Moving on from the optimum... The Firm: Demand and Supply. The firm as a black box. Overview...

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1 Prereuisites Almost essential Firm: Optimisation Frank Cowell: Microeconomics The Firm: Demand and Supply MICROECONOMICS Principles and Analysis Frank Cowell 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 October 2005 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. Overview... Response function for stage 1 optimisation Firm: Comparative Statics Output Supply Ordinary Short-run 1

2 may be a well-defined function or may be a correspondence The first response function Review the cost-minimisation and its solution Choose z to minimise m Σ w i z i subject to φ(z), z 0 i=1 The firm s cost function: C(w, ) := min Σ w i z i {φ(z) } Cost-minimising value for each input: z i* = H i (w, ), i=1,2,,m Specified vector of output level input prices The stage 1 The solution function H i is the conditional input demand function. Demand for input i, conditional on given output level A graphical approach Mapping into (, )-space z 2 Conventional case of Z. Start with any value of (the slope of the tangent to Z). Repeat for a lower value of....and again to get......the conditional demand curve Constraint set is convex, with smooth boundary Response function is a continuous map: H 1 (w,) Now try a different case Another map into (, )-space input demand function z 2 Multiple inputs at this price Now take case of nonconvex Z. Start with a high value of. Repeat for a very low value of. Points nearby work the same way. But what happens in between? A demand correspondence Constraint set is nonconvex. Response is a discontinuous map: jumps in z* Map is multivalued at the discontinuity Link to cost function 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...? no price yields a solution here 2

3 Use The the slope: cost function C(w, ) w i n Recall this relationship? C i (w, ) = z * i n So we have: C i (w, ) = H i (w, ) Optimal demand for input i conditional input demand function...yes, it's Shephard's lemma Link between conditional input demand and cost functions Simple result 1 n Use a standard property 2 ( ) 2 ( ) = w i w j w j w i n So in this case C ij (w, ) = C ji (w, ) second derivatives of a function commute The order of differentiation is irrelevant Second n Differentiate derivative this with respect to w j C ij (w, ) = H ji (w, ) Slope of input demand function Two simple results: n Therefore we have: H ji (w, ) = H ij (w, ) The effect of the price of input i on conditional demand for input j euals the effect of the price of input j on conditional demand for input i. Simple result 2 input demand curve n Use the standard relationship: C ij (w, ) = H ji (w, ) n We can get the special case: C ii (w, ) = H ii (w, ) n Because cost function is concave: C ii (w, ) 0 n Therefore: H ii (w, ) 0 Slope of conditional input demand function derived from second derivative of cost function We've just put j=i A general property The relationship of conditional demand for an input with its own price cannot be positive. and so... Link to kink figure H 1 (w,) H 11 (w, ) < 0 Consider the demand for input 1 Conseuence of result 2? Downward-sloping conditional demand In some cases it is also possible that H ii =0 Corresponds to the case where isouant is kinked: multiple w values consistent with same z*. 3

4 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) Overview... Response function for stage 2 optimisation Firm: Comparative Statics Output Supply Ordinary Short-run The second response function n Review the profit-maximisation and its solution nchoose to maximise: p C (w, ) n From the FOC: p C (w, * ) p * C(w, * ) profit-maximising value for output: * = S (w, p) input prices output price The stage 2 Price euals marginal cost Price covers average cost S is the supply function (again it may actually be a correspondence) Supply of output and output price n Use the FOC: C (w, ) = p n Use the supply function for : C (w, S(w, p) ) = p Differential of S with respect to p n Differentiate with respect to p C (w, S(w, p) ) S p (w, p) = 1 n Rearrange: 1. S p (w, p) = C (w, ) marginal cost euals price Gives an euation in w and p Use the function of a function rule Positive if MC is increasing. Gives the slope of the supply function. 4

5 The firm s supply curve p The firm s AC and MC curves. For given p read off optimal * Continue down to p What happens below p Supply of output and price of input j n Use the FOC: C (w, S(w, p) ) = p Same as before: price euals marginal cost C C/ Supply response is given by =S(w,p) n Differentiate with respect to w j C j (w, * ) + C (w, * ) S j (w, p) = 0 Use the function of a function rule again Multiple * at this price p_ no price yields a solution here _ Case illustrated is for φ with first IRTS, then DRTS. Response is a discontinuous map: jumps in * Map is multivalued at the discontinuity n Rearrange: C j (w, * ) S j (w, p) = C (w, * ) Remember, this is positive Supply of output must fall with w j if marginal cost increases with w j. 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 φ yields discontinuous S. IRTS means φ is nonconcave and so S is discontinuous. Overview... Response function for combined optimisation Firm: Comparative Statics Output Supply Ordinary Short-run 5

