PROFIT MAXIMIZATION. π = p y Σ n i=1 w i x i (2)

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1 PROFIT MAXIMIZATION DEFINITION OF A NEOCLASSICAL FIRM A neoclassical firm is an organization that controls the transformation of inuts (resources it owns or urchases into oututs or roducts (valued roducts that it sells and earns the difference between what it receives in revenue and what it sends on inuts A technology is a descrition of rocess by which inuts are converted in oututs There are a myriad of ways to describe a technology, but all of them in one way or another secify the oututs that are feasible with a given choice of inuts Secifically, a roduction technology is a descrition of the set of oututs that can be roduced by a given set of factors of roduction or inuts using a given method of roduction or roduction rocess We assume that neoclassical firms exist to make money Such firms are called for-rofit firms We then set u the firm level decision roblem as maximizing the net returns from the technologies controlled by the firm taking into account the demand for final consumtion roducts, oortunities for buying and selling roducts from other firms, and the actions of other firms in the markets in which the firm articiates In erfectly cometitive markets this means the firm will take rices as given and choose the levels of inuts and oututs that maximize rofits If the firm controls more than one roduction technology it takes into account the interactions between the technologies and the overall rofits from the grou of technologies The rofits (or net returns to a articular roduction lan are given by the revenue obtained from the lan minus the costs of the inuts or π Σ m j j y j Σ n i w i x i ( where j is the rice of the jth outut and w i is the rice of the ith inut In the case of a single outut this can be written where is the rice of the single outut y π y Σ n i w i x i (2 2 DESCRIPTIONS OF TECHNOLOGY 2 Technology Sets A common way to describe a roduction technology is with a roduction set The technology set for a given roduction rocess is defined as T {(x, y : x ɛ R n +, y ɛ R m : + x can roduce y } (3 where x is a vector of inuts and y is a vector of oututs The set consists of those combinations of x and y such that y can be roduced from the given x Date: February 7, 26

2 2 PROFIT MAXIMIZATION 22 Production Corresondence The outut corresondence P, mas inuts x ɛ R n + into subsets of oututs, ie, P: R n + 2 Rm +, or P(x R m + The set P(x is the set of all outut vectors y ɛ R m + that are obtainable from the inut vector x ɛ R n + We reresent P in terms of the technology set as P (x {y : (x, y ɛ T } (4 23 Inut Corresondence The inut corresondence V, mas oututs y ɛ R m + into subsets of inuts, ie, V: R m + 2 Rn +, or V(y R n + The set V(y is the set of all inut vectors x ɛ R n + that are able to yield the outut vector y ɛ R m + We reresent V in terms of the technology set as V (y {x : (x, y ɛ T } (5 24 Relationshis between reresentations: V(y, P(x and T(x,y The technology set can be written in terms of either the inut or outut corresondence T {(x, y : x ɛ R n +, y ɛ R m +, such that x will roduce y} (6a T {(x, y ɛ R n+m + : y ɛp (x, xɛ R n +} (6b T {(x, y ɛ R n+m + : x ɛv (y, yɛ R m + } (6c We can summarize the relationshis between the inut corresondence, the outut corresondence, and the roduction ossibilities set in the following roosition Proosition y ɛ P(x x ɛ V(y (x,y ɛ T 25 Production Function In the case where there is a single outut it is sometimes useful to reresent the technology of the firm with a mathematical function that gives the maximum outut attainable from a given vector of inuts This function is called a roduction function and is defined as f (x max [y : (x, y ɛ T ] y max [y : x ɛ V (y] y max [y] y ɛ P (x (7 26 Asymmetric Transformation Function In cases where there are multile oututs one way to reresent the technology of the firm is with an asymmetric transformation function The function is asymmetric in the sense that it normalizes on one of the oututs, treating it asymmetrically with the other oututs We usually normalize on the first outut in the outut vector, but this is not necessary If y (y, y 2,, y m we can write it in the following asymmetric fashion y (y, ỹ where ỹ (y 2, y 3,, y m The transformation function is then defined as f(ỹ, x max y {y : (y, ỹ, x T }, if it exists otherwise, ỹ m, x n (8 This gives the maximum obtainable level of y, given levels of the other oututs and the inut vector x We could also define the asymmetric transformation function based on y i In this case it would give the maximum obtainable level of y i, given levels of the other oututs (including y and the inut vector x There are additional ways to describe multiroduct technologies using functions which will be discussed in a later section

