Profit Maximization. Beattie, Taylor, and Watts Sections: 3.1b-c, 3.2c, , 5.2a-d

Transcription

1 Proit Maimization Beattie Talor and Watts Sections:.b-c.c a-d

2 Agenda Generalized Proit Maimization Proit Maimization ith One Inut and One Outut Proit Maimization ith To Inuts and One Outut Proit Maimization ith One Inut and To Oututs

3 Deining Proit Proit can be generall deined as total revenue minus total cost. Total revenue is the summation o the revenue rom each enterrise. The revenue rom one enterrise is deined as rice multilied b quantit. Total cost is the summation o all ied and variable cost.

4 4 Deining Proit Cont. Short-run roit π can be deined mathematicall as the olloing: nm m m m n n nm n n n m m m j j j n i i i m n TFC TC TR π π

5 Revenue In a erectl cometitive market revenue rom a articular enterrise can be deined as *. When the roducer can have an eect on rice then rice becomes a unction o outut hich can be reresented as *. 5

6 Marginal Revenue Marginal Revenue MR is deined as the change in revenue due to a change in outut. In a erectl cometitive orld marginal revenue equals average revenue hich equals rice. TR MR dtr d 6

7 7 Marginal Revenue Cont. When the market is not erectl cometitive then MR can be reresented as the olloing: d d MR MR d dtr MR TR ε ε * ' * ' *

8 Marginal Value O Product Marginal Value o Product MVP is deined as the change in revenue due to a change in the inut. To ind MVP ou need to substitute the roduction unction into the TR unction. TR TR MVP dtr d ' MPP 8

9 Cost Side o Proit Maimization Marginal Cost MC and Marginal Inut Cost MIC can be derived rom the cost side o the roit unction. Marginal cost is deined as the change in cost due to a change in outut. From the cost minimization roblem it as shon the dierent orms that marginal cost could take. Marginal Inut Cost is the change in cost due to a change in the inut. MIC is equal to the rice o the inut. 9

10 Standard Proit Maimization Model nm m m m n n nm n n n m m m j j j n i i i m j n i n i t r TFC Ma ij i... ;... or... or

11 Proit Maimization ith One Inut and One Outut Assume that e have one variable inut hich costs. Assume that the general roduction unction can be reresented as. Ma TFC subject to :

12 Eamining Results o Proit Maimization ith One Inut and One Outut MIC MVP MPP and d d d d d d TFC Γ Γ Γ Γ ' ' ' '

13 Notes on Proit Maimization B solving the roit maimization roblem e get the otimum decision rule here MVPMIC. With minor maniulation e can transorm the result rom the revious slide using the roduction unction into the other orm o the otimum decision MR MC.

14 Notes on Proit Maimization Cont. There are to rimar as to solve the roit maimization roblem. Solve the constrained roit ma roblem.r.t. and. Transorm the constrained roit ma roblem into an unconstrained roblem b substituting the roduction unction or its inverse into the roit ma roblem and solve.r.t. to the aroriate variable. 4

15 Solving the Proit Maimization Problem W.R.T. Inuts Assume that e have one variable inut hich costs. Assume that the general roduction unction can be reresented as. Ma TFC 5

16 Solving the Proit Maimization Problem W.R.T. Inuts Cont. Γ dγ ' d ' MPP MPP 6

17 Solving the Proit Maimization Problem W.R.T. Oututs Assume that e have one variable inut hich costs. Assume that the general roduction unction can be reresented as ith an outut rice o. Ma TFC 7

18 Solving the Proit Maimization Problem W.R.T. Oututs Cont. Γ dγ d MPP MPP MPP TFC 8

19 Proit Ma Eamle Suose that ou ould like to maimize roits given the olloing inormation: Outut Price Inut Price TFC 5-9

20 Proit Ma Eamle : Lagrangean be done in class. Solution ill s.t. Γ Γ Γ Γ d d d d d d Ma

21 Proit Ma Eamle : Unconstrained W.R.T. Inut Ma Γ 5 5 π 5 5 dγ 5 d

22 Proit Ma Eamle : Solving Using MICMVP TR 5 MIC MVP MVP 5 5 dtr d MIC 5 5

23 Proit Ma Eamle : Solving Using MPP/ MPP MPP d d 5

24 4 Proit Ma Eamle : Unconstrained W.R.T. Outut ma Γ Γ d d

25 5 Proit Ma Eamle : Solving Using MCMR * MC MR d dtc MC MR TFC TC TFC

26 Question: Ho ould ou ind the loss in roit π i ou ere a revenue maimizer instead a roit maimizer? Loss π π-ma - π Revenue-Ma 6

27 Note on Calculating Proit at Revenue Ma Proit at revenue ma can be ound b calculating roit at the revenue maimizing oint i.e. ind the inut level here MPP calculate the outut or this level o inut and then use these values to calculate roit 7

