# Prerequisites Almost essential Firm Basics Frank Cowell Microeconomics

Prerequisites Almost essential Firm: Basics Frank Cowell: Microeconomics October 2006 The Firm: Optimisation MICROECONOMICS Principles and Analysis Frank Cowell

Overview. . . Firm: Optimisation Frank Cowell: Microeconomics The setting Approaches to the firm’s optimisation problem Stage 1: Cost Minimisation Stage 2: Profit maximisation

The optimisation problem Frank Cowell: Microeconomics We want to set up and solve a standard optimisation problem. n Let's make a quick list of its components. n. . . and look ahead to the way we will do it for the firm. n

The optimisation problem Frank Cowell: Microeconomics n Objectives -Profit maximisation? n Constraints -Technology; other n Method - 2 -stage optimisation

Construct the objective function Frank Cowell: Microeconomics n n Use the information on prices… wi • price of input i p • price of output …and on quantities… zi q n • amount of input i • amount of output …to build the objective function How it’s done

The firm’s objective function Frank Cowell: Microeconomics m S w i zi i=1 Cost of inputs: n n n Revenue: Profits: pq pq – • Summed over all m inputs • Subtract Cost from Revenue to get m S w i zi i=1

Optimisation: the standard approach Frank Cowell: Microeconomics n Choose q and z to maximise P : = pq – m S w i zi i=1 . . . subject to the production constraint. . . n q (z) n . . and some obvious constraints: q 0 z 0 • Could also write this as z Z(q) • You can’t have negative output or negative inputs

A standard optimisation method Frank Cowell: Microeconomics l If is differentiable… Set up a Lagrangean to take care of the constraints l necessity Write down the First Order Conditions (FOC) l sufficiency l Check out second-order conditions l Use FOC to characterise solution L (. . . ) = 0 z 2 2 L (. . . ) z z* = …

Uses of FOC Frank Cowell: Microeconomics n n n First order conditions are crucial They are used over and over again in optimisation problems. For example: u u u n n Characterising efficiency. Analysing “Black box” problems. Describing the firm's reactions to its environment. More of that in the next presentation Right now a word of caution. . .

A word of warning Frank Cowell: Microeconomics n We’ve just argued that using FOC is useful. u u u n n n But sometimes it will yield ambiguous results. Sometimes it is undefined. Depends on the shape of the production function . You have to check whether it’s appropriate to apply the Lagrangean method You may need to use other ways of finding an optimum. Examples coming up…

A way forward Frank Cowell: Microeconomics n n We could just go ahead and solve the maximisation problem But it makes sense to break it down into two stages u u u n First stage is “minimise cost for a given output level” u u n If you have fixed the output level q… …then profit max is equivalent to cost min. Second stage is “find the output level to maximise profits” u u n The analysis is a bit easier You see how to apply optimisation techniques It gives some important concepts that we can re-use later Follows the first stage naturally Uses the results from the first stage. We deal with stage each in turn

Overview. . . Firm: Optimisation Frank Cowell: Microeconomics The setting A fundamental multivariable problem with a brilliant solution Stage 1: Cost Minimisation Stage 2: Profit maximisation

Stage 1 optimisation Frank Cowell: Microeconomics n Pick a target output level q n Take as given the market prices of inputs w n Maximise profits. . . n . . . by minimising costs m S w i zi i=1

A useful tool Frank Cowell: Microeconomics n n n For a given set of input prices w. . . …the isocost is the set of points z in input space. . . that yield a given level of factor cost. These form a hyperplane (straight line). . . because of the simple expression for factor-cost structure.

