# Prerequisites Almost essential Firm Optimisation Frank Cowell Microeconomics

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Prerequisites Almost essential Firm: Optimisation Frank Cowell: Microeconomics October 2011 The Firm: Demand Supply MICROECONOMICS Principles and Analysis Frank Cowell

Moving on from the optimum. . . Frank Cowell: Microeconomics n n We derive the firm's reactions to changes in its environment. These are the response functions. u u n We will examine three types of them Responses to different types of market events. In effect we treat the firm as a Black Box. market prices the firm output l evel; input de mands

The firm as a “black box” Frank Cowell: Microeconomics n n n 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 We need the solution function’s properties… …again and again

Overview. . . Frank Cowell: Microeconomics Firm: Comparative Statics Conditional Input Demand Response function for stage 1 optimisation Output Supply Ordinary Input Demand Short-run problem

The first response function Frank Cowell: Microeconomics Review the cost-minimisation problem and its solution l l Choose z to minimise m §The “stage 1” problem S wi zi subject to q f(z), z≥ 0 i=1 l The firm’s cost function: C(w, q) : = min S wizi §The solution function {f(z) ³q} l Cost-minimising value for each input: zi* = Hi(w, q), i=1, 2, …, m may be a well-defined function or may be a correspondence Specified vector of output level input prices § Hi is the conditional input demand function. §Demand for input i, conditional on given output level q A graphical approach

Mapping into (z 1, w 1)-space Frank Cowell: Microeconomics §Conventional case of Z. §Start with any value of w 1 (the slope of the tangent to Z). z 2 §Repeat for a lower value of w 1. §. . . and again to get. . . §. . . the conditional demand curve w 1 § Constraint set is convex, with smooth boundary l l § Response function is a continuous map: l z 1 H 1(w, q) z 1 Now try a different case

Another map into (z 1, w 1)-space Frank Cowell: Microeconomics z 2 §Now take case of nonconvex Z. §Start with a high value of w 1. §Repeat for a very low value of w 1. §Points “nearby” work the same way. § But what happens in between? w 1 §A demand correspondence l § Constraint set is nonconvex. § Response is a discontinuous map: jumps in z* l Multiple inputs at this price l l z 1 no price yields a solution here z 1 §Map is multivalued at the discontinuity

Conditional input demand function Frank Cowell: Microeconomics Link to cost function n n Assume that single-valued inputdemand functions exist. How are they related to the cost function? What are their properties? How are they related to properties of the cost function? Do you remember these. . . ?

Use. Thethe slope: cost function Frank Cowell: Microeconomics n ¶C(w, q) ———— ¶wi Optimal demand for input i Recall this relationship? Ci(w, q) = zi* conditional input demand function n So we have: Ci(w, q) = Hi(w, q) n Second derivative Differentiate this with respect to wj Cij(w, q) = Hji(w, q) §. . . yes, it's Shephard's lemma §Link between conditional input demand cost functions §Slope of input demand function Two simple results:

Simple result 1 Frank Cowell: Microeconomics n Use a standard property n So in this case ¶ 2( ) ——— = ——— ¶wi ¶wj ¶wi Cij(w, q) = Cji(w, q) n Therefore we have: Hji(w, q) = Hij(w, q) §second derivatives of a function “commute” §The order of differentiation is irrelevant §The effect of the price of input i on conditional demand for input j equals the effect of the price of input j on conditional demand for input i.

Simple result 2 Frank Cowell: Microeconomics n Use the standard relationship: Cij(w, q) = Hji(w, q) n We can get the special case: Cii(w, q) = Hii(w, q) n §We've just put j=i Because cost function is concave: §A general property Therefore: §The relationship of conditional demand for an input with its own price cannot be positive. Cii(w, q) 0 n §Slope of conditional input demand function derived from second derivative of cost function Hii(w, q) 0 and so. . .

Conditional input demand curve Frank Cowell: Microeconomics Link to “kink” figure w 1 § Consider the demand for input 1 § Consequence of result 2? H 1(w, q) § “Downward-sloping” conditional demand § In some cases it is also possible that Hii=0 H 11(w, q) < 0 z 1 § Corresponds to the case where isoquant is kinked: multiple w values consistent with same z*.

