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Prerequisites Almost essential Firm: Optimisation Frank Cowell: Microeconomics Useful, but optional Firm: Demand and Supply October 2006 The Multi-Output Firm MICROECONOMICS Principles and Analysis Frank Cowell Introduction Frank Cowell: Microeconomics This presentation focuses on analysis of firm producing more than one good For the single-output firm, some things are obvious: modelling issues production function profit maximisation the direction of production returns to scale marginal products But what of multi-product processes? Some rethinking required...? 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 “Direction” of production Ambiguity of some commodities Need a more general notation Is paper an input or an output? Aggregation over processes How do we add firm 1’s inputs and firm 2’s outputs? Net output Frank Cowell: Microeconomics Net output, written as qi, Key concept 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 in aggregation in “direction” of production We just need some reinterpretation Approaches to outputs and inputs Frank Cowell: Microeconomics NET OUTPUTS OUTPUT INPUTS q1 z1 q2 z2 ... ... qn-1 zm qn A standard “accounting” approach An approach using “net outputs” How the two are related A simple sign convention q q1 –z1 q2 –z2 ... = ... 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 Consider an industry with two firms How is total related to quantities for individual firms? qi1 = 100, qi2 = 100 qi = 200 Example 2: both firms use i as input Just add up qi = qi1 + qi2 Example 1: both firms produce i as output 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 qi1 = − 100, qi2 = − 100 qi = − 200 Example 3: firm 1 produces i that is used by firm 2 as input qi1 = 100, qi2 = − 100 qi = 0 Net output: summary Frank Cowell: Microeconomics Sign convention is common sense If i is an output… addition to overall supply of i so sign is positive If i is an inputs net reduction in overall supply of i so sign is negative If i is a pure intermediate good no change in overall supply of i 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 Reconsider single-output firm example given earlier Conventional way of writing feasibility condition: qn f(−q1, −q2, ...., −qn-1 ) qn − f(−q1, −q2, ...., −qn-1 ) Rewrite this relationship as q f(z1, z2, ...., zm ) where f is the production function Express this in net-output notation and rearrange: goods 1,…,m are inputs good m+1 is output n=m+1 F (q1, q2, ...., 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 Recall equivalence for single output firm: So, for this case: qn − f(−q1, −q2, ...., −qn-1 ) F (q1, q2, ...., qn-1, qn ) 0 F is increasing in q1, q2, ...., qn if f is homogeneous of degree 1, Fis 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 For a vector q of net outputs For all feasible q: 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 q1, q2, ...., qn if there is CRTS then Fis 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 Feasible outputs q2 Technically efficient outputs MRT at qo q° F(q) 0 F1(q°)/F2(q°) F(q)=0 q1 An example with five goods Frank Cowell: Microeconomics Goods 1 and 2 are outputs Goods 3, 4, 5 are inputs A linear technology fixed proportions of each input needed for the production of each output: q1 a1i + q2 a2i −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 q1 points satisfying q1a13 + q2a23 −q3 Draw the feasible set for the two outputs: input Constraint 3 Add Constraint 4 Add Constraint 5 points satisfying q1a14 + q2a24 −q4 Intersection is the feasible set for the two outputs points satisfying q1a15 + q2a25 −q5 q2 The resulting feasible set Frank Cowell: Microeconomics q1 The transformation curve how this responds to changes in available inputs q2 Changing quantities of inputs Frank Cowell: Microeconomics q1 points satisfying q1a13 + q2a23 −q3 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 q1a13 + q2a23 −q3 −dq3 points satisfying q1a14 + q2a24 −q4 + dq4 q2 Overview... The Multi-Output Firm Frank Cowell: Microeconomics Net outputs Integrated approach to optimisation Production possibilities Profit maximisation Profits Frank Cowell: Microeconomics The basic concept is (of course) the same But we use the concept of net output Revenue Costs this simplifies the expression exploits symmetry of inputs and outputs Consider an “accounting” presentation… Accounting with net outputs Frank Cowell: Microeconomics Suppose goods 1,...,m are inputs and goods m+1 to n are outputs n i=m+1 pi qi Revenue m pi [ qi] – Costs i=1 n 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 P0. Iso-profit lines for higher profit levels. q2 p1q1+ p2q2 = constant p1q1+ p2q2 = P use this to represent profit-maximisation q1` Frank Cowell: Microeconomics Profit maximisation: multiproduct firm (1) Feasible outputs q2 Isoprofit line Maximise profits Profit-maximising output MRTS at profit-maximising output Here q1*>0 and q2*>0 * q q* is technically efficient q1` Slope at q* equals price ratio Frank Cowell: Microeconomics Profit maximisation: multiproduct firm (2) Feasible outputs q2 Isoprofit line Maximise profits Profit-maximising output MRTS at profit-maximising output Here q1*>0 but q2* = 0 q* is technically efficient q* q1` Slope at q* ≤ price ratio Maximising profits Frank Cowell: Microeconomics Problem is to choose q so as to maximise n pi qi subject to F(q) ≤ 0 i=1 Lagrangean is n pi qi lF(q) i=1 FOC for an interior maximum is pi lFi(q) = 0 Maximised profits Frank Cowell: Microeconomics Introduce the profit function the solution function for the profit maximisation problem P(p) = max n pi qi {F(q) ≤ 0} i = 1 = pi qi* i=1 Works like other solution functions: n non-decreasing homogeneous of degree 1 continuous convex Take derivative with respect to pi : Pi(p) = qi* write qi* as net supply function qi* = qi(p) Summary Frank Cowell: Microeconomics Three key concepts Net output Transformation curve simplifies analysis key to modelling multi-output firm easy to rewrite production function in terms of net outputs summarises tradeoffs between outputs Profit function counterpart of cost function