The Firm: Comparative Statics,MicroEconomics,The Firm,Overview.,.we derive the firms reactions to changes in its environment.,These are the response functions.,We will examine three types of them, treating the firm as a.,.Black Box,Moving on from the optimum.,How it works.,Use the fact that the firm is an optimiser Behaviour can be predicted by necessary and sufficient conditions for optimum The FOC can be solved to yield behavioural response functions. Their properties derive from the solution function,The firm as a “black box”,The Firm,Production,Output Supply,Ordinary Input Demand,Optimisation,Comparative Statics,The Firm and the Market,Output Supply,Conditional Input Demand,Ordinary Input Demand,black box problems,Choose z to minimise,Q G(z),.subject to the production constraint.,z 0,.and the obvious non-negativity conditions,The solution to the first-stage problem .,C(w, Q) := min wi zi,vector of input prices,Specified output level,Yields minimised cost as a function of exogenous variables.,one for each of the m inputs,z1* = H1(w,Q) z2* = H2(w,Q) . . . . zm* = Hm(w,Q), ,.optimal input demands as a function of exogenous variables,demand for input i, conditional on output Q,We need to examine the first stage of the optimisation process,zi* = Hi(w,Q),A function of input prices,.and output level,conditional input demand function,(our first response function),Result depends on shape of Z,z,1,z,2,z,1,z,2,z,1,z,2,z,1,z,2,Take the conventional case.,Map the optimum into (z1,w1)-space,Start with an arbitrary value of w1 .,Do it again for a lower value of w1 .,.and again to get.,H1(w,Q),the conditional demand curve,In the “conventional“ case.,. the constraint set is convex, with a smooth boundary,We find the solution is a continuous map.,. that is single valued.,Points to note,Result depends on shape of Z,z,1,z,2,z,1,z,2,z,1,z,2,z,1,z,2,What about the non-convex case.?,again map the optimum into (z1,w1)-space,Z(Q),_,2,w1,z2,z1,z1,.now try a very low value of w1,But what happens in between?,Nonconvex Z : jumps in z*,w1,z1,The demand correspondence,Points to note,In this case. . the constraint set is nonconvex We find the solution is a discontinuous map The map is multivalued at the discontinuity.,Let us set this difficulty aside.,Lets take it for granted that single-valued input-demand functions exist.,How are they related to the cost function?,What are their properties?,How are they related to properties of the cost function?,Assume the existence of a conditional input demand function,Remember this.?,Ci(w, Q) = zi*,.yes, its Shephards lemma,And so.,Ci(w, Q) = Hi(w, Q),Now lets differentiate this.,conditional input demand function,Which gives us.,Cij(w, Q) = Hji(w, Q),second derivative,Cji(w, Q) = Cij(w, Q),And now for a simple result:,=,2 _ wj wi,2 _ wi wj,Second derivatives commute.,The effect of the price of input j on conditional demand for input i,Hij(w, Q) = Hji(w, Q),The effect of the price of input i on conditional demand for input j,The economic meaning.,Now for an even simpler result:,Cii(w, Q) =,Hii(w, Q),this must be negative,. so this must be negative too,. and so:,Because the cost function is concave in prices:,Consider the demand for input 1,conditional demand curve,zi,wi,Hi(w,Q),Hii(w, Q) 0,The conditional demand curve slopes downwards,Nonconvex Z yields discontinuous H,Cross-price effects are symmetric,Own-price demand slopes downward.,For the conditional demand function.,The Firm,Production,Output Supply,Ordinary Input Demand,Optimisation,Comparative Statics,The Firm and the Market,Conditional Input Demand,Conditional Input Demand,Ordinary Input Demand,black box problems,max PQ - C(w,Q) s.t. Q 0,The second -stage problem,Q* = S (w, P),supply of output,We need to examine the second stage of the optimisation process,(our second response function),For a given P read off optimal Q,Q,CQ,C/Q,Now let P fall.,Note what happens below Average Cost.,P,Q,_,_,Q=S(w,P),Supply curve,z2,Q,0,Production function with local IRTS,z1,Supply curve slopes upward,Nonconcave G yields discontinuous S,IRTS means G is nonconcave and so S is discontinuous,For the supply function.,Optimisation,The Firm,Production,Output Supply,Ordinary Input Demand,Comparative Statics,The Firm and the Market,Conditional Input Demand,Conditional Input Demand,Output Supply,black box problems,zi* = Hi(w,Q),Q* = S (w, P),Now put together the two stages of the optimisation process,By substitution:,Hi(w, S(w, P),Di(w, P),demand for input i (unconditional ),=:,(our third response function),Differentiate for the uncompensated demand,Total= Substitution effect+Output effect,Unconditional Demand can be determined from the cost function,From Shepherds lemma,And,Since,Can solve Sj(w,P)in terms of C(w,P),Hotellings Lemma,Prove Assuming one input