28 Grah o Proit and Production Production Pro it

29 Grah o Proit and Total Revenue Total Revenue Proit

30 Grah o Marginal Revenue and Marginal Cost 5 4 Marginal Revenue Marginal Cost

31 Grah o Marginal Value o Product and Marginal Inut Cost

32 Proit Ma Eamle Suose that ou ould like to maimize roits given the olloing inormation: Outut Price Inut Price TFC 5-

33 Proit Ma Eamle : Lagrangean be done in class. Solution ill s.t. Γ Γ Γ Γ d d d d d d Ma

34 Proit Ma Eamle : Unconstrained W.R.T. Inut Ma Γ π dγ 5 d 5 4

35 5 Proit Ma Eamle : Unconstrained W.R.T. Outut ma Γ Γ d d

36 6 Proit Ma Eamle : Solving Using MCMR * MC MR d dtc MC MR TFC TC TFC

37 Proit Ma Eamle : Solving Using MICMVP TR 4 MIC MVP MVP 5 4 dtr d MIC 8 4 7

38 Proit Ma Eamle : Solving Using MPP/ MPP MPP d d 5 8

39 Proit Maimization ith To Inuts and One Outut Assume that e have to variable inuts and hich cost resectivel and. Also let TFC reresent the total ied costs. Assume that the general roduction unction can be reresented as here sells at a rice o. Ma subject to : TFC 9

40 4 First Order Conditions or the Constrained Proit Maimization Problem ith To Inuts Γ Γ Γ Γ Γ MPP MPP MPP TFC

41 4 First Order Conditions or the Unconstrained Proit Maimization Problem ith To Inuts MPP MPP MPP MPP TFC π π π

42 Summar o Proit Ma Results At the otimum each inut selected ill cause the MPP ith resect to that inut to equal the ratio o inut rice to outut rice. For eamle: MPP / MPP / 4

43 Summar o Proit Ma Results Cont. From the roit ma roblem ou ill get a relationshi beteen the to inuts. This relationshi is called the eansion ath. Once ou selected a certain outut our revenue becomes triviall given to ou hen outut rice is ied. Hence ou are just minimizing cost. 4

44 Eamle o Proit Maimization ith To Variable Inuts Suose ou have the olloing roduction unction: 4 ½ ½ Suose the rice o inut is $ and the rice o inut is $6. Let the total ied cost equal $. What is the otimal amount o inut and i ou have a rice o or the outut and ou ant to roduce units? What is the roit? 44

45 Eamle o Proit Ma ith To Variable Inuts Cont. Summar o hat is knon: 6 4 ½ ½ Ma subject to :

46 46 Eamle o Proit Ma ith To Variable Inuts Cont. Solution done in class Γ Γ Γ Γ Γ

47 Eamle o Proit Ma ith To Variable Inuts Cont. Summar o hat is knon: 6 4 /4 /4 Ma subject to :

48 48 Eamle o Proit Ma ith To Variable Inuts Cont. Solution done in class Γ Γ Γ Γ Γ

49 49 Eamle : Finding the Proit Ma Inuts Using the Production Function and MPP i i / Solution done in class Set 4 4 Set MPP MPP MPP MPP TFC

50 Proit Maimization ith To Oututs and One Inut Assume that e have to roduction unctions and hich have a rice o and resectivel. Assume that ou have one inut X that can be divided beteen roduction unction and roduction unction. 5

51 Proit Maimization ith To Oututs and One Inut Cont. The amount o inut allocated to is deined as and the amount o inut allocated to is. The summation o and have to sum to X i.e. X. Note that The rice o the inut X is. 5

52 Proit Maimization ith To Oututs and One Inut Cont. Ma subject to : TFC X 5

53 5 First Order Conditions or the Constrained Proit Maimization Problem ith To Oututs in class. Solution discussed Γ Γ Γ Γ Γ Γ Γ Γ X X TFC

54 Summar o Proit Ma Results At the otimum the marginal value o roduct o the irst roduction unction ith resect to inut MVP is equal to the marginal value o roduct o the second roduction unction MVP. This gives ou the otimal allocation o inuts. For eamle: MVP MVP 54

55 Summar o Proit Ma Results Cont. With some maniulation o the revious act the otimum rule or outut selection occurs here the sloe o the PPF i.e. MRPT is equal to the negative o the outut rice ratio. This gives ou the otimal allocation o oututs. MRPT-/ 55

56 Eamle o Proit Maimization ith To Oututs and One Inut Suose ou have the olloing roduction unctions: / / We can dro nd subscrit because onl one ied inut X Suose the rice o outut is $4 and the rice o outut is $. The rice o the inut is and the total ied cost is. What is the otimal amount o outut and i ou have 9 units o inut X to allocate to both roductions? What is the roit? 56

57 Eamle o Proit Ma ith To Oututs and One Inut Cont. Summar o hat is knon: 4 X9 TFC / / Ma 4 subject to : 9 57

58 58 Eamle o Proit Ma ith To Oututs Cont. in class. Solution discussed Γ Γ Γ Γ Γ Γ Γ Γ

59 Eamle : Finding the Proit Ma Oututs Using MRPT / 9 4 9* MRPT d d Solution done in class. 59

60 Eamle : Finding the Proit Ma Inuts Using MVP MVP 4 9 MVP MVP MVP 4 8 * MPP * MPP Solution inished in class. MVP MPP MPP d d d d 4 4 6