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1. ALTERNATIVE REPRESENTATIONS OF TECHNOLOGY
The technology that is available to a firm can be represented in a variety of ways. The most general are those based on correspondences and sets.
1.1. Technology Sets. The technology set for a given production process is de- fined as
T = {(x, y) : x ∈ Rn+, y ∈ Rm :+ 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.
1.2. The Output Correspondence and the Output Set.
1.2.1. Definitions. It is often convenient to define a production correspondence and the associated output set.
1: The output correspondence P, maps inputs x ∈ Rn + into subsets of outputs, i.e., P: Rn + → 2 R
m +
. 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. 2: The output set for a given technology, P(x), is the set of all output vectors y ∈ Rm + that are obtainable from the input vector x ∈ Rn +. P(x) is then the set of all output vectors y ∈ Rm + that are obtainable from the input vector x ∈ Rn +. We often write P(x) for both the set based on a particular value of x, and the rule (correspondence) that assigns a set to each vector x.
1.2.2. Relationship between P(x) and T(x,y).
P (x) = (y : (x, y ) ∈ T )
1.2.3. Properties of P(x).
P.1a: Inaction and No Free Lunch. 0 ∈ P(x) ∀ x ∈ Rn +. P.1b: y 6 ∈ P(0), y > 0. P.2: Input Disposability. ∀ x ∈ Rn + , P(x) ⊆ P(θx), θ ≥ 1. P.2.S: Strong Input Disposability. ∀ x, x’ ∈ Rn + , x’ ≥ x ⇒ P(x) ⊆ P(x’). P.3: Output Disposability. ∀ x ∈ Rn + , y ∈ P(x) and 0 ≤ λ ≤ 1 ⇒ λy ∈ P(x). P.3.S: Strong Output Disposability. ∀ x ∈ Rn + , y ∈ P(x) ⇒ y’ ∈ P(x), 0 ≤ y’ ≤ y. P.4: Boundedness. P(x) is bounded for all x ∈ Rn +. P.5: T is a closed set P: Rn + → 2 R
m + is a closed correspondence, i.e., if [x^ → x^0 , y^ → y^0 and y^ ∈ P(x), ∀ `] then y^0 ∈ P(x^0 ). P.6: Attainability. If y ∈ P(x), y ≥ 0 and x ≥ 0, then ∀ θ ≥ 0, ∃ λθ ≥ 0 such that θy ∈ P(λθ x).
Date : August 29, 2005. 1
P.7: P(x) is convex P(x) is a convex set for all x ∈ Rn +. This is equivalent to the correspon- dence V:<n + → 2 <
m
- (^) being quasiconcave. P.8: P is quasi-concave. The correspondence P is quasi-concave on Rn + which means ∀ x, x’ ∈ Rn + , 0 ≤ θ ≤ 1, P(x) ∩ P(x’) ⊆ P(θx + (1-θ)x’). This is equivalent to V(y) being a convex set. P.9: Convexity of T. P is concave on Rn + which means ∀ x, x’ ∈ Rn + , 0 ≤ θ ≤ 1, θP(x)+(1-θ)P(x’) ⊆ P(θx + (1-θ)x’)
1.3. The Input Correspondence and Input (Requirement) Set.
1.3.1. Definitions. Rather than representing a firm’s technology with the technol- ogy set T or the production set P(x), it is often convenient to define an input corre- spondence and the associated input requirement set.
1: The input correspondence maps outputs y ∈ Rm + into subsets of inputs, V: Rm + → 2 R
n +
. 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.
2: The input requirement set V(y) of a given technology is the set of all com- binations of the various inputs x ∈ Rn + that will produce at least the level of output y ∈ Rm +. V(y) is then the set of all input vectors x ∈ Rn + that will produce the output vector y ∈ Rm +. We often write V(y) for both the set based on a particular value of y, and the rule (correspondence) that assigns a set to each vector y.
1.3.2. Relationship between V(y) and T(x,y).
V (y) = (x : (x, y) ∈ T )
1.4. Relationships between Representations: V(y), P(x) and T(x,y). The technol- ogy set can be written in terms of either the input or output correspondence.
T = {(x, y) : x ∈ Rn +, y ∈ Rm + , such that x will produce y} (1a)
T = {(x, y) ∈ Rn ++ m : y ∈ P (x), x ∈ Rn +} (1b)
T = {(x, y) ∈ Rn ++ m : x ∈ V (y), y ∈ Rm + } (1c) We can summarize the relationships between the input correspondence, the output correspondence, and the production possibilities set in the following propo- sition.
