PROFIT FUNCTIONS. 1. REPRESENTATION OF TECHNOLOGY 1.1. Technology Sets. The technology set for a given production process is defined as

PROFIT FUNCTIONS 1. REPRESENTATION OF TECHNOLOGY 1.1. Technology Sets. The technology set for a given production process is defined as T {x, y : x ɛ R n, y ɛ R m : x can produce y} where x is a vector of inputs and y is a vector of outputs. The set consists of those combinations of x and y such that y can be produced from the given x. 1.. The Output Correspondence and the Output Set. 1..1. Definitions. It is often convenient to define a production correspondence and the associated output set. 1: The output correspondence P, maps inputs x ɛ R n into subsets of outputs, i.e., P: R n Rm. A correspondence is different from a function in that a given domain is mapped into a set as compared to a single real variable or number as in a function. : The output set for a given technology, Px, is the set of all output vectors y ɛ R m that are obtainable from the input vector x ɛ R n. Px is then the set of all output vectors y ɛ R m that are obtainable from the input vector x ɛ R n. We often write Px for both the set based on a particular value of x, and the rule correspondence that assigns a set to each vector x. 1... Relationship between Px and Tx,y. P x y : x, y ɛ T 1..3. Properties of Px. P.1a: P.1 No Free Lunch. 0 ɛ Px x ɛ R. n P.1b: y P0, y > 0. P.: Input Disposability. x ɛ R n, Px Pθx, θ 1. P..S: Strong Input Disposability. x, x ɛ R n, x x Px Px. P.3: Output Disposability. x ɛ R n, y ɛ Px and 0 λ 1 λy ɛ Px. P.3.S: Strong Output Disposability. x ɛ R n, y ɛ Px y ɛ Px, 0 y y. P.4: Boundedness. Px is bounded for all x ɛ R n. P.5: Tx is a closed set P: R n Rm is a closed correspondence, i.e., if [x l x 0, y l y 0 and y l ɛ Px l, l] then y 0 ɛ Px 0. P.6: Attainability. If y ɛ Px, y 0 and x 0, then θ 0, λ θ 0 such that θy ɛ Pλ θ x. P.7: Px is convex Px is convex for all x ɛ R n. P.8: P is quasi-concave. The correspondence P is quasi-concave on R n which means x, x ɛ R n, 0 θ 1, Px Px Pθx 1-θx P.9: Convexity of Tx. P is concave on R n which means x, x ɛ R n, 0 θ 1, θpx1- θpx Pθx 1-θx Date: February 7, 006. 1

PROFIT FUNCTIONS 1..4. Properties of Tx,y. T.1a: Inaction and No Free Lunch. 0,y ɛ T x ɛ R n and y ɛ R m. This implies that Tx,y is a non-empty subset of R mn T.1b: 0,y T, y 0, y 0. T.: Input Disposability. If x,y ɛ T and θ 1 then θx, y ɛ T. T..S: Strong Input Disposability. If x,y ɛ T and x x, then x, y ɛ T. T.3: Output Disposability. x,y ɛ R nm,if x, y ɛ T and 0 < λ 1 then x, λy ɛ T. T.3.S: Strong Output Disposability. If x,y ɛ T and y y, then x, y ɛ T. T.4: Boundedness. For every finite input vector x 0, the set y ɛ Px is bounded from above. This implies that only finite amounts of output can be produced from finite amounts of inputs. T.5: Tx is a closed set. The assumption that Px and Vy are closed does not imply that T is a closed set, so it is assumed. Specifically, if [x l x 0, y l y 0 and x l, y l ɛ T, l] then x 0 y 0 ɛ T. T.9: T is a convex set. This is not implied by the convexity of Px and Vy. Specifically, a technology could exhibit a diminishing rate of technical substitution, a diminishing rate of product transformation and still exhibit increasing returns to scale..1. Profit Maximization.. PROFIT MAXIMIZATION AND THE PROFIT FUNCTION.1.1. Setup of problem. The general firm-level maximization problem can be written in a number of alternative ways. π max [ m p j y j w i x i ], such that x, y T. 1 x y j1 i1 where T is represents the graph of the technology or the technology set. The problem can also be written as π max [ m p j y j x y j1 π max [ m p j y j x y j1 w i x i ] such hat x Vy ] i1 w i x i ] such that y Px ] i1 where the technology is represented by Vy, the input requirement set, or Px,the output set. Though T is non-empty, closed and convex, profit may not attain a maximum on T, i.e. profit can be unbounded even when the technology set is well behaved if output price is higher than input price. Consider for example the production function fx x x 1/. The production function is concave but profit will go to infinity if p > w. Therefore we often write π sup x y m [ p j y j j1 a b w i x i ], such that x, y T. 3 i1 where sup stands for supremum and could be infinity. We understand that in all practical problems the sup is a max.

PROFIT FUNCTIONS 3 If we carry out the maximization in equation 1 or equation 3, we obtain a vector of optimal outputs and a vector of optimal outputs such that y is producible given x and profits cannot be increased. We denote these optimal input and output choices as yp,w and xp,w where it is implicit that y and x are vectors..1.. One output and two input example. In the case of a single output such that fx is defined and two inputs we obtain the following. π p f x 1, x w 1 x 1 w x 4 If we differentiate the expression in equation 4 with respect to each input we obtain π p f x 1, x w 1 0 x 1 x 1 π p f x 5 1, x w 0 x x If we solve the equations in 5 for x 1 and x, we obtain the optimal values of x for a given p and w. As a function of w for a fixed p, this gives the vector of factor demands for x. The optimal output is given by x xp, w 1, w x 1 p, w 1, w, x p, w 1, w 6 y f x 1 p, w 1, w, x p, w 1, w 7.. The Profit Function. If we substitute the optimal input demand from equation 6 into equation 1 or equation 4, we obtain the profit function. The profit function is usually designated by π. πp, w m p j y j p, w j1 w i x i p, w i1 p fx 1 p, w, x p, w w 1 x 1 p, w w x p, w Notice that π is a function of p and w, not x or y. The optimal x and optimal y have already been chosen. The function tells us what profits will be assuming the firm is maximizing profits given a set of output and input prices. To help understand how πp,w only depends on p and w, consider the profit function for the case of two inputs. 8 πp, w 1, w p fx 1 p, w 1, w, x p, w 1, w w 1 x 1 p, w 1, w w x p, w, w 9 Consider the derivative of πp,w with respect to p. πp, w p p fx 1p, w, x p, w x 1 p, w x 1 p fx 1 p, w, x p, w w 1 x 1 p, w p Now collect terms containing x 1p,w p and x p,w p. p fx 1p, w, x p, w x p, w x p w x p, w p 10

