Prerequisites Almost essential Firm: Optimisation THE FIRM: DEMAND AND SUPPLY MICROECONOMICS Principles and Analysis Frank Cowell July 2017 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 July 2017 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 apply them again and again July 2017 3
Overview Firm: Comparative Statics Response function for stage 1 optimisation Conditional Input Demand Output Supply Ordinary Input Demand Short-run problem July 2017 4
The first response function Review the cost-minimisation problem and its solution Choose z to minimise m Σ w i z i subject to q φ (z), z 0 i=1 The firm s cost function: C(w, q) := min Σ w i z i Cost-minimising value for each input: could be a welldefined function or a correspondence {φ(z) q} z i* = H i (w, q), i=1,2,,m Specified vector of output level input prices The stage 1 problem The solution function H i is the conditional input demand function Demand for input i, conditional on given output level q A graphical approach July 2017 5
Mapping into (z 1,w 1 )-space Conventional case of Z Start with any value of w 1 ( the slope of the tangent to Z) z 2 w 1 Repeat for a lower value of w 1 and again to get the conditional demand curve Constraint set is convex, with smooth boundary Response function is a continuous map: H 1 (w,q) z 1 z 1 Now try a different case July 2017 6
Another map into (z 1,w 1 )-space Now take case of nonconvex Z Start with a high value of w 1 z 2 w 1 Repeat for a very low value of w 1 Points nearby `work the same way But what happens in between? A demand correspondence Constraint set is nonconvex Multiple inputs at this price Response is discontinuous map: jumps in z* Map is multivalued at the discontinuity z 1 z 1 no price yields a solution here July 2017 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 July 2017 8
Use the cost function Recall this relationship? C i (w, q) = z i * The slope: C(w, q) w i So we have: C i (w, q) = H i (w, q) conditional input demand function Differentiate this with respect to w j C ij (w, q) = H ji (w, q) Second derivative Optimal demand for input i Yes, it's Shephard's lemma Link between conditional input demand and cost functions Slope of input conditional demand function: effect of w j on z i* for given q Two simple results: July 2017 9
Simple result 1 Use a standard property 2 ( ) 2 ( ) = w i w j w j w i So in this case C ij (w, q) = C ji (w,q) Therefore we have: H ji (w, q) = H ij (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 July 2017 10
Simple result 2 Use the standard relationship: C ij (w, q) = H ji (w, q) We can get the special case: C ii (w, q) = H ii (w, q) Because cost function is concave: C ii (w, q) 0 Therefore: H ii (w, q) 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 July 2017 11
Conditional input demand curve w 1 Consider the demand for input 1 H 1 (w,q) Consequence of result 2? Downward-sloping conditional demand H 11 (w, q) < 0 In some cases it is also possible that H i i = 0 z 1 Corresponds to the case where isoquant is kinked: multiple w values consistent with same z* July 2017 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) July 2017 13
Overview Firm: Comparative Statics Response function for stage 2 optimisation Conditional Input Demand Output Supply Ordinary Input Demand Short-run problem July 2017 14
The second response function Review the profit-maximisation problem and its solution Choose q to maximise: pq C (w, q) The stage 2 problem From the FOC: p = C q (w, q * ), if q * > 0 pq * C(w, q * ) profit-maximising value for output: q * = S (w, p) Price equals marginal cost Price covers average cost S is the supply function input prices output price (again it may actually be a correspondence) July 2017 15
Supply of output and output price Use the FOC: C q (w, q * ) = p Use the supply function for q: C q (w, S(w, p) ) = p marginal cost equals price Gives an equation in w and p Differential of S with respect to p Differentiate with respect to p C qq (w, S(w, p) ) S p (w, p) = 1 Rearrange: 1. S p (w, p) = C qq (w, q * ) Use the function of a function rule Positive if MC is increasing Gives slope of supply function July 2017 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 C q C/q Supply response is given by q=s(w,p) Multiple q* at this price p_ Case illustrated is for φ with first decreasing AC, then increasing AC, Response is a discontinuous map: jumps in q* no price yields a solution here q_ q Map is multivalued at the discontinuity July 2017 17
Supply of output and price of input j Use the FOC: C q (w, S(w, p) ) = p Differentiate with respect to w j C qj (w, q * ) + C qq (w, q * ) S j (w, p) = 0 Same as before: price equals marginal cost Use the function of a function rule again Rearrange: C qj (w, q * ) S j (w, p) = C qq (w, q * ) Supply of output must fall with w j if MC increases with w j Remember, this is positive July 2017 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 φ yields discontinuous S IRTS means φ is nonconcave and so S is discontinuous July 2017 19