Frank Cowell: Microeconomics For the conditional demand function. . . n n Nonconvex Z yields discontinuous H Cross-price effects are symmetric Own-price demand slopes downward (exceptional case: own-price demand could be constant)

Overview. . . Frank Cowell: Microeconomics Firm: Comparative Statics Conditional Input Demand Response function for stage 2 optimisation Output Supply Ordinary Input Demand Short-run problem

The second response function Frank Cowell: Microeconomics Review the profit-maximisation problem and its solution n. Choose q to maximise: n pq – C (w, q) n From the FOC: p Cq (w, q*) pq* ³ C(w, q*) l §The “stage 2” problem profit-maximising value for output: q* = S (w, p) input prices output price §“Price equals marginal cost” §“Price covers average cost” § S is the supply function §(again it may actually be a correspondence)

Supply of output and output price Frank Cowell: Microeconomics Use the FOC: Cq (w, q) = p n §“marginal cost equals price” Use the supply function for q: Cq (w, S(w, p) ) = p §Gives an equation in w and p n Differential of S with respect to p Differentiate with respect to p Cqq (w, S(w, p) ) Sp (w, p) = 1 n n §Use the “function of a function” rule Positive if MC is §Gives the slope increasing. Rearrange: 1. function. Sp (w, p) = ———— Cqq (w, q) of the supply

The firm’s supply curve Frank Cowell: Microeconomics p § The firm’s AC and MC curves. § For given p read off optimal q* § Continue down to p § What happens below p Cq C/q § Case illustrated is for f with first IRTS, then DRTS. Response is a discontinuous map: jumps in q* Multiple q* at this price _p – no price yields a solution here §Supply response is given by q=S(w, p) | _q q §Map is multivalued at the discontinuity

Frank Cowell: Microeconomics Supply of output and price of input j Use the FOC: Cq (w, S(w, p) ) = p §Same as before: “price equals marginal cost” Differentiate with respect to wj Cqj(w, q*) + Cqq (w, q*) Sj(w, p) = 0 §Use the “function of a function” rule again n Rearrange: Cqj(w, q*) Sj(w, p) = – ———— Cqq(w, q*) Remember, this is positive §Supply of output must fall with wj if marginal cost increases with wj.

For the supply function. . . Frank Cowell: Microeconomics Supply curve slopes upward n Supply decreases with the price of an input, if MC increases with the price of that input n Nonconcave f yields discontinuous S n IRTS means f is nonconcave and so S is discontinuous n

Overview. . . Frank Cowell: Microeconomics Firm: Comparative Statics Conditional Input Demand Response function for combined optimisation problem Output Supply Ordinary Input Demand Short-run problem

The third response function Frank Cowell: Microeconomics input prices Recall the first two response functions: * i n n zi = H (w, q) §Demand for input i, conditional on output q q* = S (w, p) §Supply of output Now substitute for q* : zi* = Hi(w, S(w, p) ) n §Stages 1 & 2 combined… Use this to define a new function: Di(w, p) : = Hi(w, S(w, p) ) output price §Demand for input i (unconditional ) §Use this relationship to analyse further the firm’s response to price changes

Demand for i and the price of output Frank Cowell: Microeconomics n Take the relationship n Differentiate with respect to p: “function of a Di(w, p) = Hi(w, S(w, p)). function” rule again Dpi(w, p) = Hqi(w, q*) Sp(w, p) n But we also have, for any q: Hi(w, q) = Ci(w, q) Hqi (w, q) = Ciq(w, q) n §Di increases with p iff Hi increases with q. Reason? Supply increases with price ( Sp>0). §Shephard’s Lemma again Substitute in the above: Dpi(w, p) = Cqi(w, q*)Sp(w, p) §Demand for input i (Di) increases with p iff marginal cost (Cq) increases with wi.