4 PROFIT FUNCTIONS πp, w p x 1p, w p x p, w p fx 1 p, w, x p, w [ p fx 1p, w, x p, w But the first order conditions in equation 5 imply that w 1 x 1 [ p fx 1p, w, x p, w w x ] ] 11 p fx 1 x x i w i 0, i 1, 1 This means that the two bracketed terms in equation 11 are equal to zero so that πp, w p which depends only on p and w. fx 1 p, w, x p, w 13 3. PROPERTIES OF, OR CONDITIONS ON, THE PROFIT FUNCTION 3.1. π.1. πp, w is an extended real valued function it can take on the value of for finite prices defined for all p, w 0 m, 0 n and πp, w pa - wb for a fixed vector a, b 0 m, 0 n. This implies that πp, w 0 if 0 m, 0 n Tx,y,which we normally assume. 3.. π.. π is nonincreasing in w 3.3. π.3. π is nondecreasing in p 3.4. π.4. π is a convex function 3.5. π.5. π is homogeneous of degree 1 in p and w. 3.6. π.6 Hotelling s Lemma. πp, w p j y j p, w 14a πp, w w i x i p, w 14b 4. DISCUSSION OF PROPERTIES OF THE PROFIT FUNCTION For ease of exposition, consider the case where there is a single output. This is easily generalized by replacing fxp,w with yp,w and letting p be a vector. 4.1. π.1. πp, w is an extended real valued function it can take on the value of for finite prices defined for all p, w 0 m, 0 n and πp, w pa - wb for a fixed vector a, b 0 m, 0 n. This implies that πp, w 0. The profit function can be infinity due to the fact that maximum profits may be infinity as discussed in section.1.1. πp, w pa - wb for a fixed vector a, b because πp, w is the maximum profit at output prices p and input prices w. Any other input combination is bounded by πp, w.

PROFIT FUNCTIONS 5 4.. π.. π is nonincreasing in w Let optimal input be xp,w with prices p and w and ˆxp,w with prices p and ŵ. Now assume that ŵ > w. It is clear that pfˆxp,w - w ˆxp,w pf xp,w - w xp,w because the optimal x with w is x. However, pfˆx - ŵ ˆx < pfˆx - w ˆx because ŵ > w by assumption. So we obtain pfˆxp, w ŵ ˆxp, w < pfˆxp, w w ˆxp, w pf xp, w w p, w xp, w pfˆxp, w ŵ ˆxp, w pf xp, w w xp, w 15 4.3. π.3. π is nondecreasing in p Let profit be π p,w with prices p and w and ˆπˆp,w with prices ˆp and w. Now assume that ˆp > p. It is clear that ˆpf xp, w w xp, w ˆp fˆxp, w w ˆxp, w 16 because ˆxp, w is optimal for prices ˆp. However, pf xp, w w xp, w < ˆp f xp, w w xp, w 17 because ˆp > p by assumption. So we obtain, pf xp, w w xp, w < ˆp f xp, w w xp, w ˆp fˆxp, w w ˆxp, w pf xp, w w xp, w ˆp fˆxp, w w ˆxp, w 18 4.4. π.4. π is a convex function Consider the definition of a convex function. A function f: R mn R 1 is said to be convex if f λ x 1 1 λ x λ f x 1 1 λ f x for each vector x 1, x R mn and for each λ [0, 1]. In the case of the profit function π maps an nm vector containing output and input prices into the real line. Alternatively, a differentiable function π is convex in R mn if and only if πp, w π p, w p π p, w p p w π p, w w w, for each distinct p, w, p, w R mn. 19 Let y,x be the profit maximizing choices of y and x when prices arep, w and let y,x be profit maximizing maximizing choices of y and x when prices arep, w. Let p w be a linear combination of the price vectors p,w and p,w, i.e. p, w λ p, w 1 λ p, w [ λ p 1 λ p, λ w 1 λ w ] Then let y,x be the profit maximizing choices of y and x when prices arep, w. This then implies that 0 πp, w p y w x 1 Now substitute the definition of p and w from equation 0 into 1

6 PROFIT FUNCTIONS πp, w p y w x [ λ p 1 λ p ] y [ λ w 1 λ w ] x λ p y λ w x 1 λ p y 1 λ w x λ [ p y w x ] 1 λ [ p y w x ] Be the definition of profit maximization, we know that when prices are p,w any output input combination other than y,x will yield lower profits. Similarly when prices are p, w. We then can write λ [ p y w x ] λ [ p y w x] λ π p, w 1 λ [ p y w x ] 1 λ [ p y w x ] 1 λ π p, w 3 Now write equation 3 without the middle terms λ [ p y w x ] λ π p, w 1 λ [ p y w x ] 1 λ π p, w 4 Now add the two inequalities in equation 4 to obtain λ [ p y w x ] 1 λ [ p y w x ] λ π p, w 1 λ π p, w 5 Now substitute for the left hand side of equation 5 from equation to obtain πp, w λ π p, w 1 λ π p, w 6 Becauuse πp,w is convex, we know that its Hessian matrix is positive semidefinite. This means that the diagonals of the Hessian matrix are all positive or zero, i.e., πp,w 0 and πp,w 0. p j wi We can visualize convexity if we hold all input prices fixed and only consider a firm with a single output, or hold all but one output price fixed. In figure 1we can see that the tangent lies below the curve. At prices p, w, profits are at the level πp, w. If we hold the output and input levels fixed at y, x and change p, we move along the tangent line denoted by Πp, w. Profits are less along this line than along the profit function because we are not adjusting x and y to account for the change in p. 4.5. π.5. π is homogeneous of degree 1 in p and w. Consider the profit maximization problem, πp, w max [ m p j y j x y j1 w i x i ], such that x, y T. 7 Now multiply all prices by λ and denote the new profit mximization problem as i1

PROFIT FUNCTIONS 7 FIGURE 1. The Profit Function is Convex Profits Π p,w p,w py w x Π p,w p p π p, w max x y [ m j1 λ p j y j λ w i x i ], such that x, y T i1 λ max x y [ m j1 p j y j w i x i ] i1 8 π p, w λ πp, w 4.6. π.6 Hotelling s Lemma. πp, w p j y j p, w 9a πp, w w i x i p, w 9b We have already shown part a of equation 9 in equation 13 for the case of a single output. We can derive part b of f equation 9 in a similar manner. Consider the derivative of πp,w 1, w with respect to w 1. The profit function is given by πp, w 1, w p fx 1 p, w 1, w, x p, w 1, w w 1 x 1 p, w 1, w w x p, w, w 30 Taking the derivative with respect to w 1 we obtain πp, w w 1 p fx 1p, w, x p, w x 1 x 1 p, w w 1 w 1 x 1 p, w w 1 Now collect terms containing x 1p,w w 1 x 1 p, w w x p, w w1 and x p,w w 1. p fx 1p, w, x p, w x x p, w w 1 31

