1 / 37

The Firm: Optimisation

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.

Albert_Lan
Download Presentation

The Firm: Optimisation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. The Firm: Optimisation Microeconomia III (Lecture 2) Tratto da Cowell F. (2004), Principles of Microeoconomics

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

  3. 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.

  4. The optimisation problem • Profit maximisation? -Technology; other - 2-stage optimisation • Objectives • Constraints • Method

  5. 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

  6. 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 –

  7. 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 zZ(q) q £f (z) • ..and some obvious constraints: • You can’t have negative output or negative inputs z³ 0 q³0

  8. 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* = …

  9. 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...

  10. 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…

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

  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 Swizi i=1

  13. 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.

  14. 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

  15. 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?

  16. z2 z1 Convex, but not strictly convex Z Any z in this set is cost-minimising • An interval of solutions

  17. 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*

  18. z2 z1 Non-convex Z • There could be multiple solutions. z* • But note that there’s no solution point between z* and z**. z**

  19. 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

  20. 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

  21. 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*)

  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 • “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

  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) ³q} vector of input prices Specified output level

  24. 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.

  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.

  26. 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

  27. 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.

  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.

  29. 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

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

  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 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.

  32. 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

  33. 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

  34. What happens if price is low... Cq C/q p • price < marginal cost q* = 0 q

  35. 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*)

  36. 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

  37. 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

More Related