Proposition 1. y ∈ P(x) ⇔ x ∈ V(y) ⇔ (x,y) ∈ T
- PRODUCTION FUNCTIONS
2.1. Definition of a Production Function. To this point we have described the firm’s technology in terms of a technology set T(x,y), the input requirement set V(y) or the output set P(x). For many purposes it is useful to represent the re- lationship between inputs and outputs using a mathematical function that maps vectors of inputs into a single measure of output. In the case where there is a single
This gives all output levels y that can be produced by the input vector x. We can show that this induced correspondence is equivalent to the output correspondence that produced f(x). We state this in a proposition. Proposition 2. Pf (x) = P (x), ∀ x ∈ Rn +. Proof. Let y ∈ Pf (x), x ∈ Rn +. By definition, y ≤ f(x). This means that y ≤ max {z: z ∈ P(x)}. Then by P.3.S, y ∈ P(x). Now show the other way. Let y ∈ P(x). By the definition of f, y ≤ max {z: z ∈ P(x)} = f(x). Thus y ∈ Pf (x). Properties P.1a, P.3, P.4 and P.5 are sufficient to yield the induced pro- duction correspondence. 2.2.1. Relationship between P(x) and f(x). We can summarize the relation- ship between P and f with the following proposition: Proposition 3. y ∈ P(x) ⇔ f(x) ≥ y, ∀ x ∈ Rn +
2.3. Examples of Production Functions.
2.3.1. Production function for corn. Consider the production technology for corn on a per acre basis. The inputs might include one acre of land and various amounts of other inputs such as tillage operations made up of tractor and implement use, labor, seed, herbicides, pesticides, fertilizer, harvesting operations made up of dif- ferent combinations of equipment use, etc. If all but the fertilizer are held fixed, we can consider a graph of the production relationship between fertilizer and corn yield. In this case the production function might be written as
y = f (land, tillage, labor, seed, fertilizer,... ) (5)
2.3.2. Cobb-Douglas production function. Consider a production function with two inputs given by y = f(x 1 , x 2 ). A Cobb-Douglas [4] [5] represention of technology has the following form.
y = Axα 1 1 xα 22
= 5 x
1 3 1 x^
1 4 2
Figure 2 is a graph of this production function.
Figure 3 shows the contours of this function.
With a single output and input, a Cobb-Douglas production function has the shape shown in figure 4.
2.3.3. Polynomial production function. We often approximate a production function using polynomials. For the case of a single input, a cubic production function would take the following form.
y = α 1 x + α 2 x^2 + α 3 x^3
= 10 x + 20 x^2 − 0. 60 x^3
The cubic production function in equation 7 is shown in figure 5.
FIGURE 2. Cobb-Douglas Production Function
x 1
x 2
f Hx 1 ,x 2 L
x 115
FIGURE 3. Contours of a Cobb-Douglas Production Function
5 10 15 20 25 30
5
10
15
20
25
30
Notice that the function first rises at an increasing rate, then increases at a de- creasing rate and then begins to fall until it reaches zero.
FIGURE 6. CES Production Function
5 10 15 20
x
5
10
15
20
x
0
25
50
75
100 f Hx 1 ,x 2 L
5 10 x1 15
FIGURE 7. CES Production Function Contours - ρ=
2.5 5 7.5 10 12.5 15 17.5 20
5
10
15
20
ln y = α 1 ln x 1 + α 2 ln x 2 + β 11 ln x^21 + β 12 ln x 1 ln x 2 + β 22 ln x^22
=
ln x 1 +
ln x 2 −
ln x^21 +
ln x 1 ln x 2 −
ln x^22
The translog can also be written with y as compared to ln y on the left hand side.
FIGURE 8. CES Production Function Contours - ρ=0.
2.5 5 7.5 10 12.5 15 17.5 20
5
10
15
20
y = A , xα 1 1 xα 2 2 eβ^11 ln, x
(^21) + β 12 ln x 1 ln x 2 + β 22 ln x (^22)
= x^11 / 3 x^12 / 10 e−.^02 ln x (^21) + 0. 1 ln x 1 ln x 2 − 0. 2 ln x 22 (10)
Figure 9 shows the translog function from equation 9 while figure 10 shows the contours of the translog function.
2.4. Properties of the Production Function. We can deduce a set of properties on f that are equivalent to the properties on P in the sense that if a particular set holds for P, it implies a particular set on f and vice versa.
2.4.1. f.1 Essentiality. f(0) = 0.
2.4.2. f.1.S Strict essentiality. f(x 1 , x 2 ,... , 0,... , xn) = 0 for all xi.