8 PROFIT FUNCTIONS πp, w x [ 1p, w p fx 1p, w, x p, w w 1 w 1 x p, w w 1 x 1 p, w 1, w But the first order conditions in equation 5 imply that w 1 x 1 [ p fx 1p, w, x p, w w x ] ] 3 p fx 1 x x i w i 0, i 1, 33 This means that the two braketed terms in equation 3 are equal to zero so that πp, w x 1 p, w 1, w 34 w1 We can show this in a more general manner as follows. Define a function gp,w as follows gp, w πp, w [ py wx ] 35 where y, x is some production plan that is not necessarily optimal at prices p,w. At prices p, w, y, x will be the profit maximizing production. The first term in equation 35 will always be greater than or equal to the second term. Thus gp,w will reach its minimum when prices are p, w. Consider then the conditions for minimizing gp,w. gp, w p j πp, w p j y j 0, j 1,,..., m gp, w πp, w w i w i x j 0, i 1,,..., n Rearranging equation 36 we obtain 36 πp, w p j y j 0, j 1,,..., m πp, w p j y j, j 1,,..., m πp, w w i x j 0, i 1,,..., n πp, w x i, i 1,,..., m w i A number of important implications come from equation 37. 1: We can obtain output supply and input demand equations by differentiating the profit function. : If we have an expression equation for the profit function, we can obtain output supply and input demand equations or functional forms for such equations by differentiating the profit function as compared to solving a maximization problem. This would allow one to find functional forms for estimating supply and demand without solving maximization problems. 37

PROFIT FUNCTIONS 9 3: Given that the output supply and input demand functions are derivatives of the profit function, we can determine many of the properties of these response functions by understanding the properties of the profit function. For example, because the profit function is homogeneous of degree one in p and w, its derivatives are homogeneous of degree zero. This means that output supply and input demand equations are homogeneous of degree zero, i.e., multiplying all prices by the same constant will not change output supply or input demand. 4: The second derivatives of the profit function are the first derivatives of the output supply and input demand functions. Properties of the second derivatives of the profit function are properties of the first derivatives of the output supply and input demand functions. 5: The symmetry of second order cross partial derivatives leads to symmetry of first cross price derivatives of the output supply and input demand functions. 5. NUMERIC EXAMPLE 5.1. Production function. Consider the following production function. y fx 1, x 4x 1 14x x 1 x 1 x x The first and second partial derivatives are given by 38 The Hessian is fx 1, x x 1 4 x 1 x fx 1, x x 14 x 1 x fx 1, x x 1 fx 1, x x 1 x 1 fx 1, x x fx 1, x 1 1 The determinant of the Hessian is given by 1 1 4 1 3 41 The rate of technical substitution is given by RT S x y, x 1 4 x 1 x 4 x 1 14 x 1 x The elasticity substitution is given by 39 40

10 PROFIT FUNCTIONS σ 1 f 1 f x 1 f 1 x f x 1 x f 11 f f 1 f 1 f f f 1 14 x 1 x 4 x 1 x x 1 1x 1 7x x 1 x x 43 x 1 x 1108 3x 1 7x 1 4x 3x 1 x 3x 5.. Profit maximization. Profit is given by π p fx 1, x w 1 x 1 w x p [ 4x 1 14x x 1 x 1 x x ] 44 w1 x 1 w x We maximize profit by taking the derivatives of 44 setting them equal to zero and solving for x 1 and x. π x 1 p [ 4 x 1 x ] w 1 0 π x p [ 14 x 1 x ] w 0 45 Rearranging 45 we obtain 4 x 1 x w 1 p 14 x 1 x w p 46a 46b Now solve equation 46a for x 1 as follows x 1 4p w 1 px p 47 Then substitute x 1 from equation 47 into equation 46b as follows

PROFIT FUNCTIONS 11 4p w1 px 14 x w p p 8p 4p w1 px 4px w p p 5p w1 x w p p x w 5p p p w1 p w 5p w 1 p x w 5p w 1 x w 5p w 1 5p w 1 w 48 If we substitute the last expression in equation 48 for x in equation 47 we obtain x 1 4p w 1 px p 4p w 1 p 4p w 1 p p p x 1 5p w1 w 7p 3w 1 5p w 1 w 6p 49 14p 4w 1 w 6p 6p w 1 w 5.3. Necessary and sufficient conditions for a maximum. Consider the Hessian of the profit equation. 1 p p πx 1, x p 1 p p For a maximum we need the diagonal elements to be negative and the determinant to be positive. The diagonal elements are negative. The determinant of the Hessian is 4p -p, which is positive. 50

1 PROFIT FUNCTIONS 5.4. Optimal output. We can obtain optimal output as a function of p, w 1 and w by substituting the optimal values of x 1 and x into the production function. yp, w 1, w fx 1 p, w, w, x p, w 1, w 6p w1 w 5p w1 w 4 14 6p w1 w 6p w1 w 5p w1 w 5p w1 w 51 Now put everything over a common denominator and multiply out as follows yp, w 1, w 7p 6p w 1 w 9p 4p 5p w 1 w 9p 6p w 1 w 9p 6p w 1 w 5p w 1 w 9p 5p w 1 w 9p 4464p 144pw 1 7pw 9p 184p 4pw 1 84pw 9p 3844p 48pw 1 4w 1 14pw 4w 1 w w 9p 34p 166pw 1 w 1 176pw 5w 1 w w 9p 704p 104pw1 w 1 08pw 4w 1 w 4w 9p 5a 5b 5c 5d 5e Notice that we have terms in the following p, pw 1 pw, w 1, w 1 w and w. Now rearrange terms in equation 5 combining like terms.

