12
***Production Technology
For simplicity, we start with the single output case. Let y denote output, and x = (x1, …, xn)’ be a (n´1) vector of inputs. The production technology is the process that transforms the inputs x into output. It is typically a complex process. This process can be represented by the production function (or production frontier) f(x):
f(x) = maxy{y| x and the best available technology}.
** Stages of Production
First, consider the simple case of a single input (n = 1) and single output. Assume that the firm intends to maximize profit
maxx,y{py – rx: y £ f(x)}
where p > 0 denotes output price, and r > 0 denotes input price.
Given p > 0, it follows that the firm would always choose y such that y = f(x). This is called technical efficiency, where the firm operates on the production frontier.
Define the stages of production as follows
- “stage 1” is the production region where ¶f/¶x > f(x)/x.
- “stage 2” is the production region where f(x)/x ³ ¶f/¶x ³ 0.
- “stage3” is the production region where ¶f/¶x < 0.
The first-order condition (FOC) for an interior solution is
p ¶f/¶x = r
or
¶f/¶x = r/p.
This states that the marginal physical product (MPP = ¶f/¶x) is equal the input/output price ratio (r/p). Since r/p > 0, it follows that the profit maximizing firm would never want to operate where MPP = ¶f/¶x < 0. Thus, the firm would never choose to be in “stage 3”.
Assuming that the firm has always the option to do nothing (and thus to obtain a zero profit), the profit maximizing firm would never choose a production plan such that pf(x) – rx < 0, or r/p > f(x)/x. But r/p = ¶f/¶x from (FOC). It follows that the firm would never choose ¶f/¶x > f(x)/x, i.e. it would never choose to be in “stage 1”.
Since the profit maximizing firm would never choose to be in either “stage 1” or “stage 3”, it would only operate in “stage 2”. For that reason, “stage 2” is called the “economic region”. In other words, only “stage 2” of production is expected to be observed in the real world if firms intend to maximize profit.
Note that, in “stage 2”:
- the marginal physical product is positive (MPP = ¶f/¶x > 0), implying that production is increasing in x in “stage 2”
- the marginal physical product is declining (¶MPP/¶x = ¶2f/¶x2 < 0), implying diminishing marginal productivity and a concave production function in “stage 2”
- the feasible set {(x, y): y £ f(x)} is convex in “stage 2”.
Note: From experimental data, it is possible to observe all three stages of production (See homework #2).
* Substitution Effects
Consider the two input case, where x = (x1, x2)’. The associated production function is
y = f(x1, x2).
Assuming that f is increasing in x2, the production function can be implicitly solved for x2, yielding
x2 = g(x1, y).
Graphing x2 as a function of x1 for a given output y gives an isoquant. The slope of an isoquant is ¶x2/¶x1 = ¶g/¶x1. It can be obtained by differentiating y = f(x1, g(x1, y)) with respect to x1, yielding
0 = ¶f/¶x1 + (¶f/¶x2)(¶g/¶x1),
or
¶g/¶x1 = -f1/f2 = MRS12,
where fi = ¶f/¶xi, i = 1, 2, and f1/f2 = the marginal rate of substitution between x1 and x2 (MRS12).
1- the Shape of an Isoquant
Assume that the production function is quasi-concave (note: a concave function is also quasi-concave). Then, by (strong) quasi-concavity of f(x1, x2), we have
(u1 u2) < 0, subject to [f1 f2] = 0, (u1 u2) ¹ 0,
where fij = ¶2f/¶xi¶xj, i, j = 1, 2.
It means that the isoquants ate strictly convex to the origin, or equivalently that the feasible technology is convex in (x1, x2).
Note that [f1 f2] = 0 implies that u2 = -(f1/f2) u1. It follows that the above expression can be alternatively written as
u1[f11 – (f1/f2)f12 + f12(-f1/f2) + (f1/f2)2f22] u1 < 0, for all u1 ¹ 0,
or
[f11 f22 – 2 f12 f1 f2) + f22 f12]/f22 < 0. (A1)
2- The Allen Elasticity of Substitution (AES)
a/ From the Cost Function
The cost function C(r, y) is given by
C(r, y) = r’xc = minx{r’x: y = f(x)},
where xc = xc(r, y) denotes the (n´1) vector of cost minimizing input demand functions.
Let the associated Lagrangean be L(x, l, r, y) = r’x + l [y-f(x)]. The first-order conditions (FOC) for cost minimization are
¶L/¶x = r’ - l ¶f/¶x = 0,
¶L/¶l = y – f(x) = 0.
The second-order condition (SOC) are satisfied under the (strong) quasi-concavity of f(x) (as stated above).
Definition: The Allen elasticity of substitution (AES) between inputs xi and xj is
sij = , i, j = 1, …, n,
or, using Shephard’s lemma (¶C/¶r = xc),
sij =, i, j = 1, …, n.
The Allen elasticity of substitution (AES) sij measures the response of the i-th input demand to a change in the j-th input price, holding output y constant (i.e., moving along an isoquant) and other input prices constant.
