#### Transcript The Firm: Optimisation

Prerequisites Almost essential Firm: Basics THE FIRM: OPTIMISATION MICROECONOMICS Principles and Analysis Frank Cowell March 2012 Frank Cowell: Firm Optimization 1 Overview... Firm: Optimisation The setting Approaches to the firm’s optimisation problem Stage 1: Cost Minimisation Stage 2: Profit maximisation March 2012 Frank Cowell: Firm Optimization 2 The optimisation problem We want to set up and solve a standard optimisation problem Let's make a quick list of its components ... and look ahead to the way we will do it for the firm March 2012 Frank Cowell: Firm Optimization 3 The optimisation problem Objectives -Profit maximisation? Constraints -Technology; other Method - 2-stage optimisation March 2012 Frank Cowell: Firm Optimization 4 Construct the objective function Use the information on prices… wi •price of input i p •price of output …and on quantities… zi q •amount of input i •amount of output …to build the objective function March 2012 How it’s done Frank Cowell: Firm Optimization 5 The firm’s objective function m Cost of inputs: S wizi Revenue: pq •Summed over all m inputs i=1 •Subtract Cost from Revenue to get Profits: m pq – S wizi i=1 March 2012 Frank Cowell: Firm Optimization 6 Optimisation: the standard approach Choose q and z to maximise m P := pq – S wizi i=1 ...subject to the production constraint... q f (z) ..and some obvious constraints: q 0 March 2012 z0 • Could also write this as zZ(q) •You can’t have negative output or negative inputs Frank Cowell: Firm Optimization 7 A standard optimisation method If f is differentiable… Set up a Lagrangean to take care L (... ) of the constraints Write down the First Order necessity L (... ) = 0 z Conditions (FOC) Check out second-order sufficiency conditions Use FOC to characterise solution March 2012 2 2L (... ) z z* = … Frank Cowell: Firm Optimization 8 Uses of FOC First order conditions are crucial They are used over and over again in optimisation problems. For example: • 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... March 2012 Frank Cowell: Firm Optimization 9 A word of warning We’ve just argued that using FOC is useful. • But sometimes it will yield ambiguous results. • Sometimes it is undefined. • Depends on the shape of the production function f. 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… March 2012 Frank Cowell: Firm Optimization 10 A way forward We could just go ahead and solve the maximisation problem But it makes sense to break it down into two stages • The analysis is a bit easier • You see how to apply optimisation techniques • It gives some important concepts that we can re-use later First stage is “minimise cost for a given output level” • 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” • Follows the first stage naturally • Uses the results from the first stage. We deal with stage each in turn March 2012 Frank Cowell: Firm Optimization 11 Overview... Firm: Optimisation The setting A fundamental multivariable problem with a brilliant solution Stage 1: Cost Minimisation Stage 2: Profit maximisation March 2012 Frank Cowell: Firm Optimization 12 Stage 1 optimisation Pick a target output level q Take as given the market prices of inputs w Maximise profits... ...by minimising costs m S wi zi i=1 March 2012 Frank Cowell: Firm Optimization 13 A useful tool 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 March 2012 Frank Cowell: Firm Optimization 14 Iso-cost lines Draw set of points where cost of input is c, a constant z2 Repeat for a higher value of the constant Imposes direction on the diagram... w1z1 + w2z2 = c" w1z1 + w2z2 = c' w1z1 + w2z2 = c z1 March 2012 Use this to derive optimum Frank Cowell: Firm Optimization 15 Cost-minimisation z2 The firm minimises cost... Subject to output constraint q Defines the stage 1 problem. Solution to the problem minimise m S wizi i=1 subject to f(z) q z* z1 March 2012 But the solution depends on the shape of the input-requirement set Z. What would happen in other cases? Frank Cowell: Firm Optimization 16 Convex, but not strictly convex Z z2 Any z in this set is cost-minimising z1 March 2012 An interval of solutions Frank Cowell: Firm Optimization 17 Convex Z, touching axis z2 Here MRTS21 > w1 / w2 at the solution. z* March 2012 z1 Input 2 is “too expensive” and so isn’t used: z2* = 0 Frank Cowell: Firm Optimization 18 Non-convex Z z2 z* There could be multiple solutions. z** But note that there’s no solution point between z* and z** z1 March 2012 Frank Cowell: Firm Optimization 19 Non-smooth Z z2 MRTS21 is undefined at z*. z* is unique costminimising point for q z* z1 March 2012 True for all positive finite values of w1, w2 Frank Cowell: Firm Optimization 20 Cost-minimisation: strictly convex Z Minimise m S wi zi Lagrange multiplier q – ff(z)] (z) + l[q i=1 Use the objective function ...and output constraint ...to build the Lagrangean Differentiate w.r.t. z1, ..., zm ; set equal to 0 ... and w.r.t l Because of strict convexity we have Denote cost minimising values by an interior solution A set of m+1 First-Order Conditions l* f1 (z* ) = w1 l*f2 (z*) = w2 … … … l*fm(z *) = wm q = f(z*) March 2012 one