PROFIT FUNCTIONS 13 yp, w 1, w 4464p 184p 3844p 34p 704p 9p 144pw 1 4pw 1 48pw 1 166pw 1 104pw 1 9p 7pw 84pw 14pw 176pw 08pw 9p 4w 1 w 1 w 1 9p 4w 1w 5w 1 w 4w 1 w 9p w w 4w 9p 53a 53b 53c 53d 53e 53f Now combine terms yp, w 1, w 334p 9p 0pw 1 9p 0pw 9p 3w 1 9p 3w 1w 9p 3w 9p 1108p w 1 w 1w w 1108 3 w 1 w 1w w 54 5.5. The example numerical profit function. We obtain the profit function by substituting the optimal values of x 1 and x into the profit equation πp, w 1, w pfx 1 p, w, w, x p, w 1, w w 1 x 1 p, w, w w x p, w, w 1108p p w 1 w 1w w 6p w1 w 5p w1 w w 1 w 55 1108p w 1 w 1 w w 6pw 1 w 1 w 1 w 5pw w 1 w w 1108p w 1 w w 1 w 6pw 1 5pw 5.6. Optimal input demands via Hotelling s lemma. We can find the optimal input demands by taking the derivative of equation 86 with respect to w 1 and w. First with respect to w 1

14 PROFIT FUNCTIONS πp, w 1, w 1108p w 1 w w 1 w 6pw 1 5pw πp, w 1, w w 1 w 1 w 6p x 1 p, w 1, w w 1 w 6p 6p w 1 w 56 Then with respect to w πp, w 1, w w w w 1 5p x p, w 1, w w w 1 5p 5p w 1 w 57 5.7. Optimal output via Hotelling s lemma. Take the derivative of the profit function with respect to p using the quotient rule πp, w 1, w 1108p w 1 w w 1 w 6pw 1 5pw πp, w 1, w p 16p 6w 1 5w 3 1108p w 1 w w 1 w 6pw 1 5pw 9p 6648p 186pw 1 156pw 334p 3w 1 3w 3w 1 w 186pw 1 156pw 9p 334p 3w 1 3w 3w 1 w 9p yp, w 1, w 1108p w 1 w w 1 w 58 6. ALGEBRAIC EXAMPLE 6.1. Production function. Consider the following production function. y fx 1, x α 1 x 1 α x β 11 x 1 β 1 x 1 x β x 59 The first and second partial derivatives are given by

PROFIT FUNCTIONS 15 The Hessian is fx 1, x x 1 α 1 β 11 x 1 β 1 x fx 1, x x α β 1 x 1 β x fx 1, x x 1 β 11 fx 1, x x 1 x β 1 fx 1, x x β fx 1, x The determinant of the Hessian is given by The rate of technical substitution is given by β11 β 1 β 1 β 60 61 β 11 β 1 β 1 β 11 4β 11β β1 6 RT S x y, x 1 α 1 β 11 x 1 β 1 x 63 x 1 α β 1 x 1 β x The elasticity substitution is given by σ 1 f 1 f x 1 f 1 x f x 1 x f 11 f f 1 f 1 f f f 1 α 1 x 1 β 11 x β 1 α x 1 β 1 x β x 1 β 11 x 1 α 1 x β 1 x α x β x 1 x α 1 x 1 β 11 x β 1 β β 1 α 1 x 1 β 11 x β 1 α x 1 β 1 x β β 11 α x 1 β 1 x β 64 6.. Profit maximization. Profit is given by π p fx 1, x w 1 x 1 w x p [ α 1 x 1 α x β 11 x 1 β 1 x 1 x β x ] 65 w1 x 1 w x We maximize profit by taking the derivatives of 65 setting them equal to zero and solving for x 1 and x. π x 1 p [ α 1 β 11 x 1 β 1 x ] w 1 0 π x p [ α β 1 x 1 β x ] w 0 Rearranging 66 we obtain 66

16 PROFIT FUNCTIONS Now solve equation 67a for x 1 as follows α 1 β 11 x 1 β 1 x w 1 p α β 1 x 1 β x w p 67a 67b x 1 w 1 pα 1 px β 1 pβ 11 68 Then substitute x 1 from equation 68 into equation 67b as follows w1 pα 1 px β 1 α β 1 pβ 11 β x w p pα β 11 w 1 β 1 pα 1 β 1 px β1 4px β 11 β w pβ 11 p pα β 11 w 1 β 1 pα 1 β 1 4px β 11 β px β1 w pβ 11 pβ 11 p 4β11 β β1 pα β 11 w 1 β 1 pα 1 β 1 pβ 11 x Now isolate x on the left hand side of equation 69 as follows. x 4β11 β β1 w β 11 p pα β 11 w 1 β 1 pα 1 β 1 pβ 11 x w β 11 p β 11 w p 4β 11 β β1 β 11 pα β 11 w 1 β 1 pα 1 β 1 pβ 11 4β 11 β β1 β 11 w p 4β 11 β β1 pα β 11 w 1 β 1 pα 1 β 1 p 4β 11 β β1 69 70 β 11w pα β 11 w 1 β 1 pα 1 β 1 p 4β 11 β β 1 β 11w w 1 β 1 p α 1 β 1 α β 11 p 4β 11 β β 1 If we substitute the last expression in equation 70 for x in equation 68 we obtain x 1 w 1 pα 1 px β 1 pβ 11 w 1 pα 1 pβ 11 pβ 1 pβ 11 x w 1 pα 1 pβ 11 pβ 1 pβ 11 β11 w pα β 11 w 1 β 1 pα 1 β 1 p 4β 11 β β 1 Now put both terms in equation 71 over a common denominator and simplify 71

PROFIT FUNCTIONS 17 x 1 p 4β 11 β β1 w1 pα 1 pβ 1 β 11 w pα β 11 w 1 β 1 pα 1 β 1 pβ 11 p 4β 11 β β1 4β11 β β1 w1 pα 1 β 1 β 11 w pα β 11 w 1 β 1 pα 1 β 1 pβ 11 4β 11 β β 1 4w 1β 11 β w 1 β 1 4pα 1 β 11 β pα 1 β 1 β 11 β 1 w pα β 11 β 1 w 1 β 1 pα 1 β 1 pβ 11 4β 11 β β 1 4w 1β 11 β 4pα 1 β 11 β β 11 β 1 w pα β 11 β 1 pβ 11 4β 11 β β 1 w 1β pα 1 β β 1 w pα β 1 p 4β 11 β β 1 w 1β β 1 w p α β 1 α 1 β p 4β 11 β β 1 7 6.3. Necessary and sufficient conditions for a maximum. Consider the Hessian of the profit equation. β11 β 1 pβ11 pβ 1 πx 1, x p β 1 β pβ 1 pβ 73 For a maximum we need the diagonal elements to be negative and the determinant to be positive. The diagonal elements will negative if β 11 and β are negative. The determinant of the Hessian is given by pβ 11 pβ 1 pβ 1 pβ 4p β 11 β p β1 p 4β 11 β β1 74 So the solution will be a maximum if β 11 and β are both less than zero and 4β 11 β > β1. Notice that if 4β 11 β β1, the test fails. 6.4. Optimal output. We can obtain optimal output as a function of p, w 1 and w by substituting the optimal values of x 1 and x into the production function.

