# Prerequisites Almost essential Firm Optimisation Frank Cowell Microeconomics

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Prerequisites Almost essential Firm: Optimisation Frank Cowell: Microeconomics Useful, but optional Firm: Demand Supply October 2006 The Multi-Output Firm MICROECONOMICS Principles and Analysis Frank Cowell

Introduction Frank Cowell: Microeconomics n This presentation focuses on analysis of firm producing more than one good u u u n For the single-output firm, some things are obvious: u u u n n modelling issues production function profit maximisation the direction of production returns to scale marginal products But what of multi-product processes? Some rethinking required. . . ? u u u nature of inputs and outputs? tradeoffs between outputs? counterpart to cost function?

Overview. . . The Multi-Output Firm Frank Cowell: Microeconomics Net outputs A fundamental concept Production possibilities Profit maximisation

Multi-product firm: issues Frank Cowell: Microeconomics n “Direction” of production u n Ambiguity of some commodities u n Need a more general notation Is paper an input or an output? Aggregation over processes u How do we add firm 1’s inputs and firm 2’s outputs?

Net output Frank Cowell: Microeconomics n Net output, written as qi, u u n Key concept u u n if positive denotes the amount of good i produced as output if negative denotes the amount of good i used up as output treat outputs and inputs symmetrically offers a representation that is consistent Provides consistency u u in aggregation in “direction” of production We just need some reinterpretation

Approaches to outputs and inputs Frank Cowell: Microeconomics NET OUTPUTS OUTPUT INPUTS q 1 z 1 q 2 z 2 . . . qn-1 zm qn §A standard “accounting” approach §An approach using “net outputs” §How the two are related §A simple sign convention q q 1 –z 1 q 2 –z 2. . . =. . . qn-1 –zm qn +q Outputs: Inputs: + net additions to the stock of a good reductions in the stock of a good

Aggregation Frank Cowell: Microeconomics n Consider an industry with two firms u u n How is total related to quantities for individual firms? u u n u qi 1 = 100, qi 2 = 100 qi = 200 Example 2: both firms use i as input u u n Just add up qi = qi 1 + qi 2 Example 1: both firms produce i as output u n Let qif be net output for firm f of good i, f = 1, 2 Let qi be net output for whole industry of good i qi 1 = − 100, qi 2 = − 100 qi = − 200 Example 3: firm 1 produces i that is used by firm 2 as input u u qi 1 = 100, qi 2 = − 100 qi = 0

Net output: summary Frank Cowell: Microeconomics n n Sign convention is common sense If i is an output… u addition to overall supply of i u so sign is positive If i is an inputs u net reduction in overall supply of i u so sign is negative If i is a pure intermediate good u no change in overall supply of i u so assign it a zero in aggregate

Overview. . . The Multi-Output Firm Frank Cowell: Microeconomics Net outputs A production function with many outputs, many inputs… Production possibilities Profit maximisation

Rewriting the production function… Frank Cowell: Microeconomics n Reconsider single-output firm example given earlier u u u n n n Conventional way of writing feasibility condition: u q f (z 1, z 2, . . , zm ) u where f is the production function Express this in net-output notation and rearrange: u qn f (−q 1, −q 2, . . , −qn-1 ) u qn − f (−q 1, −q 2, . . , −qn-1 ) 0 Rewrite this relationship as u u n goods 1, …, m are inputs good m+1 is output n=m+1 F (q 1, q 2, . . , qn-1, qn ) 0 where F is the implicit production function Properties of F are implied by those of f…

The production function F Frank Cowell: Microeconomics n Recall equivalence for single output firm: u u n So, for this case: u u n qn − f (−q 1, −q 2, . . , −qn-1 ) 0 F (q 1, q 2, . . , qn-1, qn ) 0 F is increasing in q 1, q 2, . . , qn if f is homogeneous of degree 1, F is homogeneous of degree 0 if f is differentiable so is F for any i, j = 1, 2, …, n− 1 MRTSij = Fj(q)/Fi(q) It makes sense to generalise these…

The production function F (more) Frank Cowell: Microeconomics n For a vector q of net outputs u u u n For all feasible q: u u u n q is feasible if F(q) 0 q is technically efficient if F(q) = 0 q is infeasible if F(q) > 0 F(q) is increasing in q 1, q 2, . . , qn if there is CRTS then F is homogeneous of degree 0 if f is differentiable so is F for any two inputs i, j, MRTSij = Fj(q)/Fi(q) for any two outputs i, j, the marginal rate of transformation of i into j is MRTij = Fj(q)/Fi(q) Illustrate the last concept using the transformation curve…