2.4.3. f.2 Monotonicity. ∀ x ∈ Rn + , f(θx) ≥ f(x), θ ≥ 1.
2.4.4. f.2.S Strict monotonicity. ∀ x, x’ ∈ Rn + , if x ≥ x’ then f(x) ≥ f(x’).
2.4.5. f.3 Upper semi-continuity. f is upper semi-continuous on Rn +.
2.4.6. f.3.S Continuity. f is continuous on Rn +.
2.4.7. f.4 Attainability. If f(x) > 0, f(λx) → +∞ as λ → +∞.
2.4.8. f.5 Quasi-concavity. f is quasi-concave on Rn +.
2.4.9. f.6 Concavity. f is concave on Rn +.
2.5. Discussion of the Properties of the Production Function.
This assumption is sometimes called essentiality. It says that with no inputs, there is no output.
2.5.2. f.1.S Strict essentiality. f(x 1 , x 2 ,... , 0,... , xn) = 0 for all xi.
This is called strict essentiality and says that some of each input is needed for a positive output. In this case the input requirement set doesn’t touch any axis. Consider as an example of strict essentiality the Cobb-Douglas function.
y = A xα 1 1 xα 2 2 (11) Another example is the Generalized Leontief Function with no linear terms
y = β 11 x 1 + 2β 12 x 1 x 2 + β 22 x 2 (12)
2.5.3. f.2 Monotonicity. ∀ x ∈ Rn + , f(θx) ≥ f(x), θ ≥ 1.
This is a monotonicity assumption that says with a scalar expansion of x, output cannot fall. There is also a strong version.
2.5.4. f.2 Strict monotonicity. ∀ x, x’ in Rn + , if x ≥ x’ then f(x) ≥ f(x’).
Increasing one input cannot lead to a decrease in output.
2.5.5. f.3 Upper semi-continuity. f is upper semi-continuous on Rn +.
The graph of the production function may have discontinuities, but at each point of discontinuity the function will be continuous from the right. The prop- erty of upper semi-continuity is a direct result of the fact that the output and input correspondences are closed. In fact, it follows directly from the input sets being closed.
2.5.6. f.3.S Continuity. f is continuous on Rn +.
We often make the assumption that f is continuous so that we can use calculus for analysis. We sometimes additionally assume the f is continuously differen- tiable.
2.5.7. f.4 Attainability. If f(x) > 0, f(λx) → +∞ as λ → +∞.
This axiom states that there is always a way to exceed any specified output rate by increasing inputs enough in a proportional fashion.
2.5.8. f.5 Quasi-concavity. f is quasi-concave on Rn +.
If a function is quasi-concave then
f(x) ≥ f(x^0 ) ⇒ f(λ x + (1 − λ) x^0 ) ≥ f(x^0 ) (13) If V(y) is convex then f(x) is quasi-concave because V(y) is an upper contour set of f. This also follows from quasiconcavity of P(x). Consider for example the traditional three stage production function in figure 11. It is not concave, but it is quasi-concave.
FIGURE 11. Quasi-concave Production Function
x
y
If the function f is quasi-concave the upper contour or isoquants are convex. This is useful in problems of cost minimization as can be seen in figure 12.
FIGURE 12. Convex Lower Boundary of Input Requirement Set
x (^1)
x (^2)
2.5.9. f.6 Concavity. f is concave on Rn +. If a function is concave then
f(λ x + (1 − λ) x′) ≥ λ f(x) + (1 − λ)f(x′^ ) (14) Concavity of f follows from P.9 (V.9) or the overall convexity of the output and input correspondences. This means the level sets are not only convex for a given level of output or input but that the overall correspondence is convex. Contrast the traditional three stage production function with a Cobb-Douglas one. Concavity is implied by the function lying above the chord as can be seen in figure 13 or below the tangent line as in figure 14.
2.6. Equivalence of Properties of P(x) and f(x). The properties (f.1 - f.6) on f(x) can be related to specific properties on P(x) and vice versa. Specifically the following proposition holds.
Now because P(x) ⊆ P(θx) for θ ≥ 1 it is clear that the maximum over the second set must be larger than the maximum over the first set.
To show the other way remember that if y ∈ P(x) then f(x) ≥ y. Now assume that f(θx) ≥ f(x) ≥ y. This implies that y ∈ P(θx) which implies that P(x) ⊆ P(θx).
