The "short-run approach" calculates long-run producer optima and general equilibria bybuilding on short-run solutions to the producer's profit maximization problem and onprofit-based valuation of the fixed inputs. We outline this method and illustrate it on anexample of peak-load pricing.general equilibrium, fixed-input valuation, nondifferentiable joint costs, Wong-Viner Envelope Theorem, peak-load pricing

Andrew J Wrobel

Anthony Horsley

English

Research Papers in Economics

A PRACTICAL SHORT-RUN APPROACH TO MARKET EQUILIBRIUM
by
Anthony Horsley and Andrew J. Wrobel
London School of Economics
Contents:
Abstract
1. Introduction
2. Peak-load pricing with cross-price independent demands
References
Discussion Paper
No. TE/05/488
April 2005
The Suntory-Toyota International Centres for
Economics and Related Disciplines
London School of Economics and Political Science
Houghton Street
London WC2A 2AE
Tel.: 020-7955 6679
This work was started on ESRC grant R000232822; their ﬁnancial support is gratefully
acknowledged.Abstract
The “short-run approach” calculates long-run producer optima and general equilibria by
building on short-run solutions to the producer’s proﬁt maximization problem and on
proﬁt-based valuation of the ﬁxed inputs. We outline this method and illustrate it on an
example of peak-load pricing.
Keywords: general equilibrium, ﬁxed-input valuation, nondiﬀerentiable joint costs,
Wong-Viner Envelope Theorem, peak-load pricing.
JEL Nos.:D 2 4 ,D 4 1 ,D 5 8 .
c ° by Anthony Horsley and Andrew J. Wrobel. All rights reserved. Short sections of
text, not to exceed two paragraphs, may be quoted without explicit permission, provided
that full credit, including c ° notice, is given to the source.
Contact address: Anthony Horsley, Department of Economics, London School of Eco-
nomics, Houghton Street, London WC2A 2AE, United Kingdom.
Tel.: +44-1923-442419
E-mail: LSEecon123@mac.com
URL: http://www.lseecon123.com1 Introduction
The “short-run approach” is a scheme for calculating long-run producer optima and
general equilibria by building on short-run solutions to the producer’s proﬁt maximization
problem, in which the capital inputs and natural resources are treated as ﬁxed. These
ﬁxed inputs are valued at their marginal contributions to the operating proﬁts and, where
possible, their levels are then adjusted accordingly. Since short-run proﬁti sac o n c a v e
but generally nondiﬀerentiable function of the ﬁxed inputs, their marginal values are
deﬁned as nonunique supergradient vectors.1 Also, they usually have to be obtained
as solutions to the dual programme of ﬁxed-input valuation because there is rarely an
explicit formula for the operating proﬁt: for examples of this use of duality, see [4]
and [7].2 Therefore, a general framework for the short-run approach requires the use of
subdiﬀerential calculus and duality of convex programmes. This is fully set up in [6].
Here, we illustrate the short-run approach on an example in which the operating proﬁt
function can be calculated and diﬀerentiated directly.
The short-run approach starts with ﬁxing the producer’s capacities k and optimizing
the variable quantities, viz., the outputs y and the variable inputs v. For a competitive,
price-taking producer, the optimum quantities, ˆ y and ˆ v,d e p e n do nt h e i rg i v e np r i c e s ,p
and w,a sw e l la so nk. The primal solution (ˆ y and ˆ v) is associated with the dual solution
ˆ r, which gives the unit values of the ﬁxed inputs (with ˆ r · k as their total value); the
optima are, for the moment, taken to be unique for simplicity. When the goal is limited
to ﬁnding the producer’s long-run proﬁt maxima, it can be achieved by part-inverting the
short-run solution map of (p,k,w) to (y,−v;r) so that the prices (p,r,w) are mapped
to the quantities (y,−k,−v).T h i si sd o n eb ys o l v i n gt h ee q u a t i o nˆ r(p,k,w)=r for k
and substituting any solution into ˆ y(p,k,w) and ˆ v(p,k,w) to complete a long-run proﬁt-
maximizing input-output bundle. Such a bundle may be unique but only up to scale if
the returns to scale are constant (making ˆ r h o m o g e n e o u so fd e g r e ez e r oi nk).
Even within the conﬁnes of the producer problem, this approach saves eﬀort by build-
ing on the short-run solutions that have to be found anyway: the problems of plant oper-
ation and plant valuation are of central practical interest and always have to be tackled
by producers. But the short-run approach is even more important as a practical method
for calculating market equilibria. For this, with the input prices r and w taken as ﬁxed for
simplicity, the short-run proﬁt-maximizing supply ˆ y(p,k,w) is equated to the demand for
the products ˆ x(p) to determine the short-run equilibrium output prices p?
SR (k,w).T h e
capacity values ˆ r, calculated at the equilibrium prices p?
SR (k,w) with the given k and w,
1The set of all supergradients, a.k.a. the superdiﬀerential, of a concave function Π is deﬁned by b ∂Π
:= −∂ (−Π),w h e r e∂ denotes the subdiﬀerential of a convex function. For subdiﬀerential calculus, see,
e.g., [9].
2In the two examples, the short-run proﬁt is diﬀerentiable with respect to the ﬁxed inputs (k)i ft h e
output price for the time-diﬀerentiated output good (p) is a continuous function of time. This condition
is veriﬁed in [5] for the market equilibrium price.
1are only then equated to the given capacity prices r to determine the long-run equilibrium
capacities k? (r,w), and hence also the long-run equilibrium output prices and quantities
(by substituting k? in the short-run equilibrium solution).3 In other words, the determi-
nation of investment is postponed until after the equilibrium in the product markets has
been found: the producer’s long-run problem is split into two–that of operation and that
