Prerequisites Prerequisites Almost Almost essential essential Firm: Firm:Optimisation Optimisation THE FIRM: DEMAND AND SUPPLY MICROECONOMICS Principles and Analysis Frank Cowell April 2018 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 mmarket ar ppricesket
rices April 2018 the firm ooutput utputlelevel; vel ininput d put deemmand;s ands 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 functions properties apply them again and again April 2018 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 April 2018 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 w z subject to q f i=1 i i
(z), z 0 The firms cost function: C(w, q) := min S wizi The solution function {f(z) q} Cost-minimising value for each input: Hi :conditional input demand function z = H (w, q), i=1,2,,m * i could could be be aa wellwelldefined defined function function or or aa correspondence correspondence April 2018 i Demand for input i, conditional on given output level q Specified Specified output
output level level AAgraphical graphical approach approach vector vector of of input input prices prices Frank Cowell: Firm- Demand & Supply 5 Mapping into (z1,w1)-space Left-hand panel: conventional case of Z the slope of the tangent: value of w1 Repeat for a lower value of w1 and again to get Green curve: conditional demand curve z2 w1
Constraint set is convex, with smooth boundary Response function is a continuous map: z1 H1(w,q) z1 Now Nowtry tryaa different differentcase case April 2018 Frank Cowell: Firm- Demand & Supply 6 Another map into (z1,w1)-space z2
w1 Left-hand panel: nonconvex Z Start with a high value of w1 Repeat for a very low value of w1 Points nearby `work the same way But what happens in between? A demand correspondence Constraint set is nonconvex Response is discontinuous: jumps in z* Multiple Multiple inputs inputs at this price at this price Map multivalued at discontinuity
z1 z1 no no price price yields yields aa solution solution here here April 2018 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 youre not sure about the cost function: check the presentation Firm Optimisation revise the five main properties of the function C April 2018 Frank Cowell: Firm- Demand & Supply 8
Use the cost function Recall this relationship? Yes, it's Shephard's lemma Ci (w, q) = zi* The The slope: slope: C(w, C(w, q) q) w wii Optimal Optimal demand demand for for input input ii So we have: Ci (w, q) = Hi(w, q) Link between conditional input demand and cost functions
conditional conditional input input demand demand function function Differentiate this with respect to wj Cij (w, q) = Hji(w, q) Second Second derivative derivative April 2018 Slope of input conditional demand function: effect of Dwj on zi* for given q Two Twosimple simple results: results: Frank Cowell: Firm- Demand & Supply 9 Simple result 1 Use a standard property 2() 2() =
wi wj wj wi So in this case Cij (w, q) = Cji (w,q) Therefore we have: Hji(w, q) = Hij(w, q) April 2018 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 Frank Cowell: Firm- Demand & Supply 10 Simple result 2 Use the standard relationship: Cij (w, q) = Hji(w, q) We can get the special case:
Cii (w, q) = Hii(w, q) Slope of conditional input demand function derived from second derivative of cost function We've just put j = i Because cost function is concave: A general property Cii (w, q) 0 Therefore: Hii(w, q) 0 The relationship of conditional demand for an input with its own price cannot be positive and andso so April 2018 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 possible that Hii = 0 H H1111(w, (w, q) q) << 00 z1 April 2018 Corresponds to case where isoquant is kinked: multiple w values consistent with same z* Frank Cowell: Firm- Demand & Supply 12 Conditional demand function: summary Nonconvex Z yields discontinuous H Cross-price effects are symmetric Own-price demand slopes downward (exceptional case: own-price demand could be constant)
April 2018 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 April 2018 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*), if q* > 0 Price equals marginal cost Price covers average cost pq* C(w, q*) profit-maximising value for output: q* = S (w, p) input input prices prices April 2018 output output price price S is the supply function (again it may be a correspondence) Frank Cowell: Firm- Demand & Supply 15
Supply of output and output price Use the FOC: marginal cost equals price Cq (w, q*) = p Use the supply function for q: Cq (w, S(w, p) ) = p Gives an equation in w and p Differential Differential of of SS with with respect respect to to pp Differentiate with respect to p Use the function of a function rule Cqq (w, S(w, p) ) Sp (w, p) = 1 Positive Positive ifif MC MC is is increasing increasing Rearrange: 1 Sp (w, p) = * Cqq (w, q )
April 2018 Gives slope of supply function Frank Cowell: Firm- Demand & Supply 16 The firms supply curve p AC (green) and MC (red) curves For given p read off optimal q* Continues down to p Check what happens below p Cq C/q Case illustrated is for f with first decreasing AC, then increasing AC, Response is a discontinuous map: jumps in q* Multiple Multiple q* q* at at this this price price
_p | no no price price yields yields aa solution solution here here April 2018 Supply response given by q=S(w,p) _q Multivalued at the q 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 Differentiate with respect to wj Cqj (w, q*) + Cqq (w, q*) Sj (w, p) = 0 Use the function of a function rule again