2.6.3. Definition of f → P.3. Remember that P.3 states ∀ x ∈ <n + , y ∈ P(x) and 0 ≤ λ ≤ 1 ⇒ λy ∈ P(x). So consider an input vector x and an output level y such that f(x) ≥ y. Then consider a value of λ such that 0 ≤ λ ≤ 1. Given the restrction on λ, y ≥ λy. But by Proposition 3 which follows from the definition of the producion function in equation 2 and Proposition 2, λy ∈ P (x).
2.6.4. F.3 → P.4. Recall that P.4 is that P(x) is bounded for all x ∈ Rn+. Let x ∈ <n +. The set
M (x) = {u ∈ <n + : u ≤ x} (15) is compact as it is closed and bounded. F.3 says that f(x) is upper-semicontinuous, thus the maximum
f(u∗^ ) = max{f(u) : u ∈ M (x)} ≥ f(x) (16) u∗^ ∈ M (x) exists. The closed interval [0,f(x)] is a subset of the closed interval [0,f(u∗], i.e, [0, f(x)] ⊆ [0, f(u∗)]. Therefore P(x) = [0,f(x)] is bounded.
2.6.5. P.4 → F.3. Recall that f(x) is upper semi-continuous at x^0 iff lim supn → ∞ f(xn) ≤ f(x^0 ) for all sequences xn^ → x^0.
Consider a sequence {xn} → x^0. Let yn^ ≡ f(xn). Now suppose that lim supn → ∞ yn^ = ¯y ≥ f(x^0 ). Then {yn^ } → y^0 ≥ f(x^0 ) because the maximum value of the sequence {yn} is greater than f(x^0 ). Because P : <n + → 2 <+^ is a closed correspondence, y^0 ∈ [0, f(x^0 )] and y^0 ≤ f(x^0 ), a contradiction. Thus lim supn → ∞ f(xn) ≤ f(x^0 ).
2.6.6. Other equivalencies. One can show that the following equivalences also hold.
a: P.6 ⇒ f. b: P.7 follows from the definition of P(x) in terms of f in equation 4.
2.7. Marginal and Average Measures of Production.
2.7.1. Marginal product (MP). The firm is often interested in the effect of additional inputs on the level of output. For example, the field supervisor of an irrigated crop may want to know how much crop yield will rise with an additional application of water during a particular period or a district manager may want to know what will happen to total sales if she adds another salesperson and rearranges the assigned areas. For small changes in input levels this output response is measured by the marginal product of the input in question (abbreviated MP or MPP for marginal physical product). In discrete terms the marginal product of the ith input is given as
M Pi =
∆y ∆xi
y^2 − y^1 x^2 i − x^1 i
where y^2 and x^2 are the level of output and input after the change in the input level and y^1 and x^1 are the levels before the change in input use. For small changes
in xi the marginal physical product is given by the partial derivative of f(x) with respect to xi , i.e.,
M Pi =
∂f (x) ∂xi
∂y ∂xi
This is the incremental change in f(x) as xi is changed holding all other inputs levels fixed. Values of the discrete marginal product for the production function in equation 19 are contained in table 2.7.1.
y = 10 x + 20 x^2 − 0. 60 x^3 (19) For example the marginal product in going from 4 units of input to 5 units is given by
M Pi =
∆y ∆xi
The production function in equation 19 is shown in figure 15.
FIGURE 15. Cubic Production Function
10 20 30
x
1000
2000
3000
4000
y
fHxL=10x+20x^2 - 0.6x^3
FIGURE 16. Marginal Product
x
MP
f’HxL= 40 - 0.36x
FIGURE 17. Production and Marginal Product
10 20 30
x
1000
2000
3000
4000
y
APi =
f (x) xi
y xi
For the production function in equation 19 the average product at x=5 is 475/ = 95. Figure 18 shows the average and marginal products for the production func- tion in equation 19. Notice that the marginal product curve is above the average product curve when the average product curve is rising. The two curves intersect where the average product reaches its maximum.
We can show that MP = AP at the maximum point of AP by taking the derivative of APi with respect to xi as follows.
FIGURE 18. Average and Marginal Product
5 10 15 20 25 30
x
**- 200
100
200
MP AP f’HxL= 40 - 0.36x fHxLx= 10 x^ +^ 20 x
(^2) - 0. x
f (x) xi
∂ xi
xi (^) ∂ x∂ fi − f(x) x^2 i
=
xi
∂ f ∂ xi
f(x) xi
xi
( M Pi − APi )
If we set the last expression in equation 22 equal to zero we obtain
M Pi = APi (23) We can represent MP and AP on a production function graph as slopes. The slope of a ray from the origin to a point on f(x) measures average product at that point. The slope of a tangent to f(x) at a point measures the marginal product at that point. This is demonstrated in figure 19.