18 PROFIT FUNCTIONS yp, w 1, w fx 1 p, w, w, x p, w 1, w w1 β pα 1 β β 1 w pα β 1 α 1 p 4β 11 β β1 β11 w pα β 11 w 1 β 1 pα 1 β 1 α p 4β 11 β β1 w1 β pα 1 β β 1 w pα β 1 β 11 p 4β 11 β β1 w1 β pα 1 β β 1 w pα β 1 β11 w pα β 11 w 1 β 1 pα 1 β 1 β 1 p 4β 11 β β1 p 4β 11 β β1 β11 w pα β 11 w 1 β 1 pα 1 β 1 β p 4β 11 β β1 75 We can simplify equation 75 by simplifying the squared and cross product terms. First consider x 1. x 1 w 1 β pα 1 β β 1 w pα β 1 p 4β 11 β β 1 w1 β pα 1 β β 1 w pα β 1 p 4β 11 β β1 4w1β 4pw 1 α 1 β w 1 w β 1 β pw 1 α β 1 β p 4β 11 β β1 4pw 1 α 1 β 4p α1β pw α 1 β 1 β p α 1 α β 1 β p 4β 11 β β1 w 1 w β 1 β pα 1 w β 1 β wβ 1 pw α β1 p 4β 11 β β1 pw 1 α β 1 β p α 1 α β 1 β pw α β1 p αβ 1 p 4β 11 β β 1 76 We can simplify equation 76 as follows

PROFIT FUNCTIONS 19 x 1 4w1β 8pw 1 α 1 β 4w 1 w β 1 β 4pw 1 α β 1 β p 4β 11 β β1 4p α1β 4pw α 1 β 1 β 4p α 1 α β 1 β p 4β 11 β β1 wβ 1 pw α β1 p 4β 11 β β1 p αβ 1 p 4β 11 β β 1 77 Simplifying again we obtain x 1 4p α1β 4p α 1 α β 1 β p αβ 1 p 4β 11 β β1 4pw 1 α β 1 β 8pw 1 α 1 β 4pw α 1 β 1 β pα w β1 p 4β 11 β β1 4w1β 4w 1 w β 1 β wβ 1 p 4β 11 β β 1 p 4α1β 4α 1 α β 1 β αβ1 p 4β 11 β β1 4pw1 α β 1 β α 1 β p 4β 11 β β1 pw α1 β 1 β α β1 p 4β 11 β β1 4w1β 4w 1 w β 1 β wβ 1 p 4β 11 β β 1 78 Similarly for x

0 PROFIT FUNCTIONS x 4wβ 11 8pw α β11 4p αβ 11 4w 1 w β 11 β 1 p 4β 11 β β1 4pα 1 w β 11 β 1 4pw 1 α β 11 β 1 4p α 1 α β 11 β 1 p 4β 11 β β1 w1β 1 pw 1 α 1 β1 p 4β 11 β β1 p α1β 1 p 4β 11 β β 1 p 4αβ 11 4α 1 α β 11 β 1 α1β1 p 4β 11 β β1 pw1 α β 11 β 1 α 1 β1 p 4β 11 β β1 4pw α1 β 11 β 1 α β11 p 4β 11 β β11 w1β 1 4w 1 w β 11 β 1 4wβ 11 p 4β 11 β β 1 79 And for x 1 x w1 β β 1 w p α β 1 α 1 β β11 w w 1 β 1 p α 1 β 1 α β 11 x 1 x p 4β 11 β β1 p 4β 11 β β1 4w 1 w β 11 β w 1β 1 β pw 1 β α 1 β 1 α β 11 p 4β 11 β β1 w β 11 β 1 w 1 w β1 pw β 1 α 1 β 1 α β 11 p 4β 11 β β1 pw β 11 α β 1 α 1 β pw 1 β 1 α β 1 α 1 β p α 1 β 1 α β 11 α β 1 α 1 β p 4β 11 β β1 80 Now collect terms in equation 80

PROFIT FUNCTIONS 1 p α 1 β 1 α β 11 α β 1 α 1 β pw 1 β α 1 β 1 α β 11 pw β 1 α 1 β 1 α β 11 x 1 x p 4β 11 β β1 pw β 11 α β 1 α 1 β pw 1 β 1 α β 1 α 1 β p 4β 11 β β1 w 1β 1 β w 1 w β1 4w 1 w β 11 β wβ 11 β 1 p 4β 11 β β1 p α 1 β 1 α β 11 α β 1 α 1 β p 4β 11 β β1 pw 1 β α 1 β 1 α β 11 β 1 α β 1 α 1 β p 4β 11 β β1 pw β 11 α β 1 α 1 β β 1 α 1 β 1 α 1 β 11 p 4β 11 β β1 w 1 β 1 β w 1 w 4β11 β β1 w β 11 β 1 p 4β 11 β β1 Now substitute the results from equations 70, 7 78, 79 and 81 into equation 75 81

PROFIT FUNCTIONS yp, w 1, w fx 1 p, w, w, x p, w 1, w w 1 β β 1 w p α β 1 α 1 β α 1 p 4β 11 β β1 β 11 w w 1 β 1 p α 1 β 1 α β 11 α p 4β 11 β β1 p 4α 1 β β 4α 1α β 1 β α β 11 1 4pw1 α β 1 β α 1 β p 4β 11 β β1!!!! pw α 1 β 1 β α β β 11 1 4w 1 β 4w 1w β 1 β w β 1 p 4β 11 β β1 p α 1 β 1 α β 11 α β 1 α 1 β β 1 p 4β 11 β β1 pw 1 β α 1 β 1 α β 11 β 1 α β 1 α 1 β p 4β 11 β β1! pw β 11 α β 1 α 1 β β 1 α β 1 β 1 α 1 β 11 1 p 4β 11 β β w 1β1β w 1 w 4β 11 β β1 1 p 4β 11 β β1 p 4α β β 11 4α 1α β 11 β 1 α 1 β 1 p 4β 11 β β pw1 α β 11 β 1 α 1 β1 1 p 4β 11 β β1 4pw α 1 β 11 β 1 α β11 β p 4β 11 β β11 w 1 β 1 4w 1w β 11 β 1 4w β 11 p 4β 11 β β1!! To start simplifying, we need to put everything over a common denominator! w β 11 β 1 8