for each input output constraint Frank Cowell: Firm Optimization 21 * If isoquants can touch the axes... Minimise m Swizi + l [q – f(z)] i=1 Now there is the possibility of corner solutions. A set of m+1 First-Order Conditions l*f1 (z*) w1 l*f2 (z*) w2 … … … l*fm(z*) wm q = f(z*) March 2012 Interpretation Can get “<” if optimal value of this input is 0 Frank Cowell: Firm Optimization 22 From the FOC If both inputs i and j are used and MRTS is defined then... fi(z*) wi ——— = — * fj(z ) wj MRTS = input price ratio If input i could be zero then... fi(z*) wi ——— — * fj(z ) wj MRTSji input price ratio “implicit” price = market price “implicit” price market price Solution March 2012 Frank Cowell: Firm Optimization 23 The solution... Solving the FOC, you get a cost-minimising value for each input... zi* = Hi(w, q) ...for the Lagrange multiplier l* = l*(w, q) ...and for the minimised value of cost itself. The cost function is defined as C(w, q) := min S wi zi {f(z) q} vector of input prices March 2012 Specified output level Frank Cowell: Firm Optimization 24 Interpreting the Lagrange multiplier The solution function: C(w, q) = Siwi zi* = Si wi zi*– l* [f(z*) – q] Differentiate with respect to q: Cq(w, q) = S i iwiH q(w, q) * i i fi(z ) H q(w, – l* [S Rearrange: At the optimum, either the constraint binds or the Lagrange multiplier is zero Express demands in terms of (w,q) q) – 1] Vanishes because of FOC l *fi(z*) = wi Cq(w, q) = Si [wi – l*fi(z*)] Hiq(w, q) + l* Lagrange multiplier in the stage 1 problem is just marginal cost Cq (w, q) = l* This result – extremely important in economics – is just an applications of a general “envelope” theorem. March 2012 Frank Cowell: Firm Optimization 25 The cost function is an amazingly useful concept 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 March 2012 Frank Cowell: Firm Optimization 26 Properties of C z1 * C C(w, q+q) C(w, q) Draw cost as function of w1 Cost is non-decreasing in input prices . Increasing in output, if f continuous Concave in input prices. ° Shephard’s Lemma C(tw+[1–t]w,q) tC(w,q) + [1–t]C(w,q) w1 March 2012 C(w,q) ———— = zj* wj Frank Cowell: Firm Optimization 27 What happens to cost if w changes to tw z2 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* = tC(w,q) The cost function is homogeneous of degree 1 in prices. z1 March 2012 Frank Cowell: Firm Optimization 28 Cost Function: 5 things to remember Non-decreasing in every input price • Increasing in at least one input price Increasing in output Concave in prices Homogeneous of degree 1 in prices Shephard's Lemma March 2012 Frank Cowell: Firm Optimization 29 Example Production function: q z10.1 z20.4 Equivalent form: log q 0.1 log z1 + 0.4 log z2 Lagrangean: w1z1 + w2z2 + l [log q – 0.1 log z1 – 0.4 log z2] FOCs for an interior solution: w1 – 0.1 l / z1 = 0 w2 – 0.4 l / z2 = 0 log q = 0.1 log z1 + 0.4 log z2 From the FOCs: log q = 0.1 log (0.1 l / w1) + 0.4 log (0.4 l / w2 ) l = 0.1–0.2 0.4–0.8 w10.2 w20.8 q2 Therefore, from this and the FOCs: w1 z1 + w2 z2 = 0.5 l = 1.649 w10.2 w20.8 q2 March 2012 Frank Cowell: Firm Optimization 30 Overview... Firm: Optimisation The setting …using the results of stage 1 Stage 1: Cost Minimisation Stage 2: Profit maximisation March 2012 Frank Cowell: Firm Optimization 31 Stage 2 optimisation Take the cost-minimisation problem as solved Take output price p as given • Use minimised costs C(w,q) • Set up a 1-variable maximisation problem Choose q to maximise profits First analyse components of the solution graphically • Tie-in with properties of the firm (in the previous presentation) Then we come back to the formal solution March 2012 Frank Cowell: Firm Optimization 32 Average and marginal cost p increasing returns to scale decreasing returns to scale The average cost curve Slope of AC depends on RTS Marginal cost cuts AC at its minimum Cq C/q q q March 2012 Frank Cowell: Firm Optimization 33 Revenue and profits A given market price p Revenue if output is q Cost if output is q Profits if output is q Profits vary with q Maximum profits Cq C/q p P price = marginal cost q q q March 2012 q q q q* Frank Cowell: Firm Optimization 34 What happens if price is low... Cq C/q p q* = 0 March 2012 price < marginal cost q Frank Cowell: Firm Optimization 35 Profit maximisation Objective is to choose q to max: pq – C (w, q) From the First-Order Conditions if q* > 0: p = Cq (w, q*) C(w, q*) p ———— q* In general: p Cq (w, q*) pq* C(w, q*) March 2012 “Revenue minus minimised cost” “Price equals marginal cost” “Price covers average cost” covers both the cases: q* > 0 and q* = 0 Frank Cowell: Firm Optimization 36 Example (continued) Production function: q z10.1 z20.4 Resulting cost function: C(w, q) = 1.649 w10.2 w20.8 q2 Profits: pq – C(w, q) = pq – A q2 where A:= 1.649 w10.2 w20.8 FOC: p – 2 Aq = 0 Result: q = p / 2A = 0.3031 w1–0.2 w2– 0.8 p March 2012 Frank Cowell: Firm Optimization 37 Summary Key point: Profit maximisation can be viewed in two stages: Review • Stage 1: choose inputs to minimise cost Review • Stage 2: choose output to maximise profit What next? Use these to predict firm's reactions March 2012 Frank Cowell: Firm Optimization 38