Iso-cost lines Frank Cowell: Microeconomics z 2 g n i as § Draw set of points where cost of input is c, a constant t s co § Repeat for a higher value of the constant § Imposes direction on the diagram. . . e r c in w 1 z 1 + w 2 z 2 = c" w 1 z 1 + w 2 z 2 = c' w 1 z 1 + w 2 z 2 = c z 1 Use this to derive optimum

Cost-minimisation Frank Cowell: Microeconomics z 2 § The firm minimises cost. . . § Subject to output constraint q § Defines the stage 1 problem. § Solution to the problem g cin minimise ost c m S w i zi du Re i=1 subject to (z) q l z* z 1 §But the solution depends on the shape of the inputrequirement set Z. §What would happen in other cases?

Convex, but not strictly convex Z Frank Cowell: Microeconomics z 2 Any z in this set is cost-minimising § An interval of solutions z 1

Convex Z, touching axis Frank Cowell: Microeconomics z 2 § Here MRTS 21 > w 1 / w 2 at the solution. l z* z 1 § Input 2 is “too expensive” and so isn’t used: z 2*=0.

Non-convex Z Frank Cowell: Microeconomics z 2 l §There could be multiple solutions. z* §But note that there’s no solution point between z* and z** l z 1

Non-smooth Z Frank Cowell: Microeconomics z 2 MRTS 21 is undefined at z*. § z* is unique costminimising point for q. l z* z 1 §True for all positive finite values of w 1, w 2

Frank Cowell: Microeconomics Cost-minimisation: strictly convex Z l Minimise Lagrange multiplier m S w i zi i=1 + l[q (z)] q – (z) Because of strict convexity we have an interior solution. l A set of m+1 First-Order Conditions l* 1 (z* ) = w 1 one for each input l* 2 (z* ) = w 2 … … … l* m(z* ) = wm l ü ý þ q = (z*) output constraint § Use the objective function §. . . and output constraint §. . . to build the Lagrangean § Differentiate w. r. t. z 1, . . . , zm and set equal to 0. §. . . and w. r. t l § Denote cost minimising values with a *.

If isoquants can touch the axes. . . Frank Cowell: Microeconomics l Minimise m S w i zi i=1 + l[q – f(z)] Now there is the possibility of corner solutions. l A set of m+1 First-Order Conditions l* 1 (z*) w 1 l* 2 (z*) w 2 … … … l* m(z*) wm l ü ý þ q = (z*) Can get “<” if optimal value of this input is 0 Interpretati on

From the FOC Frank Cowell: Microeconomics If both inputs i and j are used and MRTS is defined then. . . l i(z*) wi ——— = — * j(z ) wj l MRTS = input price ratio l If input i could be zero then. . . l MRTSji input price ratio § “implicit” price = market price i(z*) wi ——— — * j(z ) wj § “implicit” price market price Solution

The solution. . . Frank Cowell: Microeconomics Solving the FOC, you get a cost-minimising value for each input. . . zi* = Hi(w, q) l l . . . for the Lagrange multiplier l* = l*(w, q) . . . and for the minimised value of cost itself. l The cost function is defined as l C(w, q) : = min S wi zi { (z) q} vector of input prices Specified output level

Interpreting the Lagrange multiplier Frank Cowell: Microeconomics n The solution function: C(w, q) = Siwi zi* = Si wi zi*– l* [ (z*) – q] n Differentiate with respect to q: Cq(w, q) = Siwi. Hiq(w, q) At the optimum, either the constraint binds or the Lagrange multiplier is zero Express demands in terms of (w, q) because of i (w, q) – l* [Si i(z*) HVanishes – =1]wi q l* i(z*) FOC Rearrange: Cq(w, q) = Si [wi – l* i(z*)] Hiq(w, q) + l* Lagrange multiplier in the stage n Cq (w, q) = l* 1 problem is just marginal cost This result – extremely important in economics – is just an applications of a general “envelope” theorem.

Frank Cowell: Microeconomics The cost function is an amazingly useful concept n n n Because it is a solution function. . . it automatically has very nice properties. These are true for all production functions. And they carry over to applications other than the firm. We’ll investigate these graphically.