3 PROFIT MAXIMIZATION 3 3 THE GENERAL PROFIT MAXIMIZATION PROBLEM The general firm-level maximization roblem can be written in a number of alternative ways π max x y [Σm j j y j Σ n i w i x i ], such that (x, y T (9 where T is reresents the grah of the technology or the technology set The roblem can also be written as π max x y [Σm j j y j Σ n i w i x i ] such that x V(y ] (a π max x y [Σm j j y j Σ n i w i x i ] such that y P(x ] (b where the technology is reresented by V(y, the inut requirement set, or P(x, the outut set We can write it in terms of functions in the following two ways π max x [ f(x, x 2,, x n Σ n i w i x i ] (a π max x ỹ [ f(ỹ, x + Σ m 2 j y j Σ n i w i x i ] (b max x ỹ [ f(ỹ, x + ỹ w x] where ( 2, 3,, m, ỹ (y 2, y 3,, y m, w (w, w 2,, w n x (x, x 2,, n 4 PROFIT MAXIMIZATION WITH A SINGLE OUTPUT AND A SINGLE INPUT 4 Formulation of Problem The roduction function is given by y f(x (2 If the roduction function is continuous and differentiable we can use calculus to obtain a set of conditions describing otimal inut choice If we let π reresent rofit then we have π f (x w x (3 If we differentiate the exression in equation 3 with resect to the inut x obtain π f (x w (4 x x Since the artial derivative of f with resect to x is the marginal roduct of x this can be interreted as MP x w (5 MV P x MF C x where MVP x is the marginal value roduct of x and MFC x (marginal factor cost is its factor rice Thus the firm will continue using the inut x until its marginal contribution to revenues just covers its costs We can write equation 4 in an alternative useful way as follows

4 4 PROFIT MAXIMIZATION f (x w x f (x w (6 x This says that the sloe of the roduction function is equal to the ratio of inut rice to outut rice We can also view this as the sloe of the isorofit line Remember that rofit is given by π y w x y π + w x The sloe of the line is then w This relationshi is demonstrated in figure (7 y FIGURE Profit Maximization Point f x x 42 Inut demands If we solve equation 4 or equation 6 for x, we obtain the otimal value of x for a given and w As a function of w for a fixed, this is the factor demand for x x x(, w (8 43 Sensitivity analysis We can investigate the roerties of x(,w by substituting x(,w for x in equation 4 and then treating it as an identity f ( x(, w w (9 x If we differentiate equation 9 with resect to w we obtain As long as 2 f ( x(,w 2 f ( x(, w x(, w w, we can write x(, w w (2 (2 2 f ( x(,w

5 PROFIT MAXIMIZATION 5 If the roduction function is concave then 2 f ( x(,w Factor demand curves sloe down This then imlies that x(,w w If we differentiate equation 9 with resect to we obtain As long as 2 f ( x(,w 2 f ( x(, w x(, w, we can write + f ( x(, w x (22 x(, w f ( x(,w x 2 f ( x(,w (23 If the roduction function is concave with a ositive marginal roduct then x(,w demand rises with an increase in outut rice 44 Examle Consider the roduction function given by Factor y 5 x 5 x 2 (24 Now let the rice of outut be given by 5 and the rice of the inut be given by w The rofit maximization roblem can be written π max [5 f (x x] x max [5 (5x 5x 2 x] x max [65x 25x 2 ] x If we differentiate π with resect to x we obtain 65 5 x 5 x 65 x 3 If we write this in terms of marginal value roduct and marginal factor cost we obtain 5 (5 x MP x MF C x MP x MF C x x 3 (25 (26 (27 5 PROFIT MAXIMIZATION WITH A SINGLE OUTPUT AND TWO INPUTS 5 Formulation of Problem The roduction function is given by y f (x, x 2 (28 If the roduction function is continuous and differentiable we can use calculus to obtain a set of conditions describing otimal inut choice If we let π reresent rofit then we have π f (x, x 2 w x w 2 x 2 (29 If we differentiate the exression in equation 29 with resect to each inut we obtain

