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## The Firm: Optimisation

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**The Firm: Optimisation**Microeconomia III (Lecture 2) Tratto da Cowell F. (2004), Principles of Microeoconomics**Overview...**Firm: Optimisation The setting Approaches to the firm’s optimisation problem Stage 1: Cost Minimisation Stage 2: Profit maximisation**The optimisation problem**• We need 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.**The optimisation problem**• Profit maximisation? -Technology; other - 2-stage optimisation • Objectives • Constraints • Method**Construct the objective function**• Use the information on prices… wi • price of input i p • price of output • …and on quantities… zi • amount of input i q • amount of output How it’s done • …to build the objective function**The firm’s objective function**m Swizi i=1 • Cost of inputs: • Summed over all m inputs • Revenue: pq • Subtract Cost from Revenue to get m Swizi i=1 • Profits: pq –**Optimisation: the standard approach**• Choose q and z to maximise m Swizi i=1 P := pq – • ...subject to the production constraint... • Could also write this as zZ(q) q £f (z) • ..and some obvious constraints: • You can’t have negative output or negative inputs z³ 0 q³0**A standard optimisation method**• If is differentiable… • Set up a Lagrangean to take care of the constraints L(... ) • Write down the First Order Conditions (FOC) necessity ¶ L(... ) = 0 ¶z • Check out second-order conditions sufficiency ¶2 L (... ) ¶z2 • Use FOC to characterise solution z* = …**Uses of FOC**• First order conditions are used over and over in optimisation problems. • For example: • Characterise efficiency. • Black box problems. • Firm's reactions to its environment. • More of that in the next presentation...**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…**Overview...**Firm: Optimisation The setting A fundamental multivariable problem with a brilliant solution Stage 1: Cost Minimisation Stage 2: Profit maximisation**Stage 1 optimisation**• Pick a target output level q • Take as given the market prices of inputs w • Maximise profits... • ...by minimising costs m Swizi i=1**A useful tool**• For a given set of input prices w... • …the isocost 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.**z2**z1 Iso-cost lines • Draw set of points where cost of input is c, a constant • Repeat for a higher value of the constant increasing cost • Imposes direction on the diagram... w1z1 + w2z2 = c" w1z1 + w2z2 = c' w1z1 + w2z2 = c Use this to derive optimum**z2**Reducing cost z1 Cost-minimisation • The firm minimises cost... q • Subject to output constraint • Defines the stage 1 problem. • Solution to the problem minimise m Swizi i=1 subject to(z) q z* • But the solution depends on the shape of the input-requirement set Z. • What would happen in other cases?**z2**z1 Convex, but not strictly convex Z Any z in this set is cost-minimising • An interval of solutions**z2**z1 Convex Z, touching axis • Here MRTS21 > w1/ w2 at the solution. • Input 2 is “too expensive” and so isn’t used: z2*=0. z***z2**z1 Non-convex Z • There could be multiple solutions. z* • But note that there’s no solution point between z* and z**. z****z2**z1 Non-smooth Z MRTS21 is undefined at z*. • z* is unique cost-minimising point for q. z* • True for all positive finite values of w1, w2**Cost-minimisation: strictly convex Z**• Use the objective function • Minimise Lagrange multiplier • ...and output constraint m Swi zi i=1 • ...to build the Lagrangean + l[q – f(z)] q = f(z) • Differentiate w.r.t. z1, ..., zm and set equal to 0. • ... and w.r.t l • Because of strict convexity we have an interior solution. • Denote cost minimising values with a * . • A set of m+1 First-Order Conditions ** ** ** * lf1 (z) = w1 lf2 (z) = w2 … … … lfm(z) = wm one for each input ü ý þ q = (z) output constraint**If isoquants can touch the axes...**• Minimise m Swizi i=1 + l[q £f(z)] • 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 ü ý þ Interpretation Can get “<” if optimal value of this input is 0 q = f(z*)**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 • “implicit” price = market price • If input i could be zero then... fi(z*) wi ——— £ — fj(z*) wj • MRTS £ input price ratio • “implicit” price £ market price Solution**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) ³q} vector of input prices Specified output level**Interpreting the Lagrange multiplier**• The solution function: C(w, q) = Siwi zi* = Siwi zi*–l*[f(z*) –q] At the optimum, either the constraint binds or the Lagrange multiplier is zero • Differentiate with respect to q: Cq(w, q) = SiwiHiq(w, q) –l*[Si fi(z*) Hiq(w, q) –1] Express demands in terms of (w,q) Vanishes because of FOC l*f i(x*) = wi • Rearrange: Cq(w, q) = Si [wi – l*fi(z*)] Hiq(w, q) + l* Cq(w, q) = l* Lagrange multiplier in the stage 1 problem is just marginal cost This result – extremely important in economics – is just an applications of a general “envelope” theorem.**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.**C**C(w, q+Dq) ° w1 Properties of C z1* • Draw cost as function of w1 • Cost is non-decreasing in input prices . C(w, q) • Cost is increasing in output. • Cost is concave in input prices. • Shephard’s Lemma C(tw+[1–t]w,q) tC(w,q) + [1–t]C(w,q) C(w,q) ———— = zj* wj**z2**z1 What happens to cost if w changes to tw • Find cost-minimising inputs for w, given q q • Find cost-minimising inputs for tw, given q • So we have: • z* • z* C(tw,q) = Si twizi* = t Siwizi* = tC(w,q) • The cost function is homogeneous of degree 1 in prices.**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.**Example**Production function: q z10.1 z20.4 Equivalent form: log q 0.1log z1 + 0.4 log z2 Lagrangean: w1z1 + w2z2 + l [log q – 0.1log z1– 0.4 log z2] FOCs for an interior solution: w1– 0.1 l / z1= 0 w2– 0.4 l / z2= 0 log q = 0.1log z1 + 0.4 log z2 From the FOCs: log q = 0.1log (0.1 l / w1) + 0.4 log (0.4 l / w2 ) l=0.1–0.2 0.4–0.8w10.2 w20.8 q2 Therefore, from this and the FOCs: w1 z1+ w2 z2 = 0.5 l = 1.649w10.2 w20.8 q2**Overview...**Firm: Optimisation The setting …using the results of stage 1 Stage 1: Cost Minimisation Stage 2: Profit maximisation**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 the components of the solution graphically. • Tie-in with properties of the firm introduced in the previous presentation. • Then we come back to the formal solution.**Average and marginal cost**increasing returns to scale decreasing returns to scale • The average cost curve. p • Slope of AC depends on RTS. • Marginal cost cuts AC at its minimum Cq C/q q q**q**q q q q q* 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**What happens if price is low...**Cq C/q p • price < marginal cost q* = 0 q**Profit maximisation**• Objective is to choose q to max: pq –C (w, q) “Revenue minus minimised cost” • From the First-Order Conditions if q* > 0: p = Cq(w, q*) C(w, q*) p ³ ———— q* “Price equals marginal cost” “Price covers average cost” • In general: covers both the cases: q* > 0 and q* = 0 p £Cq(w, q*) pq* ³ C(w, q*)**Example (continued)**Production function: q z10.1 z20.4 Resulting cost function: C(w, q) = 1.649w10.2 w20.8 q2 Profits: pq – C(w, q) = pq – A q2 where A:=1.649w10.2 w20.8 FOC: p – 2 Aq = 0 Result: q = p / 2A. = 0.303 w1–0.2 w2–0.8 p**Summary**• Key point: Profit maximisation can be viewed in two stages: • Stage 1: choose inputs to minimise cost • Stage 2: choose output to maximise profit • What next? Use these to predict firm's reactions Review Review