Department of Economics Microeconomic TheoryUniversity of California, Berkeley Economics 101AFebruary 21, 2007 Spring 2007Economics 101A: Microeconomic TheorySection Notes for Week 61 The Firm’s Problem: Perfect CompetitionUnder perfect competition, the firm takes both the output price p and the input prices w1, w2as fixed.1.1 Two Stage: Cost Minimization and Profit MaximizationThe first stage is the firm’s cost minimization problemminx1,x2w1x1+ w2x2s.t. f(x1, x2) = ywhich can be solved using the LaGrangianL(x1, x2, µ) = w1x1+ w2x2− µ[f(x1, x2) − y]The solution to this problem yields the Input Requirment Functions (IRFs) also calledConditional Input Functions (CIFs) or Conditional Factor Demands (CFDs)x1= xc1(w1, w2, y)x2= xc2(w1, w2, y)Plugging these conditional input functions in to the objective function yields the cost functionC(w1, w2, y) = w1xc1(w1, w2, y) + w2xc2(w1, w2, y)The second step is to find the level of output that maximizes profits by solvingmaxypy − C(w1, w2, y)The FOC isp − Cy(w1, w2, y) = 0 or p = MC(w1, w2, y)The SOC is−Cyy(w1, w2, y) < 0 or MCy(w1, w2, y) > 0The solution to the second stage yields the optimal level of outputy = y∗(w1, w2, p)Finally, the optimal level of output can be substituted into the conditional input demandfunctions to yield the unconditional input demand functionsx1(w1, w2, p) = xc1(w1, w2, y∗(w1, w2, p))x2(w1, w2, p) = xc2(w1, w2, y∗(w1, w2, p))11.2 Direct Profit MaximizationAlternatively, the profit maximization problem can be set up and solved as an unconstrainedmaximization problem in one stepmaxx1,x2pf(x1, x2) − w1x1− w2x2which yields the FOCspfx1(x1, x2) − w1= 0pfx2(x1, x2) − w2= 02 Increasing an Input Price Always Reduces Demand for ThatInputLet’s examine the impact of an increase in input factor price w1on the unconditional amountof x1that the firm uses in production.x1(w1, w2, p) = xc1(w1, w2, y∗(w1, w2, p))∂x1(w1, w2, p)∂w1=∂xc1∂w1+∂xc1∂y·∂y∗(w1, w2, p)∂w1The first term is the substitution effect (change in x1demanded holding y constant). Thesecond term is c alled the “scale effect” and captures the fact that a change in input price w1will move the marginal cost curve, thus changing the optimal level of output y∗.To determine the sign of∂x1(w1,w2,p)∂w1first note that the substitution effect will always be lessthan or equal to zero.Now let’s consider the scale effect. First examine the case of a non-inferior input. Thismeans that demands for the input will increase as the scale of output increases or∂xc1∂y> 0The remaining thing to determine is the sign of∂y∗(w1,w2,p)∂w1. This depends upon whether themarginal cost curve moves up or down (Draw graph).Finally, note that∂MC(w1, w2, y)∂w1=∂∂w1∂C(w1, w2, y)∂y=∂∂y∂C(w1, w2, y)∂w1=∂xc1(w1, w2, y)∂y2where the last equality holds by Shepard’s Lemma. Since we know that∂xc1(w1,w2,y)∂y> 0 for anon-inferior input then MC must rise when w1increases. Finally, it is easy to see that witha fixed price and upward sloping marginal cost curve, a rise in the MC curve lowers optimaloutput y∗so that∂y∗(w1, w2, p)∂w1< 0Thus for a non-inferior input the first term of the scale effect is positive and the second termis negative so the scale effect will be negative∂xc1∂y·∂y∗(w1, w2, p)∂w1< 0For an inferior input∂xc1∂y< 0which means that MC falls as w1increases resulting in∂y∗(w1, w2, p)∂w1> 0In this case, the first term of the scale effect is negative and the second term is positive, thusagain the scale effect is negative.Thus, the scale effect will always be