Note: We know that ¶xc/¶r is a (n´n) symmetric, negative semi-definite matrix. It follows that the (n´n) matrix of Allen elasticities of substitution s = is also symmetric, negative semi-definite. This implies that sij = sji for all i, j = 1, ..., n, and sii £ 0.
Definition: Two inputs i and j are said to be substitutes (complements) if sij > 0 (< 0).
Given C > 0 and xc > 0, this means that inputs i and j are substitutes complements if ¶xic/¶rj = ¶xjc/¶ri > 0 (< 0).
Note: In the two input case (n = 2), we have ¶x1c/¶r1 £ 0. Also, by homogeneity of degree zero of xc(r, y) in r, we have
(¶x1c/¶r1) r1 + (¶x1c/¶r2) r2 = 0. (Euler equation)
It follows that
(¶x1c/¶r2) = - (¶x1c/¶r1)(r1/r2) ³ 0 (since ¶x1c/¶r1 £ 0).
Thus, in the two input case, inputs can only be substitutes. In other words, it takes at least three inputs before input complementarity can arise.
Note: The AES can also be written as
sij = C (¶xic/¶rj)/(xic×xjc)
= C [(¶xic/¶rj)(rj/xic)]/(rj×xjc)
= (¶ln xic/¶ln rj)/wj,
where wj = rjxjc/C is the j-th cost share,
or equivalently,
¶ln xic/¶ln rj = sij wj,
for all i, j = 1, ..., n. This states that the price elasticity of the cost minimizing input demand function (¶ln xic/¶ln rj) is equal to the corresponding Allen elasticity of substitution (sij) times the budget share (wj).
b/ From the Production Function
We have sij = C×(¶xic/¶rj)/(xic×xjc). Differentiating the FOC of the cost minimization problem yields the following comparative static results
= 0,
where fx = ¶f/¶x = a (1´n) vector and fxx = ¶2f/¶x2 = a (n´n) matrix. Multiplying this expression by (1/l) yields
.
It follows that
¶xc/¶r = (1/l) = (1/l) H-1 ,
where H = . This can be written as
¶xic/¶rj = (1/l), for all i, j = 1, ..., n,
where Hijc is the cofactor of fij in H. The AES can then be written as
sij = [C/(xic×xjc)] (1/l)
= [(rk xkc)/(xic×xjc)] (1/l)
or, using the (FOC), rk/l = fi, (where fi = ¶f/¶xi)
sij = , for all i, j = 1, ..., n.
This is the general formula to evaluate the AES from the production function.
Note: In the two input case (n = 2), we have
H12c = f1 f2,
and
det(H) = -[f12 f22 + f22 f11 - 2 f1 f2 f12] > 0 (from the (SOC)).
It follows that the formula for the AES in the two input case is
s12 = -.
Note: Why is the AES called an "elasticity"? To see that, consider the two input case (n = 2). While moving along a given isoquant (i.e., while holding output y constant), consider the change in the input ratio (x1/x2) due to a change in (f1/f2). This move along an isoquant can be done by changing x1, and letting x2 adjust according to x2 = g(x1, y). It follows that
-¶(x1/x2)/¶(f1/f2) = -,
or, using ¶g/¶x1 = -f1/f2,
-¶(x1/x2)/¶(f1/f2) = -.
Thus, the negative of the elasticity of (x1/x2) with respect to (f1/f2) is
-¶ln(x1/x2)/¶ln(f1/f2) = -[¶(x1/x2)/¶(f1/f2)][(f1/f2)/(x1/x2)]
= -
= -,
which the formula for the AES derived above. This shows that the AES s12 can be interpreted as the negative of the elasticity of the input ratio (x1/x2) with respect to a change in (f1/f2) obtained by moving along an isoquant. Thus, the AES is an elasticity measure of relative input change along an isoquant.
This provides the following intuitive interpretation for the AES. Consider a move along an isoquant. If the isoquant is "kinked", then the input ratio x1/x2 changes little as the slope of the isoquant (as measured by f1/f2) changes, implying a low elasticity of substitution. Alternatively, if the isoquant is "flat", then the input ratio x1/x2 changes a lot as the slope of the isoquant (as measured by f1/f2) changes, implying a high elasticity of substitution. This illustrates that the AES is an elasticity measure of the shape of isoquants.
c/ From the Profit Function
Profit maximizing behavior is given by
p(p, r) = maxx{p f(x) - r'x}
= maxx,y{p y - r'x: y = f(x)}
= maxy {p y - minx{r'x: y = f(x)}}
= maxy {p y - C(r, y)}, where C(r, y) = minx{r'x: y = f(x)} is the cost function.
This shows that profit maximizing behavior implies cost minimizing behavior (although the reverse is not necessarily true).
The first-order condition (FOC) for profit maximization is
p - ¶C/¶y = 0.
Let x*(r, p) and y*(r, p) denote the profit maximizing input demand and output supply function, respectively.
