rate, since the amount of capital is fixed. Hoever, the profit-maximizing quantity of labor demanded, L(q ( )), does depend on!.4 Short-run maximized profits Short-run profits are total revenues minus total costs: Π( ) = T R(q ( )) T C(q ( )) = q ( ) [4r + 4 (q ( )) 4] ( ) [ = 4r + ( ) 4 ] 4 = 4 4 4r.5 roducer Surplus (and confirm Hotelling s Lemma) The producer surplus at market price is the integral, from p = 0 to p =, of the quantity supplied at price p: q (p)dp = p dp = 4 p 4 = 4 4 Note that adding fixed costs to producer surplus gives us profit. The inverse of this procedure is, in effect the confirmation of Hotelling s lemma. Specifically, Hotelling s lemma states that the derivative of the profit function ith respect to, ill give the optimal q: dπ( ) = d So on the one hand, taking the integral of quantity gives us profit (if e then add fixed costs, hich don t change ith ). And on the other hand, taking the derivative of profit gives us quantity.

3 rofits and Losses Will this firm be profitable? The algebra here gets a little complex, so let s look at the case here = 8 and r = 4. With these numbers, e have T C(q) = 96 + q 4 MC(q) = 8q q ( ) = Π( ) = The firm s break-even price occurs hen profit is equal to zero: The quantity at that price is 8 4 = 96 4 = 56 = 64 q ( ) = = 4 = At that quantity, the firm s costs are T C(q) = = 8 MC(q) = 8 = 64 Furthermore, notice that the average cost of producing q = units is AC() = T C()/ = 64. So, at a quantity of, MC = AC = 64. (By the ay, note that quantity might very ell be measured in millions of units per year, so q = doesn t necessarily mean exactly to units of output...)

4 Long Run rofit Maximization No let s analyze this firm s profit-maximizing decision in the long run.. Long run total and marginal cost In the long run, the firm can freely choose labor and capital. As e ve found before, the conditional demands for labor and capital are: L (q) = r q K (q) = r q T C LR (q) = r q The long-run marginal cost curve is the derivative of the long-run total cost curve:. Long run supply function MC LR (q) = In the long run, the firm ill set = MC LR (q): dt C(q) dq = 4r q = 4r q q LR ( ) = 4 r. Long run profit-maximizing demand for labor and capital To find the long-run demand for labor, e plug the optimal quantity back into the conditional demand for labor: L LR (, r, ) = r [q LR ( )] [ = r 4 r = 6 r We can do the same thing for capital; the algebra yields the symmetric result:.4 Long-run maximized profits K LR (, r, ) = 6 r Long-run profits are total revenues minus total costs: ] Π LR ( ) = T R(q LR ( )) T C(q LR ( )) = q LR ( ) r (q LR ( )) = 4 r r = 4 r 8 r = 8 r ( 4 r ) 4

5 .5 roducer surplus and Hotelling s Lemma As before, producer surplus at market price is the integral, from p = 0 to p =, of the quantity supplied at price p: q LR (p)dp = 4 r pdp = 8 r p = 8 r In this case, because there are no fixed costs, this is exactly equal to the maximized profit. We can again confirm Hotelling s Lemma holds: dπ( ) d = 4 r = q LR ( ) 4 Comparison of SR and LR demand for labor Let s look at the age elasticity of labor in the long run and the short run. Taking the natural log of the short-run demand for labor and long-run demand for labor, e see: ln(l SR ) = ln 4 4 ln + 4 ln ln(l LR ) = ln 6 ln r ln ln We can no easily find the age elasticity of labor demand: ɛ L SR, = ln LSR (, r, ) ln ɛ L LR, = ln LLR (, r, ) ln = 4 = As expected, the demand for labor in the long run is more elastic, because in the long run the firm can substitute capital for labor. We can also find the cross-price elasticity of labor demand (i.e., the elasticity of labor demand ith respect to the price of capital, r): ɛ L SR,r = ln LSR (, r, ) ln r ɛ L LR,r = ln LLR (, r, ) ln r = 0 = Explaining hat s going on here is an excellent potential final exam question... 5