6 6 PROFIT MAXIMIZATION π f (x, x 2 w x x π f (x, x 2 w 2 x 2 Since the artial derivative of f with resect to x j is the marginal roduct of x j this can be interreted as (3 π f (x, x 2 w x w 2 x 2 (3 MP w MP 2 w 2 MV P MF C (32 MV P 2 MF C 2 where MVP j is the marginal value roduct of the jth inut and MFC j (marginal factor cost is its factor rice 52 Second order conditions The second order conditions for a maxmimum in equation 29 are given by examining the Hessian of the objective function 2 π(x H π 2 π(x 2 π(x x x x 2 π(x 2 π(x 2 π(x 2 π(x x 2 π(x 2 π(x Equation 29 has a local maximum at the oint x if 2 π (x < and 2 π 2 π [ ] π π 2 π 2 π 22 [ (33 2 π x ] 2 > at x We can say this in another way as follows The leading rincial minors of 2 π(x alternate in sign with the first leading rincial minor being negative Secifically, 2 π(x 2 f(x 2 f(x 2 f(x 2 f(x x 2 f(x 2 f(x < <, given that > > (34 53 Inut demands If we solve the equations in 3 for x and x 2, we obtain the otimal values of x for a given and w As a function of w for a fixed, this gives the vector of factor demands for x x x(, w, w 2 ( x (, w, w 2, x 2 (, w, w 2 (35

7 PROFIT MAXIMIZATION 7 54 Homogeneity of degree zero of inut demands Consider the rofit maximization roblem if we multily all rices ( w, w 2 by a constant λ as follows π f (x, x 2 w x w 2 x 2 π(λ λ f (x, x 2 λ w x λ w 2 x 2 (36a (36b λ [ f (x, x 2 w x w 2 x 2 ] (36c Maximizing 36a or 36c will give the same results for x(, w, w 2 because λ is just a constant that will not affect the otimal choice 55 Sensitivity analysis 55 Resonse of factor demand to inut rices We can investigate the roerties of x(, w, w 2 by substituting x(, w, w 2 for x in equation 3 and then treating it as an identity f ( x(, w, w 2 x w f ( x(, w, w 2 w 2 If we differentiate equation 37 with resect to w we obtain 2 f ( x(, w, w 2 x (, w, w f ( x(, w, w 2 x 2 (, w, w 2 2 f ( x(, w, w 2 x (, w, w f ( x(, w, w 2 x We can write this in more abbreviated notation as f x (, w, w 2 + f 2 x 2 (, w, w 2 x (, w, w 2 x 2 (, w, w 2 f 2 + f 22 If we differentiate equation 37 with resect to w 2 we obtain x 2 (, w, w 2 (37 (38 (39 f x (, w, w 2 + f 2 x 2 (, w, w 2 x (, w, w 2 x 2 (, w, w 2 f 2 + f 22 Now write equations 39 and 4 in matrix form ( ( x f f (, w, w 2 x (, w, w 2 2 f 2 f 22 x 2 (, w, w 2 x 2 (, w, w 2 ( If the Hessian matrix is non-singular,we can solve this equation for the matrix of first derivatives, ( x (, w, w 2 x (, w, w 2 ( f f 2 (42 f 2 f 22 x 2 (, w, w 2 x 2 (, w, w 2 (4 (4

