THE FIRM: OPTIMISATION

Transcription

1 Prerequisites Almost essential Firm: Basics THE FIRM: OPTIMISATION MICROECONOMICS Principles and Analysis Frank Cowell Note: the detail in slides marked * can only be seen if you run the slideshow July 2017 Frank Cowell: Firm Optimization 1

2 Overview Firm: Optimisation The setting Approaches to the firm s optimisation problem Stage 1: Cost Minimisation Stage 2: Profit maximisation July 2017 Frank Cowell: Firm Optimization 2

3 The optimisation problem We set up and solve a standard optimisation problem make a quick list of its components look ahead to the way we will do it for the firm Objectives - Profit maximisation? Constraints - Technology; other Method - 2-stage optimisation July 2017 Frank Cowell: Firm Optimization 3

4 Construct the objective function Use info on prices and on quantities to build objective function w i price of input i z i amount of input i p price of output q amount of output Cost of inputs: mm ii=1 Revenue: pppp ww ii zz ii summed over all m inputs Profits: pppp mm ii=1 ww ii zz ii subtract Cost from Revenue to get July 2017 Frank Cowell: Firm Optimization 4

5 Optimisation: the standard approach Choose q and z to maximise Π := pq m Σ w i z i i=1...subject to the production constraint... q φ (z) Could also write this as z Z(q)..and some obvious constraints: q 0 z 0 You can t have negative output or negative inputs July 2017 Frank Cowell: Firm Optimization 5

6 A standard optimisation method If φ is differentiable Set up a Lagrangian to take care of the constraints Write down the First Order Conditions (FOC) Check out second-order conditions necessity sufficiency Use FOC to characterise solution L (... ) L (... ) = 0 z 2 L (... ) z 2 z * = July 2017 Frank Cowell: Firm Optimization 6

7 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... July 2017 Frank Cowell: Firm Optimization 7

8 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 φ You have to check whether it s appropriate to apply the Lagrangian method You may need to use other ways of finding an optimum July 2017 Frank Cowell: Firm Optimization 8

9 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 July 2017 Frank Cowell: Firm Optimization 9

10 Overview Firm: Optimisation The setting A fundamental multivariable problem with a brilliant solution Stage 1: Cost Minimisation Stage 2: Profit maximisation July 2017 Frank Cowell: Firm Optimization 10

11 Stage 1 optimisation Pick a target output level q Take as given the market prices of inputs w Maximise profits......by minimising costs m Σ w i z i i=1 July 2017 Frank Cowell: Firm Optimization 11

12 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 July 2017 Frank Cowell: Firm Optimization 12

13 Iso-cost lines z 2 w 1 z 1 + w 2 z 2 = c" Draw set of points where cost of input is c, a constant Repeat for a higher value of the constant Imposes direction on the diagram... 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 July 2017 Frank Cowell: Firm Optimization 13

14 Cost-minimisation z 2 q The firm minimises cost... Subject to output constraint Defines the stage 1 problem Solution to the problem minimise z* z 1 m Σ w i z i i=1 subject to φ(z) q But the solution depends on the shape of the input-requirement set Ζ What would happen in other cases? July 2017 Frank Cowell: Firm Optimization 14

15 Convex, but not strictly convex Z z 2 Any z in this set is cost-minimising z 1 An interval of solutions July 2017 Frank Cowell: Firm Optimization 15

16 Convex Z, touching axis z 2 z 1 Here MRTS 21 > w 1 / w 2 at the solution z* Input 2 is too expensive and so isn t used: z 2 * = 0 July 2017 Frank Cowell: Firm Optimization 16

17 Non-convex Z z 2 z* There could be multiple solutions z** But note that there s no solution point between z* and z** z 1 July 2017 Frank Cowell: Firm Optimization 17

18 * Non-smooth Z z 2 z* MRTS 21 is undefined at z* z* is unique costminimising point for q z 1 True for all positive finite values of w 1, w 2 July 2017 Frank Cowell: Firm Optimization 18

19 Cost-minimisation: strictly convex Z Minimise m Σ w i z i i=1 * * λ φ 1 (z ) = w 1 * * λ φ 2 (z ) = w 2 * * λ φ m (z ) = w m q = φ(z *) Lagrange multiplier + λ[q q φ(z)] φ Because of strict convexity we have an interior solution A set of m+1 First-Order Conditions output constraint one for each input Use the objective function...and output constraint...to build the Lagrangian Differentiate w.r.t. z 1,..., z m ; set equal to 0... and w.r.t λ Denote cost minimising values by * July 2017 Frank Cowell: Firm Optimization 19

