***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  f/x  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”
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- 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
f11
f12
(u1 u2) 
f12  u 1 
u 
  < 0, subject to [f1 f2]  1  = 0, (u1 u2)  0,

f 22  u 2 
u2 
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).
 u1 
 = 0 implies that u2 = -(f1/f2) u1. It follows that the above expression
u2 
Note that [f1 f2] 
can be alternatively written as
u1[f11 – (f1/f2)f12 + f12(-f1/f2) + (f1/f2)2f22] u1 < 0, for all u1  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, , r, y) = r’x +  [y-f(x)]. The first-order
conditions (FOC) for cost minimization are
L/x = r’ -  f/x = 0,
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3
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
ij =
 2C
C
, i, j = 1, …, n,
ri r j (C / ri )(C / r j )
or, using Shephard’s lemma (C/r = xc),


ij =
x ic C
, i, j = 1, …, n.
r j x ic x cj
The Allen elasticity of substitution (AES) ij 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  =
 11  1n 
 . 
  is also symmetric, negative semi-definite. This implies that ij =

 n1 .  nn 
ji for all i, j = 1, ..., n, and ii  0.
Definition: Two inputs i and j are said to be substitutes (complements) if ij > 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)  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
ij
= C (xic/rj)/(xicxjc)
= C [(xic/rj)(rj/xic)]/(rjxjc)
= (ln xic/ln rj)/wj,
c
where wj = rjxj /C is the j-th cost share,
or equivalently,
ln xic/ln rj = ij wj,
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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 (ij) times the budget share (wj).
b/ From the Production Function
We have ij = C(xic/rj)/(xicxjc). Differentiating the FOC of the cost minimization
problem yields the following comparative static results
 f xx
 f
x

 f x ' x c / r  I n 
= 0,


0    / r   0 
where fx = f/x = a (1n) vector and fxx = 2f/x2 = a (nn) matrix. Multiplying this
expression by (1/) yields
f xx
f
 x
f x '  x c / r 
I n 

  (1 /  )  .

0  (  / r ) /  
0
It follows that
1
f '  I 
I 
f
x /r = (1/)  xx x   n  = (1/) H-1  n  ,
0
 fx 0   0 
f '
f
where H =  xx x  . This can be written as
0
 fx
c
xi /rj = (1/)
H cij
c
det( H )
, for all i, j = 1, ..., n,
where Hijc is the cofactor of fij in H. The AES can then be written as
ij
= [C/(xi xj )] (1/)
c
c
H cij
det( H )
= [(  nk1 rk xkc)/(xicxjc)] (1/)
H ijc
det( H )
or, using the (FOC), rk/ = fi, (where fi = f/xi)
c
 nk 1f k x k H ij
ij =
, for all i, j = 1, ..., n.
x i x j det( H )
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
12 = -
f1x1  f 2 x 2
f1f2
.
2
2
x 1x 2
f1 f 22  f 2 f11  2f1f 2 f12
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
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5
along an isoquant can be done by changing x1, and letting x2 adjust according to
x2 = g(x1, y). It follows that
1 x1 g

x 2 x 22 x1
-(x1/x2)/(f1/f2) = ,
f11 f1f 21  f12 f1f 22  g
 2   2 
f2
f2
f 2  x1
 f2
or, using g/x1 = -f1/f2,
-(x1/x2)/(f1/f2) = -

1
f2 x 2  f1x1 
f 2 x 22
1
f f 2  f 22 f12  2f12 f1f 2
3 11 2
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)]
=-

1
f2 x 2  f1x1 
f 2 x 22

f1
f2
x1
x2
1
f11f 22  f 22 f12  2f12 f1f 2
f 23
f x f x
f1f2
=- 1 1 2 2 2
,
2
x 1x 2
f1 f 22  f 2 f11  2f1f 2 f12
which the formula for the AES derived above. This shows that the AES 12 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, 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.
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It follows that
p=
C
(r, y*(r, p)).
y
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 from Hotelling's lemma. It follows that
1 = (2C/y2)(2/p2), or (2C/y2) = (2/p2)-1,
and
0 = (2C/yr) + (2C/y2)(2/pr), or (2C/yr) = -(2C/y2)(2/pr).
Combining these two results yields
(2C/yr) = -(2/p2)-1(2/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 (/r = -x*), we obtain
-2/r2 = 2C/r2 + (C/ry)(/pr)
or, using the transpose of (B1),
-2/r2 = 2C/r2 - (2/rp)(2/p2)-1(/pr),
or
2C/r2 = -2/r2 + (2/rp)(2/p2)-1(/pr)
(B2)
Expression (B2) is a "Le Chatelier" result. The convexity of  in prices implies that
2/p2 > 0. It follows that (2/rp)(2/p2)-1(/pr) is a (nn) positive semidefinite matrix. Note that the matrix [-2/r2] = x*/r is symmetric, negative semidefinite, and that the matrix 2C/r2 = xc/r is also symmetric, negative semi-definite.
Then, (B2) states that the symmetric, negative semi-definite matrix [-2/r2] = x*/r
exceeds the symmetric, negative semi-definite matrix 2C/r2 = xc/r by a positive semidefinite matrix. This means that the negative of the profit function -(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
ij = C  (2C/rirj)/(xic  xjc)
= [  nk 1 rk xk*][-2/rirj + (2/rip)(2/p2)-1(/prj)]/(xi*  xj*),
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7
or, using Hotelling's lemma (/ri = -xi*),