Demand for i and the price of j Frank Cowell: Microeconomics n Again take the relationship Di(w, p) = Hi(w, S(w, p)). n Differentiate with respect to wj: Dji(w, p) = Hji(w, q*) + Hqi(w, q*)Sj(w, p) n Use Shephard’s Lemma again: Hqi(w, q) = Ciq(w, q) = Cqi(w, q) “substitution Use this and the previous result on Sj(w, p) to give a effect” decomposition into a “substitution effect” and an “output effect”: n Dji(w, p) = Hji(w, q*) Cjq(w, q*) Ciq(w, q*) Cqq(w, q*). “output effect”

Results from decomposition formula Frank Cowell: Microeconomics n Take the general relationship: Ciq(w, q*)Cjq(w, q*) Dji(w, p) = Hji(w, q*) Cqq(w, q*). We already know this is symmetric in i and j. n Obviously symmetric in i and j. §The effect wi on demand for input j equals the effect of wj on demand for input i. Now take the special case where j = i: Ciq(w, q*)2 Dii(w, p) = Hii(w, q*) Cqq(w, q*). We already know this is negative or zero. cannot be positive. §If wi increases, the demand for input i cannot rise.

Frank Cowell: Microeconomics Input-price fall: substitution effect w 1 § The initial equilibrium § price of input falls conditional demand curve original § value to firm of price fall, given a fixed output level H 1(w, q) price fall initial price level Change in cost Notional increase in factor input if output target is held constant z 1* z 1

Input-price fall: total effect Frank Cowell: Microeconomics Conditional demand at original output w 1 § The initial equilibrium § Substitution effect of inputprice of fall. § Total effect of input-price fall Conditional demand at new output price fall initial price level Change in cost ordinary demand curve z 1* z** 1 z 1

The ordinary demand function. . . Frank Cowell: Microeconomics Nonconvex Z may yield a discontinuous D n Cross-price effects are symmetric n Own-price demand slopes downward n n Same basic properties as for H function

Overview. . . Frank Cowell: Microeconomics Firm: Comparative Statics Conditional Input Demand Optimisation subject to sideconstraint Output Supply Ordinary Input Demand Short-run problem

The short run. . . Frank Cowell: Microeconomics This is not a moment in time but… n … is defined by additional constraints within the model n Counterparts in other economic applications where we sometimes need to introduce side constraints n

The short-run problem Frank Cowell: Microeconomics We build on the firm’s standard optimisation problem n Choose q and z to maximise n P : = pq – n m S w i zi i=1 subject to the standard constraints: q f (z) q ³ 0, z ³ 0 n But we add a side condition to this problem: zm = `zm Let `q be the value of q for which zm =`zm would have been freely chosen in the unrestricted cost-min problem… n

The short-run cost function Frank Cowell: Microeconomics ~ _ C(w, q, zm ) : = min S wi zi {zm =`zm } Short-run demand for input i: n ~ _ i H (w, q, zm) =Ci(w, q, zm ) Compare with the ordinary cost function ~ _ §The solution function with the side constraint. §Follows from Shephard’s Lemma n C(w, q) C(w, q, zm ) n So, dividing by q: ~ _ C(w, q) C(w, q, zm ) _________ q q §By definition of the cost function. We have “=” if q =`q. §Short-run AC ≥ long-run AC. SRAC = LRAC at q =`q Supply curves

Frank Cowell: Microeconomics MC, AC and supply in the short and long run § AC if all inputs are variable § MC if all inputs are variable § Fix an output level. p § AC if input m is now kept fixed ~ Cq § MC if input m is now kept fixed § Supply curve in long run Cq ~ C/q § Supply curve in short run C/q §SRAC touches LRAC at the given output §SRMC cuts LRMC at the given output q q §The supply curve is steeper in the short run

Conditional input demand Frank Cowell: Microeconomics w 1 § The original demand curve for input 1 § The demand curve from the problem with the side constraint. H 1(w, q) § “Downward-sloping” conditional demand § Conditional demand curve is steeper in the short run. ~ _ H 1(w, q, zm) z 1

Key concepts Frank Cowell: Microeconomics Basic functional relations n price signals firm input/output responses n Review n Hi(w, q) demand for input i, conditional on output Review n S (w, p) supply of output Di(w, p) demand for input i (unconditional ) Review n And they all hook together like this: n Hi(w, S(w, p)) = Di(w, p)

What next? Frank Cowell: Microeconomics Analyse the firm under a variety of market conditions. n Apply the analysis to the consumer’s optimisation problem. n

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