PROFIT FUNCTIONS 3 yp, w 1, w fx 1 p, w, w, x p, w 1, w! pw 1α 1 β 4β 11 β β1 pw α 1 β 1 4β 11 β β1 p α 1 4β 11 β β1 α β 1 α 1 β p 4β 11 β β1 pw α β 11 4β 11 β β1 pw1 α β 1 4β 11 β β1 p α 4β 11 β β1 α1 β 1 α β 11 p 4β 11 β β1 p β 11 4α 1 β 4α 1α β 1 β α 1 β 4pw1 β 11 α β 1 β α 1 β p 4β 11 β β1!!! pw β 11 α 1 β 1 β α β1 4w 1 β 11 β 4w 1w β 11 β 1 β w β 11β1 p 4β 11 β β1 p β 1 α 1 β 1 α β 11 α β 1 α 1 β p 4β 11 β β 1! pw β 1 β 11 α β 1 α 1 β β 1 α 1 β 1 α 1 β 11 p 4β 11 β β1! w 1 β 1 β w 1 w β 1 4β 11 β β1 w β 11 β1 p 4β 11 β β1 pw 1β 1 β α 1 β 1 α β 11 β 1 α β 1 α 1 β p 4β 11 β β1! p β 4α β 11 4α 1α β 11 β 1 α 1 β 1 p 4β 11 β β pw1β α β 11 β 1 α 1 β1 1 p 4β 11 β β1! 4pw β α 1 β 11 β 1 α β11 p 4β 11 β β11 w 1 β 1 β 4w 1 w β 11 β 1 β 4w β 11 β p 4β 11 β β1 83 Notice that we have terms in the following p, pw 1 pw, w 1, w 1 w and w. Now rearrange terms in equation 83 combining like terms. After some manipulations we obtain! yp, w 1, w w β 11 w 1 w β 1 w 1β p α β 11 α 1 α β 1 α 1β p 4β 11 β β 1 84 6.5. The example profit function. We obtain the profit function by substituting the optimal values of x 1 and x into the profit equation πp, w 1, w pfx 1 p, w, w, x p, w 1, w w 1 x 1 p, w, w w x p, w, w w β 11 w 1 w β 1 w1β p αβ 11 α 1 α β 1 α1β p p 4β 11 β β1 w1 β β 1 w p α β 1 α 1 β β11 w w 1 β 1 p α 1 β 1 α β 11 w 1 p 4β 11 β β1 w p 4β 11 β β1 85 We can simplify equation 85 as follows

4 PROFIT FUNCTIONS w β 11 w 1 w β 1 w1β p αβ 11 α 1 α β 1 α1β πp, w 1, w p 4β 11 β β1 w1 β β 1 w p α β 1 α 1 β β11 w w 1 β 1 p α 1 β 1 α β 11 w 1 p 4β 11 β β1 w p 4β 11 β β1 w β 11 w 1 w β 1 w1β p αβ 11 α 1 α β 1 α1β p 4β 11 β β1 w 1 β w 1 w β 1 pw 1 α β 1 α 1 β p 4β 11 β β1 w β 11 w 1 w β 1 pw α 1 β 1 α β 11 p 4β 11 β β1 w β 11 w 1 w β 1 w1β p αβ 11 α 1 α β 1 α1β p 4β 11 β β1 pw1 α β 1 α 1 β p 4β 11 β β1 pw α 1 β 1 α β 11 p 4β 11 β β1 We can write equation 86 in the following useful fashion πp, w 1, w w β 11 w 1 w β 1 w1 β pw 1 α β 1 α 1 β pw α 1 β 1 α β 11 p α β 11 α 1 α β 1 α 1 β p 4β 11 β β1 87 6.6. Optimal input demands. We can find the optimal input demands by taking the derivative of equation 86 with respect to w 1 and w. First with respect to w 1 Then with respect to w πp, w 1, w β1 w β w 1 p α β 1 α 1 β w 1 p 4β 11 β β1 88 β1 w β w 1 p α β 1 α 1 β x 1 p 4β 11 β β1 πp, w 1, w w β 11 w 1 β 1 p α 1 β 1 α β 11 w p 4β 11 β β1 89 w β 11 w 1 β 1 p α 1 β 1 α β 11 x p 4β 11 β β1 6.7. Optimal output. Take the derivative of equation 87 with respect to p 86!

PROFIT FUNCTIONS 5 πp, w 1, w πp, w 1, w p 0 1 @ w β 11 w 1 w β 1 w1 β pw 1 α β 1 α 1 β pw α 1 β 1 α β 11 p α β 11 α 1 α β 1 α 1 β A p 4β 11 β β 1 p 4β 11 β β1 w 1 α β 1 α 1 β w α 1 β 1 α β 11 p α β 11 α 1 α β 1 α 1 β p 4β 11 β β 1 w β 11 w 1 w β 1 w 1 β pw 1 α β 1 α 1 β pw α 1 β 1 α β 11 p α β 11 α 1 α β 1 α 1 β p 4β 11 β β 1 pw 1 α β 1 α 1 β pw α 1 β 1 α β 11 p α β 11 α 1 α β 1 α 1 β p 4β 11 β β 1 4β 11 β β 1 w β 11 w 1 w β 1 w1 β pw 1 α β 1 α 1 β pw α 1 β 1 α β 11 p α β 11 α 1 α β 1 α 1 β p 4β 11 β β 1 p α β 11 α 1 α β 1 α 1 β w β 11 w 1 w β 1 w1 β p α β 11 α 1 α β 1 α 1 β p 4β 11 β β 1 p 4β 11 β β 1 w β 11 w 1 w β 1 w 1 β w β 11 w 1 w β 1 w1 β p α β 11 α 1 α β 1 α 1 β p 4β 11 β β1 4β 11 β β 1 90 4β 11 β β 1 4β 11 β β 1 7. SINGULARITY OF THE HESSIAN MATRIX We can show that the Hessian matrix of the profit function is singular. This is a direct implication of linear homogeneity. To show this first write the identity πt p, tw t πp, w, where t is a scaler greater than zero 91 Differentiate equation 91 with respect to t. This will yield m j1 π t p j p j i1 π t w i w i πp, w 9 Differentiate equation 9 with respect to t again. Each term in equation 9 will yield a sum of derivatives so the result will be a double sum as follows m m j1 k1 π t p k t p j p k p j i1 m k1 m j1 l1 π t p k t w i p k w i π t w l t p j w l p j i1 l1 We can factor t out of each terms in equation 93 because π t w l t w i w l w i 0 93 This will give fx t x 1 t fx x 94