2.7.3. Elasticity of ouput. The elasticity of output for a production function is given by
i =
∂ f ∂ xi
xi y
3. ECONOMIES OF SCALE
3.1. Definitions. Consider the production function given by
y = f (x 1 , x 2 , ... xn) = f (x) (25) where y is output and x is the vector of inputs x 1 ...xn. The rate at which the amount of output, y, increases as all inputs are increased proportionately is called the degree of returns to scale for the production function f(x). The function f is said to exhibit nonincreasing returns to scale if for all x ∈ Rn + , λ ≥ 1, and μ ≤ 1,
f (λ x) ≤ λf(x) and μf(x) ≤ f (μx) (26)
∂ ln f(λx) ∂ ln λ
|λ=1 =
∂f( λx) ∂λ
λ f(λx)
|λ=
∂f ∂λx
x
λ f(λx)
|λ = 1
∂f ∂x
λ
· x
λ f(λx)
|λ = 1
∂f ∂x
x f(λx)
|λ = 1
∂f ∂x
x f(x)
So the elasticity of scale is simply the elasticity of the marginal product of x, i.e.
∂f (x) ∂x
x f(x)
∂y ∂x
x y
∂ln y ∂ln x
In the case of multiple inputs, the elasticity of scale can also be represented as
∑ (^) n i=
∂f ∂xi
xi y
∑^ n
i=
∂f ∂xi y xi
∑^ n
i=
M Pi APi
This can be shown as follows where x is now an n element vector:
∂ ln f( λx) ∂ ln λ
|λ=1 =
∂f( λx) ∂λ
λ f(λx)
∑^ n
i=
∂f ∂λxi
xi
λ f(λx)
|λ = 1
∑^ n
i=
∂f ∂xi
λ · xi
λ f(λx) |λ = 1
∑^ n
i=
∂f ∂xi
xi f(λx)
|λ = 1
∑^ n
i=
∂f(x) ∂xi
xi f(x)
∑^ n
i=
M Pi
xi y
Thus elasticity of scale is the sum of the output elasticities for each input. If is less than one, then the technology is said to exhibit decreasing returns to scale and isoquants spread out as output rises; if it is equal to one, then the technol- ogy exhibits constant returns to scale and isoquants are evenly spaced; and if is greater than one, the technology exhibits increasing returns to scale and the iso- quants bunch as output expands. The returns to scale from increasing all of the inputs is thus the average marginal increase in output from all inputs, where each input is weighted by the relative size of that input compared to output. With de- creasing returns to scale, the last expression in equation 33 implies that MPi < APi for all i.
3.2. Implications of Various Types of Returns to Scale. If a technology exhibits constant returns to scale then the firm can expand operations proportionately. If the firm can produce 5 units of output with a profit per unit of $20, then by dou- bling the inputs and producing 10 units the firm will have a profit of $40. Thus the firm can always make more profits by expanding. If the firm has increasing returns to scale, then by doubling inputs it will have more than double the output. Thus if it makes $20 with 5 units it will make more than $40 with 10 units etc. This assumes in all cases that the firm is increasing inputs in a proportional manner. If the firm can reduce the cost of an increased output by increasing inputs in a manner that is not proportional to the original inputs, then its increased economic returns may be larger than that implied by its scale coefficient.
3.3. Multiproduct Returns to Scale. Most firms do not produce a single product, but rather, a number of related products. For example it is common for farms to produce two or more crops, such as corn and soybeans, barley and alfalfa hay, wheat and dry beans, etc. A flour miller may produce several types of flour and a retailer such as Walmart carries a large number of products. A firm that produces several different products is called a multiproduct firm. Consider the production possibility set of the multi-product firm
T = {(x, y) : x ∈ Rn+, y ∈ Rm :+ x can produce y}
where y and x are vectors of outputs and inputs, respectively. We define the mul- tiproduct elasticity of scale by
m = sup{r : there exists a δ > 1 such that (λ x, λ yr^ ) ∈ T for 1 ≤ λ ≤ δ} (35)
For our purposes we can regard the sup as a maximum. The constant of pro- portion is greater than or equal to 1. This gives the maximum proportional growth rate of outputs along a ray, as all inputs are expanded proportionally [2]. The idea is that we expand inputs by some proportion and see how much outputs can pro- portionately expand and still be in the production set. If r = 1, then we have con- stant returns to scale. If r < 1 then we have decreasing returns, and if r > 1, we have increasing returns to scale.
