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Prerequisites Almost essential Firm: Optimisation THE FIRM: DEMAND AND SUPPLY MICROECONOMICS Principles and Analysis Frank Cowell March 2012 Frank Cowell: Firm- Demand & Supply 1 Moving on from the optimum… We derive the firm's reactions to changes in its environment These are the response functions • We will examine three types of them • Responses to different types of market events In effect we treat the firm as a black box the firm March 2012 Frank Cowell: Firm- Demand & Supply 2 The firm as a “black box” 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 March 2012 Frank Cowell: Firm- Demand & Supply 3 Overview… Firm: Comparative Statics Conditional Input Demand Response function for stage 1 optimisation Output Supply Ordinary Input Demand Short-run problem March 2012 Frank Cowell: Firm- Demand & Supply 4 The first response function Review the cost-minimisation problem and its solution Choose z to minimise The “stage 1” problem m S wi zi subject to q f (z), z ≥ 0 i=1 The firm’s cost function: The solution function C(w, q) := min S wizi {f(z) q} Cost-minimising value for each input: zi* = Hi(w, q), i=1,2,…,m could be a welldefined function or a correspondence March 2012 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 Frank Cowell: Firm- Demand & Supply 5 Mapping into (z1,w1)-space Conventional case of Z Start with any value of w1 ( the slope of the tangent to Z) Repeat for a lower value of w1 …and again to get… z2 w1 …the conditional demand curve Constraint set is convex, with smooth boundary Response function is a continuous map: z1 H1(w,q) z1 Now try a different case March 2012 Frank Cowell: Firm- Demand & Supply 6 Another map into (z1,w1)-space Now take case of nonconvex Z Start with a high value of w1 Repeat for a very low value of w1 Points “nearby” work the same way z2 But what happens in between? w1 A demand correspondence Constraint set is nonconvex Response is discontinuous map: jumps in z* Multiple inputs at this price z1 Map is multivalued at the discontinuity z1 no price yields a solution here March 2012 Frank Cowell: Firm- Demand & Supply 7 Conditional input demand function Assume that single-valued input-demand functions exist How are they related to the cost function C? What are their properties? How are their properties related to those of C? • tip if you’re not sure about the cost function… • …check the presentation “Firm Optimisation” • …revise the five main properties of the function C March 2012 Frank Cowell: Firm- Demand & Supply 8 Use the cost function Recall this relationship? …yes, it's Shephard's lemma Ci(w, q) = zi* The slope: C(w, q) ———— wi Optimal demand for input i So we have: Ci(w, q) = Hi(w, q) Link between conditional input demand and cost functions conditional input demand function Differentiate this with respect to wj Cij(w, q) = Hji(w, q) Slope of input demand function Second derivative March 2012 Two simple results: Frank Cowell: Firm- Demand & Supply 9 Simple result 1 Use a standard property second derivatives of a function “commute” So in this case The order of differentiation is irrelevant 2() 2() ——— = ——— wi wj wj wi Cij(w, q) = Cji(w,q) Therefore we have: Hji(w, q) = Hij(w, q) March 2012 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 Frank Cowell: Firm- Demand & Supply 10 Simple result 2 Cij(w, q) = Hji(w, q) Slope of conditional input demand function derived from second derivative of cost function We can get the special case: We've just put j = i Use the standard relationship: Cii(w, q) = Hii(w, q) Because cost function is concave: A general property Therefore: Cii(w, q) 0 Hii(w, q) 0 The relationship of conditional demand for an input with its own price cannot be positive and so… March 2012 Frank Cowell: Firm- Demand & Supply 11 Conditional input demand curve w1 Consider the demand for input 1 Consequence of result 2? H1(w,q) “Downward-sloping” conditional demand In some cases it is also possible that Hii = 0 H11(w, q) < 0 z1 March 2012 Corresponds to the case where isoquant is kinked: multiple w values consistent with same z* Frank Cowell: Firm- Demand & Supply 12 For the conditional demand function… Nonconvex Z yields discontinuous H Cross-price effects are symmetric Own-price demand slopes downward (exceptional case: own-price demand could be constant) March 2012 Frank Cowell: Firm- Demand & Supply 13 Overview… Firm: Comparative Statics Conditional Input Demand Response function for stage 2 optimisation Output Supply Ordinary Input Demand Short-run problem March 2012 Frank Cowell: Firm- Demand & Supply 14 The second response function Review the profit-maximisation problem and its solution Choose q to maximise: The “stage 2” problem pq – C (w, q) From the FOC: p Cq (w, q*) pq* C(w, q*) “Price equals marginal cost” “Price covers average cost” profit-maximising value for output: S is the supply function q* = S (w, p) input prices March 2012 output price (again it may actually be a correspondence) Frank Cowell: Firm- Demand & Supply 15 Supply of output and output price Use the FOC: Cq (w, q) = p “marginal cost equals price” Use the supply function for q: Cq (w, S(w, p) ) = p Gives an equation in w and p Differential of S with respect to p Differentiate with respect to p Use the “function of a function” rule Cqq (w, S(w, p) ) Sp (w, p) = 1 Positive if MC is increasing Rearrange: 1 . Sp (w, p) = ———— Gives slope of supply function Cqq (w, q) March 2012 Frank Cowell: Firm- Demand & Supply 16 The firm’s supply curve 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 March 2012 Supply response is given by q=S(w,p) | _q q Map is multivalued