6 PROFIT FUNCTIONS 1 t m m j1 k1 1 t n This then implies that i1 π p k p j p k p j 1 t m k1 m j1 π p k w i p k w i 1 t l1 π w l p j w l p j i1 l1 π w l w i w l w i 0 95 m m j1 k1 i1 π p k p j p k p j m k1 m j1 π p k w i p k w i l1 i1 π w l p j w l p j l1 π w l w i w l w i 0 The last expression in equation 96 is actually a quadratic form involving the Hessian of πp,w. To see this write out the Hessian and then pre and post multiply by a vector containing output and input prices. p1 p... p m w 1... w n 6 4 π p 1 π p 1 p. π p 1 p m π p 1 w 1. π p 1 w n π p p 1... π p. π... π p m p 1 π p m p π w 1 p 1... π w 1 p............... π p m π p m w 1 π p p m... w 1 p m... π p w 1... π... w1............. π p w n... π p m w n π w 1 w n... π w n p 1 π w n p π w n p m π w n w 1 97 Equation 96 implies that this quadratic form is zero for this particular vector p,w. This means that the Hessian matrix is singular. The profit function is convex so we know that its Hessian is positive semi-definite. This implies that it is positive semidefinite only and cannot be positive definite. 8. SENSITIVITY ANALYSIS FOR THE PROFIT MAXIMIZATION PROBLEM The Hessian of the profit function is positive semidefinite so all its diagonal elements are nonnegative Hadley [9, p. 117]. 8.1. Response of Output Supply. Consider first the response of any output to its own price. y j p j π p j π w n 3 7 5 6 4 96 p 1 p. p n w 1 w. w n 0 98 Supply curves derived from profit maximization will always slope upwards because this derivative is on the diagonal of the Hessian. Now consider the response of an output to another product price. y j p k π p k p j 99 3 7 5

PROFIT FUNCTIONS 7 This element is not on the diagonal of the Hessian of the profit function and so we cannot determine its sign. Similarly we cannot sign the response of an output to an input price. y j π 100 w i w i p j 8.. Response of Input Supply. Consider first the response of any input to its own price. x i w i π w i 0 101 Demand curves derived from profit maximization will always slope downwards because this derivative is on the diagonal of the Hessian and input demand is the negative of the derivative of the profit function with respect to input price. Now consider the response of an input to another input price. x i π 10 w l w l w i This element is not on the diagonal of the Hessian of the profit function and so we cannot determine its sign. Similarly we cannot sign the response of an input to an output price. x i π 103 p k p k w i 8.3. Homogeneity. Because output supply and input demand are derivatives of a function which is homogeneous of degree one, they are homogeneous of degree zero. y j λ p, λ w y j p, wx i λ p, λ w x i p, w 104 8.4. Symmetry of response. If πp, w is twice differentiable then Young s theorem on second cross partial derivatives implies π π p i p j p j p i This then implies π π p k w i w i p k π π w i w j w j w i 105 y j p i w i p k x i w j y i p j y k w i x j w i 106

8 PROFIT FUNCTIONS The response of any output to a different output price is symmetric, i.e. cross price derivatives are equal. Similarly with input demand response. The way an output responds to a particular input price is the same as the response of that input to the output s price. 8.5. Own and cross price response. Because any principal submatrix of a positive semi-definite matrix is also positive semidefinite Hadley [9, p. 6], we also have the following results. x i w i x j w j x i w j x j w i 107 To see this note that if the submatrix above is positive semi-definite then the determinant must be non-negative which means that the product of the diagonal elements - the product of the off diagonal elements must be greater than or equal to zero. We also have y j p j y k p k y j p k y k p j 108 The product of own price responses is larger than the product in cross price responses. We also have y j p j x k w k y j w k x k p j 109 9. SUBSTITUTES AND COMPLEMENTS 9.1. Gross substitutes. We say that inputs are gross substitutes if x i p, w w j x j p, w w i 0 110 9.. Gross complements. We say that inputs are gross complements if x i p, w w j x j p, w w i 0 111 10. RECOVERING PRODUCTION RELATIONSHIPS FROM THE PROFIT FUNCTION 10.1. Marginal Products. We can find marginal products from the first order conditions for profit maximization. The easiest way to see this is when technology is represented by an asymmetric transformation function. π max x ỹ [p 1 fỹ, x m 1 p j y j w i x i ] i1 11a max x ỹ [p 1 fỹ, x p ỹ w x] where p p, p 3,..., p m, ỹ y, y 3,..., y m, w w 1, w,..., w n x x 1, x,..., p n If fỹ,x is differentiable, then the first order conditions for maximizing profit are as follows.

PROFIT FUNCTIONS 9 π f ỹ, x p 1 w i 0, i 1,,..., n x i x i π f ỹ, x p 1 p j 0, j, 3,..., m y j y j We can then obtain marginal products by rearranging the first order conditions. We can find the impact on y 1 of an increase in any of the inputs as 113 f ỹ, x w i, i 1,,..., n 114 x i p 1 We can find the impact on y 1 of an increase in any of the other outputs as f ỹ, x y j p j p 1, j, 3,..., m 115 We can find other marginal products by solving the profit maximization problem with a different normalization than the one on y 1. In the case of a single output we obtain f x w i, i 1,,..., n 116 x i p This is clear from figure. The isoprofit line is given by Solving for y we obtain π py wx y π p w p x FIGURE. Marginal Product is Equal to Price Ratio y f x x

30 PROFIT FUNCTIONS 10.. Output elasticity. We normally think of output elasticity in terms of one output and many inputs so define it as follows. If we substitute for fx x i ɛ i fx x i x i y from equation 116 we obtain ɛ i w i p 10.3. Elasticity of scale. Elasticity of scale is given by the sum of the output elasticities,i.e., x i y 117 118 ɛ i1 fp, w x i x i y i1 w i p x i y 119 n i1 w i x i p y Cost Revenue A profit maximizing firm will never produce in a region of the production function with increasing returns to scale assuming the technology eventually becomes locally concave and remains concave. As the result for a profit maximizing firm, returns to scale is just the ratio of cost to revenue. 11. DUALITY BETWEEN THE PROFIT FUNCTION AND THE TECHNOLOGY We can show that if we have an arbitrarily specified profit function that satisfies the conditions in section 3, we can construct from it a reasonable technology that satisfies the conditions we specified in section 1..4 and that would generate the specified profit function. A general discussion of this duality is contained in Diewert [] 11.1. Constructing a technology from the profit function. Let T* be defined as T {x, y : py wx πp, w, p, w 0, 0} 10 To see the intuition of why we can construct T this way consider the case of one input and one output as in figure 3 If we pick a particular set of prices then T consists of all points below the line that is tangent to Tx,y at the optimal input output combination. The equation {x, y : py wx πp, w for a particular set of prices defines a line in R or a hyperplane in R nm. Points below the line are considered to be in T. Now pick a different set of prices with w lower and construct a different line as in figure 4. The number of points in T is now less than before. If we add a third set of prices with w higher than in the original case as in figure 5, we can reduce the size of T even more. Continuing in this fashion we can recover Tx,y. This is an application of what is often called Minkowski s theorem. To show that T is in fact equal to T, we need to show that T T. We do this by assuming that some particular input output combination x 0, y 0 T but x 0, y 0 / T. If x 0, y 0 / T, then x 0,