Properties of C Frank Cowell: Microeconomics z 1* C C(w, q+Dq) C(w, q) ° § Draw cost as function of w 1 § Cost is non-decreasing in input prices. § Cost is increasing in output. § Cost is concave in input prices. § Shephard’s Lemma C(tw+[1–t]w , q) t. C(w, q) + [1–t]C(w , q) w 1 C(w, q) ———— = zj* wj

Frank Cowell: Microeconomics What happens to cost if w changes to tw z 2 § Find cost-minimising inputs for w, given q q § Find cost-minimising inputs for tw, given q • §So we have: z* C(tw, q) = Si t wizi* = t Siwizi* = t. C(w, q) z 1 §The cost function is homogeneous of degree 1 in prices.

Cost Function: 5 things to remember Frank Cowell: Microeconomics n Non-decreasing in every input price. u Increasing in at least one input price. Increasing in output. n Concave in prices. n Homogeneous of degree 1 in prices. n Shephard's Lemma. n

Example Frank Cowell: Microeconomics Production function: q z 10. 1 z 20. 4 Equivalent form: log q 0. 1 log z 1 + 0. 4 log z 2 Lagrangean: w 1 z 1 + w 2 z 2 + l [log q – 0. 1 log z 1 – 0. 4 log z 2] FOCs for an interior solution: w 1 – 0. 1 l / z 1 = 0 w 2 – 0. 4 l / z 2 = 0 log q = 0. 1 log z 1 + 0. 4 log z 2 From the FOCs: log q = 0. 1 log (0. 1 l / w 1) + 0. 4 log (0. 4 l / w 2 ) l = 0. 1– 0. 2 0. 4– 0. 8 w 10. 2 w 20. 8 q 2 Therefore, from this and the FOCs: w 1 z 1 + w 2 z 2 = 0. 5 l = 1. 649 w 10. 2 w 20. 8 q 2

Overview. . . Firm: Optimisation Frank Cowell: Microeconomics The setting …using the results of stage 1 Stage 1: Cost Minimisation Stage 2: Profit maximisation

Stage 2 optimisation Frank Cowell: Microeconomics n n Take the cost-minimisation problem as solved. Take output price p as given. u u n n Choose q to maximise profits. First analyse the components of the solution graphically. u n Use minimised costs C(w, q). Set up a 1 -variable maximisation problem. Tie-in with properties of the firm introduced in the previous presentation. Then we come back to the formal solution.

Average and marginal cost Frank Cowell: Microeconomics p increasing returns to scale decreasing returns to scale Cq q § The average cost curve. § Slope of AC depends on RTS. § Marginal cost cuts AC at its minimum C/q q

Revenue and profits Frank Cowell: Microeconomics § A given market price p. § Revenue if output is q. § Cost if output is q. § Profits vary with q. Cq p § Maximum profits C/q P §price = marginal cost q q q q*

What happens if price is low. . . Frank Cowell: Microeconomics Cq p q* = 0 C/q §price < marginal cost q

Profit maximisation Frank Cowell: Microeconomics Objective is to choose q to max: n pq – C (w, q) From the First-Order Conditions if q* > 0: p = Cq (w, q*) “Revenue minus minimised cost” n n C(w, q*) p ———— q* In general: p Cq (w, q*) pq* C(w, q*) “Price equals marginal cost” “Price covers average cost” covers both the cases: q* > 0 and q* = 0

Example (continued) Frank Cowell: Microeconomics Production function: q z 10. 1 z 20. 4 Resulting cost function: C(w, q) = 1. 649 w 10. 2 w 20. 8 q 2 Profits: pq – C(w, q) = pq – A q 2 where A: = 1. 649 w 10. 2 w 20. 8 FOC: p – 2 Aq = 0 Result: q = p / 2 A. = 0. 3031 w 1– 0. 2 w 2– 0. 8 p

Summary Frank Cowell: Microeconomics n Key point: Profit maximisation can be viewed in two stages: Review u Stage 1: choose inputs to minimise cost Review u Stage 2: choose output to maximise profit n What next? Use these to predict firm's reactions

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