at the discontinuity Frank Cowell: Firm- Demand & Supply 17 Supply of output and price of input j Use the FOC: Cq (w, S(w, p) ) = p Same as before: “price equals marginal cost” Use the “function of a function” Differentiate with respect to wj Cqj(w, q*) + Cqq (w, q*) Sj(w, p) = 0 rule again Rearrange: Cqj(w, q*) Sj(w, p) = – ———— Cqq(w, q*) Supply of output must fall with wj if marginal cost increases with wj Remember, this is positive March 2012 Frank Cowell: Firm- Demand & Supply 18 For the supply function… Supply curve slopes upward Supply decreases with the price of an input, if MC increases with the price of that input Nonconcave f yields discontinuous S IRTS means f is nonconcave and so S is discontinuous March 2012 Frank Cowell: Firm- Demand & Supply 19 Overview… Firm: Comparative Statics Conditional Input Demand Response function for combined optimisation problem Output Supply Ordinary Input Demand Short-run problem March 2012 Frank Cowell: Firm- Demand & Supply 20 The third response function Recall the first two response functions: zi* = Hi(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) ) Use this to define a new function: Di(w,p) := Hi(w, S(w, p) ) input prices March 2012 Stages 1 & 2 combined… output price Demand for input i (unconditional ) Use this relationship to analyse further the firm’s response to price changes Frank Cowell: Firm- Demand & Supply 21 Demand for i and the price of output Take the relationship Di(w, p) = Hi(w, S(w, p)) Differentiate with respect to p: Dpi(w, p) = Hqi(w, q*) Sp(w, p) Di increases with p iff Hi increases with q. Reason? Supply increases with price ( Sp>0) “function of a function” rule again But we also have, for any q: Hi(w, q) = Ci(w, q) Hqi (w, q) = Ciq(w, q) 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 March 2012 Frank Cowell: Firm- Demand & Supply 22 Demand for i and the price of j Again take the relationship Di(w, p) = Hi(w, S(w, p)) Differentiate with respect to wj: Dji(w, p) = Hji(w, q*) + Hqi(w, q*)Sj(w, p) Use Shephard’s Lemma again: Hqi(w, q) = Ciq(w, q) = Cqi(w, q) Use this and the previous result on Sj(w, p) to give a decomposition into a “substitution effect” and an “output effect”: Dji(w, p) = Hji(w, q*) Cjq(w, q*) Ciq(w, q*) Cqq(w, q*) “output effect” . “substitution effect” March 2012 Frank Cowell: Firm- Demand & Supply 23 Results from decomposition formula 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. The effect wi on demand for input j equals the effect of wj on demand for input i Obviously symmetric in i and j. 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. March 2012 If wi increases, the demand for input i cannot rise cannot be positive. Frank Cowell: Firm- Demand & Supply 24 Input-price fall: substitution effect The initial equilibrium w1 price of input falls conditional demand curve original value to firm of price fall, given a fixed output level output level H1(w,q) price fall initial price level Change in cost Notional increase in factor input if output target is held constant z1* March 2012 z1 Frank Cowell: Firm- Demand & Supply 25 Input-price fall: total effect w1 Conditional demand at original output 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 ordinary demand curve z1* March 2012 z** 1 z1 Frank Cowell: Firm- Demand & Supply 26 The ordinary demand function… Nonconvex Z may yield a discontinuous D Cross-price effects are symmetric Own-price demand slopes downward Same basic properties as for H function March 2012 Frank Cowell: Firm- Demand & Supply 27 Overview… Firm: Comparative Statics Conditional Input Demand Optimisation subject to side-constraint Output Supply Ordinary Input Demand Short-run problem March 2012 Frank Cowell: Firm- Demand & Supply 28 The short run… This is not a moment in time but… … is defined by additional constraints within the model Counterparts in other economic applications where we sometimes need to introduce side constraints March 2012 Frank Cowell: Firm- Demand & Supply 29 The short-run problem We build on the firm’s standard optimisation problem Choose q and z to maximise m P := pq – S wizi i=1 subject to the standard constraints: q f (z) q 0, z 0 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… March 2012 Frank Cowell: Firm- Demand & Supply 30 The short-run cost function ~ _ C(w, q, zm ) := min S wi zi {zm =`zm } Short-run demand for input i: ~ _ ~ _ i H (w, q, zm) =Ci(w, q, zm ) Compare with the ordinary cost function ~ _ C(w, q) C(w, q, zm ) So, dividing by q: ~ _ C(w, q) C(w, q, zm ) _______ _________ q q March 2012 The solution function with the side constraint Follows from Shephard’s Lemma 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: Firm- Demand & Supply 31 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 March 2012 q The supply curve is steeper in the short run Frank Cowell: Firm- Demand & Supply 32 Conditional input demand The original demand curve for input 1 w1 H1(w,q) The demand curve from the problem with the side constraint “Downward-sloping” conditional demand Conditional demand curve is steeper in the short run ~ _ H1(w, q, zm) z1 March 2012 Frank Cowell: Firm- Demand & Supply 33 Key concepts Basic functional relations price signals firm input/output responses Hi(w,q) demand for input i, conditional on output Review S (w,p) supply of output Review Di(w,p) demand for input i (unconditional ) Review And they all hook together like this: Hi(w, S(w,p)) = Di(w,p) March 2012 Frank Cowell: Firm- Demand & Supply 34 What next? Analyse the firm under a variety of market conditions Apply the analysis to the consumer’s optimisation problem March 2012 Frank Cowell: Firm- Demand & Supply 35