8 8 PROFIT MAXIMIZATION We can comute the various artial derivatives on the left hand side of equation 42 by inverting the Hessian or using Cramer s rule in connection with equation 4 The inverse of the Hessian of a two variable roduction function gan be comuted by using the adjoint The adjoint is the transose of the cofactor matrix of the Hessian For a square nonsingular matrix A, its inverse is given by We comute the inverse by first comuting the cofactor matrix A A A+ (43 2 f(x, x 2,, x n ( f f 2 f 2 f 22 cofactor[f ] ( 2 f 22 cofactor[f 2 ] ( 3 f 2 cofactor[f 2 ] ( 3 f 2 (44 cofactor[f 22 ] ( 4 f cofactor [ 2 f(x, x 2,, x n ] ( f22 f 2 f 2 f We then find the adjoint by taking the transose of the cofactor matrix adjoint [ 2 f(x, x 2,, x n ] ( f22 f 2 f 2 f (45 We obtain the inverse by dividing the adjoint by the determinant of [ 2 f(x, x 2,, x n ] ( f22 f 2 ( f f 2 f 2 f (46 f 2 f 22 f f 22 f 2 f 2 Referring back to equation 42, we can comute the various artial derivatives ( f22 f 2 ( x (, w, w 2 x (, w, w 2 x 2 (, w, w 2 x 2 (, w, w 2 f 2 f ( f f 22 f 2 f 2 (47 We then obtain

9 PROFIT MAXIMIZATION 9 x (, w, w 2 f 22 ( f f 22 f2 2 x (, w, w 2 f 2 ( f f 22 f2 2 x 2 (, w, w 2 f 2 ( f f 22 f2 2 x 2 (, w, w 2 f ( f f 22 f2 2 (48a (48b (48c (48d The denominator is ositive by second order conditions in eqaution 34 or the fact that f( is concave Because we have a maximum, f and f 22 are less than zero Own rice derivatives are negative and so factor demand curves sloe downwards The sign of the cross artial derivatives deends on the sign of f 2 If x and x 2 are gross substitutes, then f 2 is negative and the second cross artials are ositive This means that the demand for x goes u as the rice of x 2 goes u 552 Finding factor demand resonse using Cramer s rule When we differentiate the first order conditions we obtain ( ( x f f (, w, w 2 2 f 2 f 22 ( f f 2 f 2 f 22 x (, w, w 2 x 2(, w, w 2 x 2(, w, w 2 ( x (, w, w 2 x 2(, w, w 2 ( ( ( To find x (, w, w 2 we relace the first column of the Hessian with the right hand side vector and then form the ratio of the determinant of this matrix to the determinant of the Hessian First for the determinant of the matrix with the righthand side relacing the first column Then find the determinant of the Hessian Forming the ratio we obtain (49 f 2 f 22 f 22 (5 f f 2 f 2 f 22 f f 22 f 2 f 2 (5 which is the same as in equation 48 x (, w, w 2 f 22 ( f f 22 f 2 f 2 ( Resonse of factor demand to outut rice If we differentiate equation 37 with resect to we obtain

10 PROFIT MAXIMIZATION f x (, w, w 2 f 2 x (, w, w 2 Now write equation 53 in matrix form + f 2 x 2 (, w, w 2 + f 22 x 2 (, w, w 2 f f 2 (53 ( ( x(, w, w2 f f 2 f 2 f 22 x 2 (, w, w 2 ( f f 2 (54 Multily both sides of equation 54 by 2 f(x, x 2,, x n to obtain ( x(, w, w 2 x 2(, w, w 2 ( ( f f 2 f f 22 f 2 f 2 ( f22 f 2 f 2 f ( f f 22 f 2 f 2 ( f f 22 + f 2 f 2 ( f f 2 (55 f f 2 f 2 f ( f f 22 f 2 f 2 This imlies that x (, w, w 2 x 2 (, w, w 2 f f 22 + f 2 f 2 ( f f 22 f 2 2 f f 2 f 2 f ( f f 22 f 2 2 (56 These derivatives can be of either sign Inuts usually have a ositive derivative with resect to outut rice 56 Examle 56 Production function Consider the following roduction function y f(x, x 2 3x + 6x 2 x 2 + x x 2 2 x 2 2 (57 The first and second artial derivatives are given by