PROFIT FUNCTIONS 31 FIGURE 3. Half-Spaces and Technology Set y T x,y x FIGURE 4. Two Half-Spaces and Technology Set y T x,y x y 0 can be separated from T by a hyperplane py 0 - wx 0. But T consists of points lying below the hyperplane defined by the maximal profits with prices p and w. This means that the point x 0, y 0 cannot be in T. We can show this in figure 6. Consider point a in figure 6 which is not in Tx,y. It is in T if we only consider the half spaces defined by the hyperplanes intersecting Tx,y at points c and d. But at the prices defined by a hyperplane with slope as the one passing through b, point a is not in T. If we consider any other point not in T, there exists a hyperplane that also excludes it from T. This implies than that T T and so T T. For more on this topic consult McFadden [1] or Fare and Primont [6]. 11.. Using the properties of the profit function to recover properties of the technology. We repeat here for convenience the properties of Tx,y and πp,w. 11..1. Properties of the technology.

3 PROFIT FUNCTIONS FIGURE 5. Three Half-Spaces and Technology Set y T x,y x FIGURE 6. Separating Hyperplanes and the Technology Set y c d a b T x,y x T.1a: Inaction and No Free Lunch. 0,y ɛ T x ɛ R n and y ɛ R m. This implies that Tx,y is a non-empty subset of R mn T.1b: 0,y T, y 0, y 0. T.: Input Disposability. If x,y ɛ T and θ 1 then θx, y ɛ T. T..S: Strong Input Disposability. If x,y ɛ T and x x, then x, y ɛ T. T.3: Output Disposability. x,y ɛ R nm,if x, y ɛ T and 0 < λ 1 then x, λy ɛ T. T.3.S: Strong Output Disposability. If x,y ɛ T and y y, then x, y ɛ T. T.4: Boundedness. For every finite input vector x 0, the set y ɛ Px is bounded from above. This implies that only finite amounts of output can be produced from finite amounts of inputs. T.5: Tx is a closed set. The assumption that Px and Vy are closed does not imply that T is a closed set, so it is assumed. Specifically, if [x l x 0, y l y 0 and x l, y l ɛ T, l] then x 0 y 0 ɛ T.

PROFIT FUNCTIONS 33 T.9: T is a convex set. This is not implied by the convexity of Px and Vy. Specifically, a technology could exhibit a diminishing rate of technical substitution, a diminishing rate of product transformation and still exhibit increasing returns to scale. 11... Properties of the profit function. π.1: πp, w is an extended real valued function it can take on the value of for finite prices defined for all p, w 0 m, 0 n and πp, w pa - wb for a fixed vector a, b 0 m, 0 n. This implies that πp, w 0 if 0 m, 0 n Tx,y,which we normally assume. π.: π is nonincreasing in w π.3: π is nondecreasing in p π.4: π is a convex function π.5: π is homogeneous of degree 1 in p and w. We will consider only a few of the properties of Tx,y 11..3. Tx,y is non-empty. By π.1, πp, w pa - wb for a fixed vector a, b 0 m, 0 n or πp, w 0 if 0 m, 0 n Tx,y. Given that πp, w 0, with x zero, there are obviously values of y for example 0 which make py - wx less than πp, w. So T x,y is not empty and 0,y T x,y. 11..4. Strong Input Disposability. If x,y ɛ T and x x, then x, y ɛ T.. Suppose that x, y ɛ T, 0 x x and p, w 0. Then py - wx py - wx πp, w p, w 0. Thus y, x ɛ T. 11..5. Strong Output Disposability. If x,y ɛ T and y y, then x, y ɛ T.. Suppose y, x ɛ T and 0 y y then py - wx py - wx πp, w p, w 0. Thus y, x ɛ T. 11..6. Tx is a closed and convex set. From the definition, T* is the intersection of a family of closed half-spaces because we have a hyperplane in nm space dividing the space. Thus T is a closed convex set Rockafellar [13, p. 10].

34 PROFIT FUNCTIONS REFERENCES [1] Chambers, R. G. Applied Production Analysis. Cambridge: Cambridge University Press, 1988. [] Diewert, W. E. Functional Forms for Profit and Transformation Functions. Journal of Economic Theory.61973:84-316. [3] Diewert, W. E. Applications of Duality Theory in Frontiers of Quantitative Economics, Vol., ed. M. Intriligator and D. Kendrick. Amsterdam: North Holland, 1974. [4] Diewert, W.E. Duality Approaches to Microeconomic Theory in Handbook of Mathematical Economics, Vol, ed. K.J. Arrow and M.D. Intriligator. Amsterdam: North Holland, 198 [5] Fare, R. Fundamentals of Production Theory. New York: Springer-Verlag, 1988. [6] Fare, R. and D. Primont. Multi-output Production and Duality: Theory and Applications. Boston: Kluwer Academic Publishers, 1995. [7] Ferguson, C. E. The Neoclassical Theory of Production and Distribution. Cambridge: Cambridge University Press, 1971. [8] Fuss, M. and D. McFadden. Production Economics: A Dual Approach to Theory and Application. Amsterdam: North Holland, 1978. [9] Hadley, G. Linear Algebra. Reading, MA: Addison-Wesley, 1961. [10] Lau, L. J. Characterization of the Normalized Restricted Profit Function Journal of Economic Theory1Feb. 1976:131-163. [11] Lau, L. J. Applications of Profit Functions in Production Economics: A Dual Approach to Theory and Applications, ed. M. Fuss and D. McFadden, Amsterdam: North-Holland, 1978. [1] McFadden, D. Duality of Production, Cost and Profit Functions in Production Economics: A Dual Approach to Theory and Applications, ed. M. Fuss and D. McFadden, Amsterdam: North-Holland, 1978. [13] Rockafellar, R. T. Convex Analysis. Princeton New Jersey: Princeton University Press, 1970. [14] Shephard, R. W. Theory of Cost and Production Functions. Princeton New Jersey: Princeton University Press, 1970. [15] Varian, H. R. Microeconomic Analysis. 3 rd Edition. New York: Norton, 199.