ij = -[  nk 1 rk /rk][-2/rirj + (2/rip)(2/p2)1
(/prj)]/[(/ri)(/rj)],
for all i, j = 1, ..., n. This formula allows the estimation of the AES solely from knowing
the profit function (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))}.
The associated Lagrangean is L = r'x +  [y - F(g(x))]. The (FOC) are
r' =  (F/g)(g/x).
This yields
C = r'xc =  (F/g)(g/x) xc
= (C/y)(F/g)(g/x) xc, since(C/y) =  from the envelope
theorem.
The linear homogeneity of g(x) gives [(g/x) x] = g, implying that
C = (C/y)(F/g) g,
= (C/y) G(y), where G(y) = [(F/g) g],
or
(C/y)/C = 1/G(y),
or
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ln(C)/y = 1/G(y).
Integrating with respect to y gives
ln(C) = ln[H(r)] + ln[K(y)],
where ln[H(r)] is the constant of integration, and ln[K(y)] = [1/G(y)] dy. This
implies that the cost function C(r, y) takes the from
C(r, y) = K(y)  H(r).
Q.E.D.
Thus, under a homothetic technology, C(r, y) = K(y)  H(r) implies that
xic = C/ri = K(y)  H/ri, (from Shephard's lemma)
or
ln(xic) = ln[K(y)] + ln[H/ri], i = 1, ..., n.
Thus,
ln(xic)/ln(y) = ln[K(y)]/ln(y) = independent of i, for all i = 1, ..., n.
It follows that, under a homothetic technology, the output elasticities ln(xic)/ln(y) are
the same for all inputs.
* Homogenous Technology (= a special case of homotheticity)
The production function is homogenous of degree k if f(x) = [g(x)]k, where g(x) is linear
homogenous in x.
A homogenous function is always homothetic. Thus, under a homogenous technology,
the marginal rate of substitution MRSij is also constant along a ray through the origin, i.e.
that all isoquants have basically the "same shape".
Under a homogenous technology of degree k, the cost function takes the form C(r, y) =
y1/k  H(r).
Proof: We have C(r, y) = r' xc = minx{r'x: subject to y = g(x)k}.
The associated Lagrangean is L = r'x +  [y - g(x)k]. The (FOC) are
r' =  k gk-1 (g/x).
This yields
C = r'xc =  k gk-1 (g/x) xc
= (C/y) k gk-1 (g/x) xc, since(C/y) =  from the envelope
theorem.
The linear homogeneity of g(x) gives [(g/x) x] = g, implying that
C = (C/y) k gk,
= (C/y) k y, since y = g(x)k,
or
(C/y)(y/C) = 1/k,
or
ln(C)/ln(y) = 1/k.
Integrating with respect to ln(y) gives
ln(C) = ln[H(r)] + (1/k) ln(y),
where ln[H(r)] is the constant of integration. This implies that the cost function
C(r, y) takes the from
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C(r, y) = y1/k  H(r).
Q.E.D.
Under a homogenous technology of degree k, C(r, y) = y1/k  H(r) implies that
xic = C/ri = y1/k  H/ri, (from Shephard's lemma)
or
ln(xic) = (1/k) ln(y) + ln[H/ri], i = 1, ..., n.
Thus,
ln(xic)/ln(y) = 1/k, for all i = 1, ..., n.
It follows that, under a homogenous technology of degree k, the output elasticities
ln(xic)/ln(y) are all equal to (1/k).
This implies that ln(xic)/ln(y) > (=, or <) 1 when k < (=, or >) 1.
Note: Under a homogenous technology of degree k, the average cost function AC(r, y) =
C(r, y)/y becomes
AC(r, y) = [y1/k H(r)]/y = y(1-k)/k H(r).
It follows that the average cost function AC(r, y) is
- increasing in y if k < 1
- constant with respect to y if k = 1
- decreasing in y if k > 1.
** The Spacing of the Isoquants
* Returns to Scale
The concept of "returns to scale" involves measuring the spacing of the isoquants along a
ray through the origin.
Definition: The production technology is said to exhibit
increasing returns to scale (IRTS) at x if f(t  x) > t  f(x), for all scalars t >
1,
constant returns to scale (CRTS) at x if f(t  x) = t  f(x), for all scalars t >
1,
decreasing returns to scale (DRTS) at x if f(t  x) < t f(x), for all scalars t >
1,
where y = f(x) is the production function.
Note that this definition is “global” since it applies for all t > 1.
A local measure: Note that differentiating ln(y) = ln[f(t  x)] with respect to the scalar t