Rearrange: Cqj (w, q*) Sj (w, p) = Cqq (w, q*) Supply of output must fall with wj if MC increases with wj Remember, Remember, this this is is positive positive April 2018 Frank Cowell: Firm- Demand & Supply 18 Supply function: summary 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 April 2018 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 April 2018 Frank Cowell: Firm- Demand & Supply 20 The third response function Recall the first two response functions: zi* = Hi(w,q) q* = S (w, p)
Now substitute for q* : zi* = Hi(w, S(w, p) ) April 2018 Supply of output Stages 1 & 2 combined Use this to define a new function: Di(w,p) := Hi(w, S(w, p) ) input input prices prices Demand for input i, conditional on output q output output price price Demand for input i (unconditional ) Use this relationship to analyse firms 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: Di increases with p iff Hi increases Dp (w, p) = Hq (w, q ) Sp(w, p) i i * with q. Reason? Supply increases with price ( Sp > 0 ) function functionofofaa function functionrule ruleagain again But we also have, for any q: Hi(w, q) = Ci (w, q) Hqi (w, q) = Ciq (w, q) Substitute in the above: Dpi(w, p) = Cqi (w, q*) Sp (w, p) April 2018 Shephards Lemma again Demand for input i (Di) increases with p iff marginal cost (Cq) increases with wi 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 Shephards Lemma again: Hqi(w, q) = Ciq (w, q) Use this and the previous result on Sj (w, p) to give a decomposition into a substitution effect and an output effect: Cjq(w, q*) Dji(w, p) = Hji(w, q*) Ciq(w, q*) * Csubstitution (w, q ) substitution qq output output effect effect . effect effect Substitution effect is just slope of conditional input demand curve Output effect is [effect of Dwj on q][effect of Dq on demand for 23 April 2018 i] Frank Cowell: Firm- Demand & Supply
Results from decomposition formula Take the general relationship: Ciq(w, q*)Cjq(w, q*) Dji(w, p) = Hji(w, q*) Cqq(w, q*) . We We know know this this is is symmetric symmetric in in ii and and jj The effect wi on demand for input j equals the effect of wj on demand for input i Symmetric Symmetric in in ii and and jj Now take the special case where j = i: Ciq(w, q*)2 Dii(w, p) = Hii(w, q*) Cqq(w, q*) We We know know this this is
is negative or zero negative or zero April 2018 If wi increases, the demand for input i cannot rise . cannot cannot be be positive positive Frank Cowell: Firm- Demand & Supply 24 Input-price fall: substitution effect w1 conditional demand curve original z1* : initial equilibrium
grey arrow: fall in w1 shaded area: value of price fall original output outputlevel level H1(w,q) price fall initial initialprice price level level Change in cost z1* April 2018 Notional Notionalincrease increaseininfactor factor input if output
target is input if output target isheld held constant constant z1 Frank Cowell: Firm- Demand & Supply 25 Input-price fall: total effect w1 Conditional Conditional demand demandatat original originaloutput output Conditional Conditional demand demandatatnew new output output z1* : initial equilibrium
green line: substitution effect z1** : new equilibrium price fall initial initialprice price level level ordinary ordinarydemand demand curve curve z1* April 2018 z1** z1 Frank Cowell: Firm- Demand & Supply 26 Ordinary demand function: summary
Nonconvex Z may yield a discontinuous D Cross-price effects are symmetric Own-price demand slopes downward Same basic properties as for H function April 2018 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 April 2018 Frank Cowell: Firm- Demand & Supply 28 The short run: concept This is not a moment in time
It is defined by additional constraints within the model Counterparts in other economic applications where one may need to introduce side constraints April 2018 Frank Cowell: Firm- Demand & Supply 29 The short-run problem We build on the firms standard optimisation problem Choose q and z to maximise m P := pq S wi zi 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 April 2018 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 April 2018 The solution function with the side constraint
Follows from Shephards Lemma By definition of the cost function. We have = if q =`q Short-run AC long-run AC. SRAC = LRAC at q =`q Supply Supplycurves curves Frank Cowell: Firm- Demand & Supply 31 MC, AC and supply in the short and long run green curve: AC if all inputs variable red curve: MC if all inputs variable `q : given output level p ~ Cq C/q black curve: AC if input m kept fixed
brown curve: MC if input m kept fixed LR supply curve follows LRMC SR supply curve follows SRMC Cq ~ C/q SRAC touches LRAC at given output SRMC cuts LRMC at given output q April 2018 q Supply curve steeper in the short run Frank Cowell: Firm- Demand & Supply 32 Conditional input demand w1
Brown curve: demand for input 1 H1(w,q) Purple curve: demand for input 1 in problem with the side constraint Downward-sloping conditional demand Conditional demand curve is steeper in the short run ~ _ 1 H (w, q, zm) z1 April 2018 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
S (w,p) supply of output D (w,p) demand for input i (unconditional ) i And they all hook together like this: Hi(w, S(w,p)) = Di(w,p) April 2018 Frank Cowell: Firm- Demand & Supply 34 What next? Analyse the firm under a variety of market conditions Apply the analysis to the consumers optimisation problem April 2018 Frank Cowell: Firm- Demand & Supply 35