20 If isoquants can touch the axes... Minimise m Σw i z i i=1 + λ [q φ(z)] Now there is the possibility of corner solutions A set of m+1 First-Order Conditions λ * φ 1 (z * ) w 1 λ * φ 2 (z * ) w 2 λ * φ m (z * ) w m q = φ(z * ) Can get < if optimal value of this input is 0 Interpretation July 2017 Frank Cowell: Firm Optimization 20

21 From the FOC If both inputs i and j are used and MRTS is defined then... φ i (z ) w = i φ j (z ) w j MRTS = input price ratio implicit price = market price If input i could be zero then... φ i (z ) w i φ j (z ) w j MRTS ji input price ratio implicit price market price Solution July 2017 Frank Cowell: Firm Optimization 21

22 The solution... Solving the FOC, you get a cost-minimising value for each input... z i* = H i (w, q)...for the Lagrange multiplier λ * = λ * (w, q)...and for the minimised value of cost itself. The cost function is defined as C(w, q) := min Σ w i z i vector of input prices {φ(z) q} Specified output level July 2017 Frank Cowell: Firm Optimization 22

23 Interpreting the Lagrange multiplier The solution function: C(w, q) = Σ i w i z i * = Σ i w i z i* λ * [φ(z * ) q] Differentiate with respect to q: C q (w, q) = Σ i w i H i q(w, q) λ * [Σ i φ i (z * ) H i q(w, q) 1] Vanishes because of FOC λ *φ i (z*) = w i Rearrange: C q (w, q) = Σ i [w i λ * φ i (z * )] H i q(w, q) + λ * At the optimum, either the constraint binds or the Lagrange multiplier is zero Express demands in terms of (w,q) Lagrange multiplier in the stage 1 problem is just marginal cost C q (w, q) = λ * Result is just an applications of a general envelope theorem It holds for the untransformed, original version of the problem If we use a transformed version of the constraint (for example log q logφ (z)) we get a different Lagrange multiplier) July 2017 Frank Cowell: Firm Optimization 23

24 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 July 2017 Frank Cowell: Firm Optimization 24

25 Properties of C C z 1 * C(w, q+ q) C(w, q) Draw cost as function of w 1 Cost is non-decreasing in input prices Increasing in output, if φ continuous Concave in input prices Shephard s Lemma C(tw+[1 t]w,q) tc(w,q) + [1 t]c(w,q) w 1 C(w,q) = z j * w j July 2017 Frank Cowell: Firm Optimization 25

26 What happens to cost if w changes to tw z 2q Find cost-minimising inputs for w, given q Find cost-minimising inputs for tw, given q So we have: z* C(tw,q) = Σ i t w i z i* = t Σ i w i z i* = tc(w,q) The cost function is homogeneous of degree 1 in prices z 1 July 2017 Frank Cowell: Firm Optimization 26

27 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 July 2017 Frank Cowell: Firm Optimization 27

28 Example Production function: q z z Equivalent form: log q 0.1 log z log z 2 Lagrangian: w 1 z 1 + w 2 z 2 + λ [log q 0.1 log z log z 2 ] FOCs for an interior solution: w λ / z 1 = 0 w λ / z 2 = 0 log q = 0.1 log z log z 2 From the FOCs: log q = 0.1 log (0.1 λ / w 1 ) log (0.4 λ / w 2 ) λ = w w q 2 Therefore, from this and the FOCs: w 1 z 1 + w 2 z 2 = 0.5 λ = w w q 2 July 2017 Frank Cowell: Firm Optimization 28

29 Overview Firm: Optimisation The setting using the results of stage 1 Stage 1: Cost Minimisation Stage 2: Profit maximisation July 2017 Frank Cowell: Firm Optimization 29

30 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 July 2017 Frank Cowell: Firm Optimization 30

31 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 C q C/q q q July 2017 Frank Cowell: Firm Optimization 31

32 * Revenue and profits C q C/q 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 p Π price = marginal cost q q q q* q q July 2017 Frank Cowell: Firm Optimization 32

33 What happens if price is low... C q C/q p price < average cost q* = 0 q July 2017 Frank Cowell: Firm Optimization 33

34 Profit maximisation Objective is to choose q to max: pq C (w, q) From the First-Order Conditions if q* > 0: p = C q (w, q*) C(w, q*) p q* In general: pq* C(w, q*) Revenue minus minimised cost Price equals marginal cost Price covers average cost covers both the cases: q* > 0 and q* = 0 July 2017 Frank Cowell: Firm Optimization 34

35 Example (continued) Production function: q z z Resulting cost function: C(w, q) = w w q 2 Profits: pq C(w, q) = pq A q 2 where A:= w w FOC: p 2 Aq = 0 Result: q = p / 2A = w w p July 2017 Frank Cowell: Firm Optimization 35

36 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 July 2017 Frank Cowell: Firm Optimization 36