Firm’s transformation curve Frank Cowell: Microeconomics §Goods 1 and 2 are outputs q 2 §Feasible outputs §Technically efficient outputs §MRT at qo q° F(q) 0 F 1(q°)/F 2(q°) F(q)=0 q 1

An example with five goods Frank Cowell: Microeconomics n n n Goods 1 and 2 are outputs Goods 3, 4, 5 are inputs A linear technology u u n fixed proportions of each input needed for the production of each output: q 1 a 1 i + q 2 a 2 i −qi where aji is a constant i = 3, 4, 5, j = 1, 2 given the sign convention −qi > 0 Take the case where inputs are fixed at some arbitrary values…

The three input constraints Frank Cowell: Microeconomics q 1 points satisfying q 1 a 13 + q 2 a 23 −q 3 § Draw the feasible set for the two outputs: § input Constraint 3 § Add Constraint 4 § Add Constraint 5 points satisfying q 1 a 14 + q 2 a 24 −q 4 § Intersection is the feasible set for the two outputs points satisfying q 1 a 15 + q 2 a 25 −q 5 q 2

The resulting feasible set Frank Cowell: Microeconomics q 1 The transformation curve how this responds to changes in available inputs q 2

Changing quantities of inputs Frank Cowell: Microeconomics q 1 points satisfying q 1 a 13 + q 2 a 23 −q 3 §The feasible set for the two consumption goods as before: § Suppose there were more of input 3 § Suppose there were less of input 4 points satisfying q 1 a 13 + q 2 a 23 −q 3 −dq 3 points satisfying q 1 a 14 + q 2 a 24 −q 4 + dq 4 q 2

Overview. . . The Multi-Output Firm Frank Cowell: Microeconomics Net outputs Integrated approach to optimisation Production possibilities Profit maximisation

Profits Frank Cowell: Microeconomics n The basic concept is (of course) the same u n But we use the concept of net output u u n Revenue Costs this simplifies the expression exploits symmetry of inputs and outputs Consider an “accounting” presentation…

Accounting with net outputs Frank Cowell: Microeconomics n Suppose goods 1, . . . , m are inputs and goods m+1 to n are outputs n å i=m+1 pi qi Revenue m å pi [ qi] i=1 – Costs n å pi qi i=1 = Profits § Cost of inputs (goods 1, . . . , m) § Revenue from outputs (goods m+1, . . . , n) § Subtract cost from revenue to get profits

Iso-profit lines. . . Frank Cowell: Microeconomics §Net-output vectors yielding a given P 0. § Iso-profit lines for higher profit levels. q 2 g n i s a re c in ofit pr p 1 q 1+ p 2 q 2 = constant p 1 q 1+ p 2 q 2 = P 0 use this to represent profitmaximisation q 1`

Frank Cowell: Microeconomics Profit maximisation: multi-product firm (1) § Feasible outputs q 2 § Isoprofit line § Maximise profits §Profit-maximising output §MRTS at profit-maximising output § Here q 1*>0 and q 2*>0 * q § q* is technically efficient g sin a re c in ofit pr q 1` §Slope at q* equals price ratio

Frank Cowell: Microeconomics Profit maximisation: multi-product firm (2) § Feasible outputs q 2 § Isoprofit line § Maximise profits §Profit-maximising output §MRTS at profit-maximising output increasing profit § Here q 1*>0 but q 2* = 0 § q* is technically efficient q* q 1` §Slope at q* ≤ price ratio

Maximising profits Frank Cowell: Microeconomics n Problem is to choose q so as to maximise n å pi qi i=1 n subject to F(q) ≤ 0 Lagrangean is n å pi qi i=1 n l F(q) FOC for an interior maximum is u pi l Fi(q) = 0

Maximised profits Frank Cowell: Microeconomics n Introduce the profit function u the solution function for the profit maximisation problem n P(p) = max å pi qi {F(q) ≤ 0} i = 1 n = å pi qi* i=1 Works like other solution functions: u u n n non-decreasing homogeneous of degree 1 continuous convex Take derivative with respect to pi : u u u Pi(p) = qi* write qi* as net supply function qi* = qi(p)

Summary Frank Cowell: Microeconomics n n Three key concepts Net output u u u n Transformation curve u n simplifies analysis key to modelling multi-output firm easy to rewrite production function in terms of net outputs summarises tradeoffs between outputs Profit function u counterpart of cost function

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