Chapter 9
The Theory of Decreasing Cost Production
1. Decreasing cost services/Natural monopolies:
a. Average cost decreases all the way to market demand.
b. A number of important examples: public utilities (electricity distribution,
water and sewage), transportation facilities (highways, bridges, tunnels, mass transit),
recreation (parks, beaches), telecommunications, software
2. A simple model sufficient to analyze decreasing cost services
a. First-best models dichotomize in two ways:
1). between the interpersonal equity and pareto optimal conditions and
2). within the pareto optimal conditions, between markets with problems
and all other goods and services that can be marketed competitively
b. Assuming the government achieves the interpersonal equity conditions implies
a one-consumer equivalent economy.
c. The dichotomy within the pareto-optimal conditions allows a focus on the
decreasing cost good.
d. Assume a single consumer who consumes good X and supplies labor L, with
utility function U = U(X,L), L entering U negatively, and prices PX and PL.
e. The production function is X=f(L), with increasing returns: f'>0, f"">0, and
the Inada conditions hold.
1). ACX= PL/APL = PL/ f/L = PLL/f > PL/f' = MCX, from Euler's equation
on homogenous functions: Lf'> f, with increasing returns. (At any (L,X), the slope of f is
greater than the slope of a ray from the origin.
2). With MC<AC, AC is continuously decreasing
3). Therefore, the cheapest provision is to have one firm supply the
service—a natural monopoly
f. Social welfare maximization (equivalent to maximizing the consumer's utility)
MAX
(X, L)
U(X, L)
s, t. X  f(L)
g. Pareto- optimal condition
Uxf' + UL = 0
1). Second-order condition can't be ignored with decreasing cost services
 U 
d f'  L 
 UX   0
dL
for a utility maximum-- the curvature of the indifference curves for X and L must
be greater than the curvature of the production function
Insert Figure 9.3
3. Decreasing cost services incompatible with a competitive market allocation
2
a. Assume the owner of the single firm is a price taker and maximizes profit.
max PX f(L)  PL L
(L)
1). The FOC: Pxf’ – PL = 0 or Px = PL/f’ = MCX
2). Appears to be an pareto optimal equilibrium since consumer sets
UX  UL