6 input prices The third response function n Recall the first two response functions: z i* = H i (w,) * = S (w, p) n Now substitute for * : z i* = H i (w, S(w, p) ) n Use this to define a new function: D i (w,p) := H i (w, S(w, p) ) output price Demand for input i, conditional on output Supply of output Stages 1 & 2 combined 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 n Take the relationship D i (w, p) = H i function of a (w, S(w, p)). function rule again n Differentiate with respect to p: D pi (w, p) = H i (w, * ) S p (w, p) n But we also have, for any : H i (w, ) = C i (w, ) H i (w, ) = C i (w, ) n Substitute in the above: D i increases with p iff H i increases with. Reason? Supply increases with price ( S p >0). Shephard s Lemma again D pi (w, p) = C i (w, * )S p (w, p) Demand for input i (D i ) increases with p iff marginal cost (C ) increases with w i. Demand for i and the price of j n Again take the relationship D i (w, p) = H i (w, S(w, p)). n Differentiate with respect to w j : function of a function rule yet again D ji (w, p) = H ji (w, * ) + H i (w, * )S j (w, p) n Use Shephard s Lemma again: H i (w, ) = C i (w, ) = C i (w, ) n Use this and the previous result substitution S j (w, p) to give a effect decomposition into a substitution effect and an output effect : C i (w, * )C j (w, * ) D ji (w, p) = H ji (w, * ) C (w, * ). output effect Results from decomposition formula n Take the general relationship: C i (w, * )C j (w, * ) D ji (w, p) = H ji (w, * ) C (w, * ). We already know this is symmetric in i and j. n Now take the special case where j = i: We already know this is negative or zero. Obviously symmetric in i and j. C i (w, * ) 2 D ii (w, p) = H ii (w, * ) C (w, * ). cannot be positive. The effect w i on demand for input j euals the effect of w j on demand for input i. If w i increases, the demand for input i cannot rise. 6

7 initial price level Input-price fall: substitution effect conditional demand curve H 1 (w,) original output level The initial euilibrium price of input falls value to firm of price fall, given a fixed output level initial price level Input-price fall: total effect demand at original output demand at new output The initial euilibrium Substitution effect of inputprice of fall. Total effect of input-price fall price fall Change in cost Notional increase in factor input if output target is held constant price fall Change in cost ordinary demand curve z* 1 z* 1 z** 1 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 Overview... Optimisation subject to sideconstraint Firm: Comparative Statics Output Supply Ordinary Short-run 7

8 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 The short-run n We build on the firm s standard optimisation n Choose and z to maximise Π := p m Σ w i z i n subject to the standard constraints: φ (z) 0, z 0 i=1 n But we add a side condition to this : z m = z m n Let be the value of for which z m = z m would have been freely chosen in the unrestricted cost-min The short-run cost function ~ _ C(w,, z m ) := min Σ w i z i {z m = z m } ncompare with the ordinary cost function ~ _ C(w, ) C(w,, z m ) n So, dividing by : ~ _ C(w, ) C(w,, z m ) The solution function with the side constraint. nshort-run demand for input i: ~ _ ~ _ Follows from Shephard s H i (w,, z m ) =C i (w,, z m ) Lemma By definition of the cost function. We have = if =. Short-run AC long-run AC. SRAC = LRAC at = Supply curves MC, AC and supply in the short and long run p ~ C/ ~ C C C/ AC if all inputs are variable MC if all inputs are variable Fix an output level. AC if input m is now kept fixed MC if input m is now kept fixed Supply curve in long run Supply curve in short run SRAC touches LRAC at the given output SRMC cuts LRMC at the given output The supply curve is steeper in the short run 8

9 input demand Key concepts H 1 (w,) The original demand curve for input 1 The demand curve from the with the side constraint. Basic functional relations price signals firm input/output responses Downward-sloping conditional demand demand curve is steeper in the short run. Review Review Review n H i (w,) n S (w,p) n D i (w,p) demand for input i, conditional on output supply of output demand for input i (unconditional ) ~ _ H 1 (w,, z m ) n H i (w, S(w,p)) = D i (w,p) And they all hook together like this: What next? Analyse the firm under a variety of market conditions. Apply the analysis to the consumer s optimisation. 9