11 PROFIT MAXIMIZATION f(x, x 2 x 3 2 x + x 2 f(x, x x 4 x 2 2 f(x, x f(x, x 2 x 2 f(x, x 2 4 (58 The Hessian is 2 f(x, x 2 2 f(x, x 2 2 f(x, x 2 2 f(x, x 2 x 2 f(x, x 2 ( 2 4 (59 The determinant of the Hessian is given by (6 562 Profit maximization Profit is given by π f(x, x 2 w x w 2 x 2 [ 3x + 6x 2 x 2 + x x 2 2 x 2 ] (6 2 w x w 2 x 2 We maximize rofit by taking the derivatives of 6 setting them equal to zero and solving for x and x 2 Rearranging 62 we obtain π x [ 3 2 x + x 2 ] w π [ 6 + x 4 x 2 ] w 2 ( x + x 2 w 6 + x 4 x 2 w 2 (63a (63b Multily equation 63b by 2 and add them together to obtain

12 2 PROFIT MAXIMIZATION 62 7 x 2 w + 2 w 2 7 x 2 62 w 2 w 2 (64 x w 7 2w 2 7 Now substitute x 2 in equation 63b and solve for x 6 + x 4 ( 62 7 w 7 2w 2 7 w 2 x w w ( 62 7 w 7 2w 2 7 4w 7 8w 2 7 4w 7 w ( Necessary and sufficient conditions for a maximum Consider the Hessian of the rofit equation 2 π(x, x 2 2 π(x, x 2 2 π(x, x 2 2 π(x, x 2 x 2 π(x, x 2 ( 2 4 (66 For a maximum we need the diagonal elements to be negative and the determinant to be ositive The diagonal elements are negative The determinant of the Hessian is given by ( Inut demand derivatives comuted via first order conditions and formulas Consider the change in inut demand with a change in inut rice Remember the Hessian of the roduction function from equation 59 is given by ( 2 2 f(x, x Now substitute in the formulas for inut demand derivatives from equation 48

13 PROFIT MAXIMIZATION 3 x (, w, w 2 f 22 ( f f 22 f x (, w, w 2 f 2 ( f f 22 f2 2 7 x 2 (, w, w 2 f 2 ( f f 22 f2 2 7 x 2 (, w, w 2 f ( f f 22 f Inut demand derivatives comuted from the otimal inut demand equations x w 7 w 2 7 x w 7 2w 2 7 x 4 7 x (68a (68b (68c (68d (69 6 PROFIT MAXIMIZATION WITH A SINGLE OUTPUT AND MULTIPLE INPUTS 6 Formulation of Problem The roduction function is given by y f (x, x 2,, x n (7 If the roduction function is continuous and differentiable we can use calculus to obtain a set of conditions describing otimal inut choice If we let π reresent rofit then we have π f (x, x 2,, x n n w i x i, j, 2, n (7 If we differentiate the exression in equation 7 with resect to each inut we obtain π f (x w j, j, 2, n (72 x j x j Since the artial derivative of f with resect to x j is the marginal roduct of x j this can be interreted as j MP j MV P j w j, j, 2, n MF C j, j, 2, n (73