(evaluated at t = 1) gives
ln(y)/ln(t) =  in1 ln(y)/ln(xi) =  in1 (y/xi)(xi/y)
Definition: At point x, the scale elasticity (SE) is defined as
SE =  in1 ln(y)/ln(xi) =  in1 (y/xi)(xi/y).
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The scale elasticity provides a local measure of "returns to scale". More specifically, in
the neighborhood of a point x, technology exhibits
increasing returns to scale (IRTS) if SE > 1,
constant returns to scale (CRTS) if SE = 1,
decreasing returns to scale (DRTS) if SE < 1.
In general, "returns to scale" can vary over different regions of the production
technology.
The case of a homogenous technology: Under a homogenous technology of degree k, we
have y = f(x) = [h(x)]k, where h(x) is linear homogenous. Then, ln[f(t  x)] = k
ln[h(t  x)], and
ln(y)/ln(t) = k ln(h)/ln(t)
= k  in1 ln(h)/ln(xi) = k,
since  in1 (h/xi) = 1 from the linear homogeneity of h (Euler equation). It
follows that the scale elasticity is
SE =  in1 ln(y)/ln(xi) = k.
Thus, for homogenous technology, the scale elasticity SE is equal to the degree of
homogeneity k.
a/ Measuring SE from the cost function
Under cost minimizing behavior, we have
SE =  in1 (y/xi)(xi/y)
=  in1 ri xi/( y), using the (FOC) to cost minimization (y/xi = ri/)
=
 in1 ( ri x i ) / y
, using the envelope theorem (C/y = ).
C / y
Note that  in1 ri xi/y = AC is the average cost, and C/y = MC is the marginal cost. It
follows that
SE = AC/MC.
This implies that, in the neighborhood of a point x, technology exhibits
increasing returns to scale (IRTS) if AC/MC > 1, or AC > MC,
constant returns to scale (CRTS) if AC/MC = 1, or AC = MC,
decreasing returns to scale (DRTS) if AC/MC < 1, or AC < MC.
Note: We have
ln(C)/ln(y) = (C/y)/(C/y) = MC/AC.
It implies that MC/AC is the elasticity of the cost C with respect to output y. This
yields
ln(C)/ln(y) = 1/SE,
Thus, the elasticity of cost with respect to output is the inverse of the scale
elasticity. It follows that in the neighborhood of a point x, technology exhibits
increasing returns to scale (IRTS) if ln(C)/ln(y) < 1,
constant returns to scale (CRTS) if ln(C)/ln(y) = 1,
decreasing returns to scale (DRTS) if ln(C)/ln(y) > 1.
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b/ Measuring SE from the profit function
Under profit maximizing behavior, we have
SE =  in1 (y/xi)(xi/y)
=  in1 ri xi/(p y), using the (FOC) to profit maximization (y/xi = ri/p).
Since  in1 ri xi = C = cost, and p y = R = revenue, it follows that
SE = C/R.
This implies that, in the neighborhood of a point x, technology exhibits
increasing returns to scale (IRTS) if C/R > 1, or C > R, or  < 0,
constant returns to scale (CRTS) if C/R = 1, or C = R, or  = 0
decreasing returns to scale (DRTS) if C/R < 1, or C < R, or  > 0.
Note: Under free entry and exit, a long-run equilibrium can exist only under zero profit
( = 0). Indeed,  > 0 would stimulate entry, while  < 0 would stimulate exit.
Thus, under free entry and exit, a long-run equilibrium can exist only if all firms
produce at a size y corresponding to (local) CRTS.
* Returns to Size
The concept of "returns to size" involves measuring the spacing of the isoquants along
the "expansion path", i.e. moving from one isoquant to another while keeping the
marginal rate of substitution (MRSij) constant.
Note: Under homothetic technology, the expansion path is a straight line through the
origin. In this case, "returns to scale" and "returns to size" are equivalent.
However, for non-homothetic technologies, "returns to scale" and "return to size"
can differ.
Let AC(r, y) = C(r, y)/y be the average cost function.
Definition: The production technology is said to exhibit
increasing returns to size if AC(r, y) is decreasing in output,
constant returns to size if AC(r, y) does not change with output y,