Overview Firm: Comparative Statics Response function for combined optimisation problem Conditional Input Demand Output Supply Ordinary Input Demand Short-run problem July 2017 20
The third response function Recall the first two response functions: z i* = H i (w,q) q * = S (w, p) Now substitute for q* : z i* = H i (w, S(w, p) ) Use this to define a new function: D i (w,p) := H i (w, S(w, p) ) Demand for input i, conditional on output q Supply of output Stages 1 & 2 combined Demand for input i (unconditional ) input prices output price Use this relationship to analyse further the firm s response to price changes July 2017 21
Demand for i and the price of output Take the relationship D i (w, p) = H i (w, S(w, p)) Differentiate with respect to p: D pi (w, p) = H qi (w, q * ) S p (w, p) D i increases with p iff H i increases with q. Reason? Supply increases with price ( S p >0) function of a function rule again But we also have, for any q: H i (w, q) = C i (w, q) H qi (w, q) = C iq (w, q) Substitute in the above: D pi (w, p) = C qi (w, q * ) S p (w, p) Shephard s Lemma again Demand for input i (D i ) increases with p iff marginal cost (C q ) increases with w i July 2017 22
Demand for i and the price of j Again take the relationship D i (w, p) = H i (w, S(w, p)) Differentiate with respect to w j : D ji (w, p) = H ji (w, q * ) + H qi (w, q * )S j (w, p) Use Shephard s Lemma again: H qi (w, q) = C iq (w, q) Use this and the previous result on S j (w, p) to give a decomposition into a substitution effect and an output effect : C jq (w, q * ) D ji (w, p) = H ji (w, q * ) C iq (w, q * ) C qq (w, q * ). output effect substitution effect Substitution effect is just slope of conditional input demand curve Output effect is [effect of w j on q] [effect of q on demand for i] July 2017 23
Results from decomposition formula Take the general relationship: C iq (w, q * )C jq (w, q * ) D ji (w, p) = H ji (w, q * ) C qq (w, q * ). The effect w i on demand for input j equals the effect of w j on demand for input i We already know this is symmetric in i and j. Now take the special case where j = i: We already know this is negative or zero. Obviously symmetric in i and j. C iq (w, q * ) 2 D ii (w, p) = H ii (w, q * ) C qq (w, q * ). cannot be positive. If w i increases, the demand for input i cannot rise July 2017 24
Input-price fall: substitution effect w 1 conditional demand curve H 1 (w,q) original output level The initial equilibrium price of input falls value to firm of price fall, given a fixed output level initial price level price fall Change in cost Notional increase in factor input if output target is held constant z* 1 z 1 July 2017 25
Input-price fall: total effect w 1 Conditional demand at original output Conditional demand at new output The initial equilibrium Substitution effect of inputprice of fall Total effect of input-price fall initial price level price fall ordinary demand curve z* 1 z** 1 z 1 July 2017 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 July 2017 27
Overview Firm: Comparative Statics Optimisation subject to side-constraint Conditional Input Demand Output Supply Ordinary Input Demand Short-run problem July 2017 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 July 2017 29
The short-run problem We build on the firm s standard optimisation problem Choose q and z to maximise Π := pq m Σ w i z i i=1 subject to the standard constraints: q φ (z) q 0, z 0 But we add a side condition to this problem: z m = z m Let q be the value of q for which z m = z m would have been freely chosen in the unrestricted cost-min problem July 2017 30
The short-run cost function ~ _ C(w, q, z m ) := min Σ w i z i {z m = z m } The solution function with the side constraint Short-run demand for input i: ~ _ ~ _ Follows from Shephard s Lemma H i (w, q, z m ) =C i (w, q, z m ) Compare with the ordinary cost function ~ _ C(w, q) C(w, q, z m ) So, dividing by q: ~ C(w, q) C(w, q, z m ) 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 July 2017 31
MC, AC and supply in the short and long run AC if all inputs are variable MC if all inputs are variable p ~ C q Fix an output level AC if input m is now kept fixed MC if input m is now kept fixed ~ C/q C q C/q Supply curve in long run Supply curve in short run 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 July 2017 32
Conditional input demand w 1 The original demand curve for input 1 H 1 (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 ~ _ H 1 (w, q, z m ) z 1 July 2017 33
Key concepts Basic functional relations price signals firm input/output responses H i (w,q) S (w,p) D i (w,p) demand for input i, conditional on output supply of output demand for input i (unconditional ) And they all hook together like this: H i (w, S(w,p)) = D i (w,p) July 2017 34
What next? Analyse the firm under a variety of market conditions Apply the analysis to the consumer s optimisation problem July 2017 35