14 4 PROFIT MAXIMIZATION where MVP j is the marginal value roduct of the jth inut and MFC j (marginal factor cost is its factor rice 62 Inut demands If we solve the equations in 72 for x j, j, 2,, n, we obtain the otimal values of x for a given and w As a function of w for a fixed, this gives the vector of factor demands for x x x(, w (x (, w, x 2 (, w,, x n (, w (74 63 Second order conditions The second order conditions for a maxmimum in equation 7 are given by examining the Hessian of the objective function 2 π(x H π 2 π(x 2 π(x x x x 2 π(x 2 π(x 2 π(x 2 π(x x n x x n 2 π(x x x n 2 π(x x n 2 π(x x n x n π π 2 π n π 2 π 22 π 2n π n π n2 π nn Equation 7 has a local maximum at the oint x if the leading rincial minors of 2 π(x alternate in sign with the first leading rincial minor being negative, the second ositive and so forth Thus π < and π π 22 π 2 π 2 > and so on Secifically, (75 2 π(x 2 f(x 2 f(x 2 f(x 2 f(x x 2 f(x 2 f(x 2 f(x 2 f(x x 2 f(x x x 3 2 f(x 2 f(x 2 f(x x 3 2 f(x x 3 x 2 f(x x 3 2 f(x 3 < <, given that > > < (76 64 Sensitivity analysis We can investigate the roerties of x(,w by substituting x(,w for x in equation 72 and then treating it as an identity f ( x(, w x j w j, j, 2, n (77 If we differentiate the first equation in 77 with resect to w j we obtain

15 PROFIT MAXIMIZATION 5 2 f(x(, w x (, w + 2 f(x(, w (, w + 2 f(x(, w x 3 (, w + x 3 x ( 2 f(x(, w 2 f(x(,w 2 f(x(,w x 3 x 2 f(x(,w x n x x (, w (, w x 3(, w x n (, w (78 If we differentiate the second equation in 77 with resect to w j we obtain 2 f(x(, w x x (, w + 2 f(x(, w ( 2 f(x(, w x 2 f(x(,w (, w + 2 f(x(, w x 3 x 3 (, w + 2 f(x(,w x 3 2 f(x(,w x n x (, w (, w x 3 (, w x n(, w (79 If we differentiate the j th equation in 77 with resect to w j we obtain 2 f(x(, w x x j x (, w + 2 f(x(, w x j (, w ( 2 f(x(, w x x j Continuing in the same fashion we obtain f(x(, w 2 f(x(,w x j 2 f(x(,w j j 2 f(x(,w x n x j (, w + x (, w (, w x j(, w x n (, w (8

16 6 PROFIT MAXIMIZATION 2 f(x(, w 2 f(x(, w x 2 f(x(, w x x j 2 f(x(, w x x n x j x n x j x x j j x j x n x n x x n x n x j n x (, w (, w x j (, w x n(, w If we then consider derivatives with resect to each of the w j we obtain B@ 2 f(x(, w 2 f(x(, w x 2 f(x(, w x x j 2 f(x(, w x xn We can then write x j xn x (, w x (, w (, w (, w x n (, w x n (, w xn x xn xn x j n x (, w w n (, w w n x n (, w w n CA B@ C A x (, w x (, w (, w (, w xn(, w xn(, w 2 f(x(, w 2 f(x(, w x 2 f(x(, w x x j 2 f(x(, w x x n x (, w wn (, w wn xn(, w wn x j x n CA B@ x n x x n x n x j n If the roduction function is concave, the Hessian will be at least negative semidefinite This means that its characteristics roots are all negative or zero If the Hessian is negative definite then its characteristics roots are all negative A negative definite matrix is invertible The inverse of an invertible matrix has characteristics roots which are recirocals of the characteristic roots of the original matrix So if the roots of the original matrix are all negative, the roots of the inverse will also be negative If the characteristics roots of a matrix are all negative then the matrix is negative definite And the diagonal elements of negative definite matrix are all negative So own rice derivatives are negative The second order conditions for rofit maximization also imly that the Hessian is negative definite as is clear from equation 76 C A CA (8 (82 (83