decreasing returns to size if AC(r, y) is increasing in output y.
A local measure: Note that differentiating ln(AC) = ln[C/y] with respect to output y gives
ln(AC)/ln(y) = (AC/y)(y/AC) = [(C/y)/y - C/(y2)](y/AC)
= [(C/y) - C/y](1/AC)
= [MC - AC]/AC
= sign(MC - AC).
It follows that the technology exhibits
increasing returns to size if (MC - AC) < 0, or AC > MC,
constant returns to size if (MC - AC) = 0, or AC = MC,
decreasing returns to size if (MC - AC) > 0, or AC < MC.
Note that these results are similar to the ones obtained with "returns to scale".
Indeed, we have seen that, when AC = MC, the technology exhibits (at least local)
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CRTS, implying that it is (at least locally) linear homogenous, and thus (at least
locally) homothetic.
Note: Assume profit maximization. This implies that
ln(AC)/ln(y) = [(C/y) - C/y](y/C)
= [p y - C]/C, (from the (FOC): p = C/y)
= /C = sign().
It follows that  > (=, or <) 0 when the firm produces in a region of decreasing
(constant, or increasing) returns to size. Under free entry and exit, a long-run
equilibrium can exist only if  = 0 (since  > 0 would stimulate entries and  < 0
would stimulate exits). If the AC(y, ) function has a U-shape, this implies that
the long-run equilibrium for firms is at the point which miny{AC(r, y}. Then, the
long equilibrium price pe is
pe = miny{AC(r, y)}.
In other words, the efficient firm size is the output level that minimizes AC(, y).
At that point,
pe = MC (i.e., marginal cost pricing, from profit maximization)
and
pe = AC (i.e., average cost pricing in long-run equilibrium, with  = 0).
** Multi-output Technology
So far, we have focused on the single output case (m = 1). All the arguments presented
above can be extended to the multi-output case, where y is a (m1) output vector, and p
is a (m1) output price vector, m > 1.
- Production function: The implicit joint production function can be written as f(y, x) = 1,
where f(y, x) is increasing in x, decreasing in y, and concave in (x, y).
This generalizes the single output case (where f(y, x) = fo(x) – y +1 = 1 when m = 1, y =
fo(x) being the single output production function).
- Cost function: C(r, y) = minx{r’x: subject to f(y, x) = 1)},
which has for solution xc(r, y), the cost minimizing input demand functions. The cost
function C(r, y) is linear homogeneous, increasing and concave in r. And it satisfies
Shephard’s lemma: C/r = xc(r, y).
- Scale elasticity: (when m > 1)
SE
=-
 in1 ln( f ) /  ln( x i )
 mj1 ln( f ) /  ln( y j )
,
m
= 1/[  i
1 ln(C)/ln(yi)].
This generalizes the single output case (where m = 1 and ln(f)/ln(y) = -1).
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* Economies of scope
The concept of “economies of scope” is relevant only in the multi-output case where y =
(y1, …, ym)’ is a (m1) vector. Let M = {1, 2, …, m}be the set of all outputs. Partition
the set M into s mutually exclusive subsets: M = {M1, M2, …, Ms}, s  m. Let
Yk
= {y: such that yj > 0 for j  Mk, and yj = 0 for j  Mk}
= the k-th product line,
k = 1, …, s. It follows that C(r, Yk) is the cost of production for a firm that specializes in
the k-th product line.
Definitions: The technology exhibits economies of scope if
 sk1 C(r, Yk) > C(r, y), where y =  sk1 Yk.
It exhibits diseconomies of scope if
 sk1 C(r, Yk) < C(r, y), where y =  sk1 Yk.
And the technology is said to be non-joint if
 sk1 C(r, Yk) = C(r, y), where y =  sk1 Yk.
Note that economies of scope can motivate the existence of multi-product firms since
they imply that producing more than one product at a time can reduce production cost.
Alternatively, there is no economic incentive for multi-product firms when the
technology exhibits either non-jointness or diseconomies of scope.
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