It follows that
p = (r, y*(r, p)).
Differentiating this identity with respect to p and r gives
1 = (¶2C/¶y2)(¶y*/¶p)
and
0 = (¶2C/¶y¶r) + (¶2C/¶y2)(¶y*/¶r).
But y* = ¶p/¶p from Hotelling's lemma. It follows that
1 = (¶2C/¶y2)(¶2p/¶p2), or (¶2C/¶y2) = (¶2p/¶p2)-1,
and
0 = (¶2C/¶y¶r) + (¶2C/¶y2)(¶p2/¶p¶r), or (¶2C/¶y¶r) = -(¶2C/¶y2)(¶p2/¶p¶r).
Combining these two results yields
(¶2C/¶y¶r) = -(¶2p/¶p2)-1(¶p2/¶p¶r). (B1)
Next, let xc(r, y) be the cost minimizing input demand functions. Profit maximizing behavior implying cost minimizing behavior gives the identity
x*(r, p) = xc(r, y*(r, p)).
Differentiating this identity with respect to r yields
¶x*/¶r = ¶xc/¶r + (¶xc/¶y)(¶y*/¶r).
Using Shephard's lemma (¶C/¶r = xc) and Hotelling's lemma (¶p/¶r = -x*), we obtain
-¶2p/¶r2 = ¶2C/¶r2 + (¶C/¶r¶y)(¶p/¶p¶r)
or, using the transpose of (B1),
-¶2p/¶r2 = ¶2C/¶r2 - (¶p2/¶r¶p)(¶2p/¶p2)-1(¶p/¶p¶r),
or
¶2C/¶r2 = -¶2p/¶r2 + (¶p2/¶r¶p)(¶2p/¶p2)-1(¶p/¶p¶r) (B2)
Expression (B2) is a "Le Chatelier" result. The convexity of p in prices implies that ¶2p/¶p2 > 0. It follows that (¶p2/¶r¶p)(¶2p/¶p2)-1(¶p/¶p¶r) is a (n´n) positive semi-definite matrix. Note that the matrix [-¶2p/¶r2] = ¶x*/¶r is symmetric, negative semi-definite, and that the matrix ¶2C/¶r2 = ¶xc/¶r is also symmetric, negative semi-definite. Then, (B2) states that the symmetric, negative semi-definite matrix [-¶2p/¶r2] = ¶x*/¶r exceeds the symmetric, negative semi-definite matrix ¶2C/¶r2 = ¶xc/¶r by a positive semi-definite matrix. This means that the negative of the profit function -p(r, ×) is "more concave" in r than the cost function C(r, ×). Equivalently, this means that the magnitude of the input response to an input price change is larger under profit maximization than under cost minimization. This has the more general and intuitive interpretation that restricting choices (in this case, restricting output to be equal to y under cost minimization) tends to reduce optimal quantity adjustments to changing market prices.
Another use of (B2) is in the measurement of the AES from the profit function. Substituting (B2) into the definition of the AES from the cost function, and using the consistency of x* with xc, we obtain
sij = C × (¶2C/¶ri¶rj)/(xic × xjc)
= [rk xk*][-¶2p/¶ri¶rj + (¶p2/¶ri¶p)(¶2p/¶p2)-1(¶p/¶p¶rj)]/(xi* × xj*),
or, using Hotelling's lemma (¶p/¶ri = -xi*),
sij = -[rk ¶p/¶rk][-¶2p/¶ri¶rj + (¶p2/¶ri¶p)(¶2p/¶p2)-1(¶p/¶p¶rj)]/[(¶p/¶ri)(¶p/¶rj)],
for all i, j = 1, ..., n. This formula allows the estimation of the AES solely from knowing the profit function p(p, r).
** Output Effects
Definition: The "output elasticities" are defined as
¶ln(xic)/¶ln(y) = ¶(xic/¶y)(y/xic),
where xic is the cost minimizing input demand function for the i-th input, i = 1, ..., n.
The output elasticities are elasticities of cost minimizing input demand functions with respect to output, holding input prices constant. They measure the shape of the "expansion path" obtained as one moves from one isoquant to another, holding the marginal rate of substitution constant.
Definition: The i-th input is said to be
- inferior if ¶ln(xic)/¶ln(y) < 0
- normal if 0 < ¶ln(xic)/¶ln(y) < 1
- superior if ¶ln(xic)/¶ln(y) > 1.
* Homothetic Technology
The production function is homothetic if f(x) = F(g(x)), where ¶F/¶d > 0, and g(x) is linear homogenous in x.
Under the homotheticity of f(x), we have seen that the marginal rate of substitution MRSij = fi/fj is homogenous of degree zero in x. This means that the marginal rate of substitution MRSij is constant along a ray through the origin, i.e. that all isoquants have basically the "same shape".
Under a homothetic technology, the cost function takes the form C(r, y) = K(y) × H(r).
Proof: We have C(r, y) = r' xc = minx{r'x: subject to y = F(g(x))}.