PX
PL
U
PX   X  PL
UL
U X f   U L with Px = PL/f’, the pareto-optimal condition
3). Problem: Pxf’ – PL = 0 or Px = PL/f’ = MCX , at X0, is a profit
minimum, since PL/f’ declines continuously with increasing returns. Profits are earned
only by pushing beyond that point, to the entire market.
{Insert Figure 9.4}
4. Optimal pricing
a. Price must equal marginal cost to satisfy the pareto-optimal condition,
however—where the market demand curve and MC intersect
{Insert Figure 9.5}
b. First-best pricing solution: marginal cost pricing plus a lump-sum subsidy to
cover the monopolist's losses at the optimum—the subsidy is the shaded portion in the
figure
1). Requires regulation since the monopolist would set MR=MC to
maximize its profit.
2). In a many-person economy, the required lump-sum subsidy gets
folded in with the lump-sum taxes and transfers to satisfy the interpersonal equity
conditions-- the lump-sum taxes collected must be sufficient to cover both the lump-sum
transfers to other individuals and the lump-sum transfers to decreasing cost firms.
3). Payment of the subsidy depends not on usage of the service but on
people's relative social marginal utilities of income
5. Optimal investment rule
a. There is an all-or-none investment decision for decreasing cost services that is
absent from standard private investment analysis: Is having the service better than not
having it at all—profit is not a direct guide since optimal pricing generates a loss
b. Two cases-easy case and hard case
1). Easy case: a profit maximizing monopolist charging a single price
could at least break even—Demand curve above or tangent to AC curve
Sufficient condition for operating the service
Examples: public utilities, some but not all transportation service, some
but not all recreational facilities, telecommunications, software
2). Hard case: a profit maximizing monopolist charging a single price
3
could not break even—Demand curve below AC curve
Necessary condition for operating the service comes from this case
Examples: some remote recreational facilities, rural highways and
bridges, mass rail transit in cities
6. The Easy Case
a. Need to analyze in X-L space
b. A straight line from the origin acts as both a representation of break-even
production (X,L_) combinaitons and a budget line for the consumer with PXX = PLL
1). Holding PL constant and varying PX traces out the price-consumption
(P-C) locus for the consumer
{Insert Figure 9.10}
c. If the break-even (P-C) locus intersects the production function, then breakeven production is feasible, at point B in the figure
{Insert Figure 9.11}
1). Indifference curve I1 must lie above the indifference curve through the
origin (no-service)—break-even production preferred to no service
d. Setting PX = MC is even better for the consumer
{Insert Figure 9.12}
1). I2 is the highest indifference curve attainable on f(L)
2). At tangency point A, PX = MC = PL/f', and the consumer transfers 0b
of income (with PL=1) to the firm: PXX – PLL = (-)0b
7. The hard case
a. The break-even P-C locus is everywhere inside f(L)—break-even production is
not possible
{Insert Figure 9.13}
b. Whether the consumer wants the service depends on whether the indifference
curve through the origin, I0, cuts f(L)
{Figure 9.14 here}
1). I0 is everywhere inside f(L)—no service preferred to any possible
level of service.
{Figure 9.14 here)
2). I0 cuts f(L)—consumer can reach I2 at tangency point A with PX=MC
and a transfer of 0b units of income to the firm
c. A valid income test for wanting the service
{Figure 9.16 here}
1). Tangency lines to I0 represent budget lines in which PX decreases and
the consumer pays amounts od to remain on I0—the distances 0d represent the
consumer's willingness to pay for the price decreases-the Hicks Compensating Variation
(HCV)
{Figure 9.17 here}
2). If I0 cuts f(L), then the amount the consumer is willing to pay to be
able to purchase the service at PX=MC, distance 0c in the figure, exceeds the amount of
subsidy, ob, the firm requires to provide the service at that price
The necessary condition for wanting the service
4
d. Marshallian versus compensated demand curves
{Figure 9.18 here }
1). In PX-X space, the necessary condition is often represented as follows:
the consumer surplus at the marginal cost price, area aEP10 in the figure exceeds the
required lump-sum subsidy, area aP10CB. But D is the actual, or Marshallian demand
curve, and consumer surplus areas behind Marshallian demand curves are not valid
measures of willingness to pay
2). The equivalent to the distances 0d in X-L space are areas behind the
compensated demand curve.
e. Compensated demand curves and the expenditure function
1). Consider two utility levels, UA without the service and UB with the
service



2). UB > UA iff M( P ; UB) > M( P ; UA) for any price vector P


3). Let M(P A ; U A )  T A , possibly zero, and M(P B ; U B )  T B , with TB
including the subsidy the consumer pays to the firm
4). The necessary condition for wanting the service is
r
r
T B - T A < M P A ;U A - M P B;U A
(
)
(
)
5). The LHS is the subsidy the consumer pays to the firm.
6). The RHS gives the income the consumer is willing to sacrifice to face
B

prices P instead of P A , the consumer's willingness to pay. Thus, it corresponds to
distance 0c above.
7). The RHS also represents a summation of areas under compensated
demand (supply) curves.
r A
N
rA A
rB A
PiA ¶M P;U
M P ;U - M P ;U = å ò PB
ds
i
¶s
i=1
(
)
(
(
)
N
)
= å ò PB X ci dP
i=1
PiA
i
--When the ith compensated demand (supply) is integrated, it is
evaluated at the prices PB for the 1 to (i - 1) goods and factors that have already been
integrated, and at prices PA for the (i + 1) to N goods and factors that have yet to be
integrated. Since the Xi are the compensated demands (supplies), the order of integration
makes no difference.
8). If the new product is the first good and it is “small” so that prices Pi
for i ≥ 2 remain unchanged, then:


PiA
M PA ; UA  M PB ; U A   B XicdPl


A
1


Pi
-- P is the price at which the demand curve intersects the price
axis, and P is the marginal cost price. Hence, the area defined by the stand figure above
is E P1B A providing D refers to the compensated demand curve
B
1
5
--The compensated demand curve lies to the left of the actual
demand curve because the consumer sacrifices income as price is lowered to remain at
the initial utility level.
9). Numerical techniques now exist for calculating compensated demand
curves from estimates of actual demand curves. No need to use Marshaliian consumer
surplus.
10). No need to estimate the compensated demand curve in the easy case
because the ability to break even is a sufficient condition for wanted the service.
8. Decreasing cost services and nonexclusive public goods
a. Decreasing cost services with zero marginal costs are often called public goods,
as are nonexclusive goods
b. The connection is that both goods exhibit nonrivalry in consumption—use by
any one person does not in any way diminish the ability of others to use the good.
c. They are quite different and the distinction between them is worth keeping
{Figure 9.19 here}
d. Consider a two person economy in which the demands of the two people for
the two types of goods are the same across the goods, d1 and d2 in the figure
opt
e. The optimal allocation of the nonexclusive good is XPG
, the output at which
the vertical summation of d1 and d2 hits the horizontal axis.
opt
f. The optimal allocation of the decreasing cost service is XDC
, the output at
which the horizontal summation of d1 and d2 hits the horizontal axis.
9. U.S. regulatory policy for decreasing cost services
{Figure 19.20 here}
a. Three natural choices for regulating natural monopolies such as public utilities
1). Give the franchise to a monopolist without regulation—no regulatory
administrative costs--rejected for fear monopolist will set the profit-maximizing output
and price (XM, PM)
2). Set the efficient marginal cost price and output (Xopt, PMC) and
subsidize the monopolist out of general tax revenues so that it covers the full cost of the
service
--Rejected since the subsidy breaks the link between payment and
use of the service-violates the benefits-received principle of paying for public services, a
deeply held view of fairness in the U.S.
3). Set price equals average cost, (XADF, PAC)-- used for almost all
decreasing cost services---Accords with the benefits-received principle
--Seen as a reasonable compromise between equity (the benefitsreceived principle) and efficiency (PAC likely to be close to PMC)
b. A difficulty with hard case services
{Figure 19.21 here}
1). Can't satisfy benefits-received principle at any price
2). Closest approximation is to minimize the deficit—same as
maximizing profit— (XMD,PMD)—e.g. in setting mass rail transit fares
6
3). Danger: at (XMD,PMD), the service may fail the all-or–none test even if
it could pass the test at PMC.
10. Appendix-Increasing returns implies decreasing average cost in many factor case
a. Consider Y = f(X1, …, XN) = f(Xi), where Xi = input i, i = 1, ..., N, and Y =
output.
b. Increasing returns implies
λ BY  f(λ  Xi ) ,  > 1
c. Differentiating with respect to Xk,
f(λX i )
λ Bf k (X i )=
=λf K (λX i ) , or
X K
λβ – 1 fK = fK (λXi) k = 1, …, N
d. The firm's cost minimization problem is
min  Pi X i
(X i )
s.t. Y  f(X i )
e. The FOC imply
Pi fi (X i )
=
i = 2,K,N
Pl fl (X i )
f. This condition is satisfied if the firm increases its use of the Xi by the scalar .
f i (λX i ) λ β –1f i (X i ) f i (X i ) Pi



f i (λX i ) λ β –1f1 (X i ) f i (X i ) P1
g. A scalar increase in all factors by  increases total cost by  but output by l b .
Therefore, the total cost function has the form
1
b
TC = kY
h. AC = TC/Y = kY
= kY(1-b )/b
AC 1  β
i.

k  Y ((1– β)/β –1)  0, for β  1
Y
β
(1/ b -1)
1). Average cost declines continuously as output increases with increasing
returns to scale.