of investment–and the equilibrium problem is “inserted” in-between. Since the operat-
ing solutions usually have relatively simple forms, doing things in this order can greatly
ease the ﬁxed-point problem of solving for equilibrium: in our example the problem is
elementary if approached in this way. Furthermore, unlike the optimal investment of the
pure producer problem, the equilibrium investment k? has a deﬁnite scale (determined by
demand for the products). Put another way: ˆ r(p?
SR (k,w),k,w),t h ev a l u et ob ee q u a t e d
to r,i snot homogeneous of degree zero in k like ˆ r(p,k,w). Thus one can keep mostly
to single-valued maps and avoid dealing with multi-valued correspondences. And ﬁnally,
like the short-run producer optimum, the short-run market equilibrium is of interest in
itself.
This approach to long-run market equilibrium is illustrated in Figure 1, on the as-
sumption that iterative methods are used to solve the demand-supply equation for p and
the price-value equation for k (which correspond to the inner and the outer loops in
Figure 1).
As an easy but instructive introduction to this method, we rehearse Boiteux’s treat-
ment of the simplest peak-load pricing problem, viz., the problem of pricing the services
of a homogeneous capacity that produces a nonstorable good with cyclic demands, such
as electricity. A direct calculation of the long-run equilibrium poses a ﬁxed-point problem
but, with cross-price independent demands, the short-run equilibrium can be obtained
by intersecting the supply and demand curves for each time instant separately, and the
long-run equilibrium is found by adjusting the capacity so that its unit cost equals the
unit operating proﬁt (which is the total capacity charge over the cycle). With discretized
time, this solution is given in [1, 3.2—3.3];4 we give its continuous-time version. Elsewhere,
in [6], we develop this idea into a frame for the analysis of investment and pricing by any
industry that supplies a range of commodities–such as a good diﬀerentiated over time,
locations or events–and we apply it to augment the rudimentary one-station model to
a full-blown continuous-time equilibrium model of electricity pricing with a diverse tech-
nology, including energy storage and hydro as well as thermal generation, and with a
3The short-run approach to equilibrium might also be based on short-run cost minimisation, in which
not only the capital inputs (k) but also the outputs (y)a r ek e p tﬁxed and are shadow-priced in the dual
problem, but such cost-based calculations are usually much more complicated than those using proﬁt
maximisation: see [6] for a comparison.
4Boiteux’s work is also presented by Drèze [2, pp. 10—16], but the short-run character of the approach
is more evident in the original [1, 3.2—3.3] because Boiteux discusses the short-run equilibrium ﬁrst, before
using it as part of the long-run equilibrium system. Drèze mentions the short-run equilibrium on its own
only as an afterthought [2, p. 16].
2general, cross-price dependent demand.
Figure 1: Flow chart for iterative implementation of the short-run approach to long-run
market equilibrium. For simplicity, all input prices are assumed to be ﬁxed (in terms of
the numeraire).
2 Peak-load pricing with cross-price independent de-
mands
Consider the problem of pricing, over the demand cycle, the services of a homogeneous
productive capacity with a unit capital cost r and a unit running cost w. The technology
can be interpreted as, e.g., electricity generation from a single type of thermal station
with a fuel cost w (in $/kWh) and a capacity cost r (in $/kW) per period. The cycle
is represented by a continuous time interval [0,T]. Demand for the time-diﬀerentiated,
nonstorable product, Dt (p),i sa s s u m e dt od e p e n do n l yo nt h et i m et and the current price
p. As a result, the short-run equilibrium can be found separately at each instant t,b y
intersecting the demand and supply curves in the price-quantity plane. This is because,
with this technology, short-run supply is cross-price independent: given a capacity k,t h e
3supply is
S (p,k,w)=
⎧
⎨
⎩
0 for p <w
[0,k] for p = w
k for p >w
(1)
where p is the current price. That is, given a time-of-use (TOU) tariﬀ p (i.e., given a
price p(t) at each time t), a proﬁt-maximizing output trajectory is any selection from the
correspondence t 7→ S (p(t),k,w).W h e n Dt (w) >k , the short-run equilibrium TOU
price, p?
SR (t,k,w), exceeds w by whatever is required to bring the demand down to k
(Figure 2a). The total premium over the cycle is the unit operating proﬁt ,w h i c hi nt h e
long run should equal the unit capacity cost r–i.e., the long-run equilibrium capacity,
k? (r,w), can be determined by solving for k the equation
r =
Z T
0
(p
?
SR (t,k,w) − w)
+dt (2)
where π+ =m a x {π,0} is the nonnegative part of π (i.e., by equating to r the shaded
area in Figure 2b). Put into the short-run equilibrium price function, the equilibrium
capacity gives the long-run equilibrium price
p
?
LR (t;r,w)=p
?
SR (t,k
? (r,w),w).( 3 )
For comparison, to calculate the long-run equilibrium directly requires timing the
capacity charges so that they are borne entirely by the resulting demand peaks–i.e., it
requires ﬁnding a density function γ ≥ 0 such that
Z T
0
γ (t)dt =1 and if γ (t) > 0 then y(t)=s u p
τ
y(τ) (4)
where: y(t)=Dt (p(t)) and p(t)=w + rγ (t).
This poses a ﬁxed-point problem that, unlike the short-run approach, is not much sim-
pliﬁed by cross-price independence of demands.5
Since the operating proﬁti sΠSR (p,k,w)=k
R T
0 (p(t) − w)+dt, the break-even con-
dition (2) can be rewritten as r = ∂ΠSR/∂k, i.e., it can be viewed as equating the
capital input’s price to its proﬁt-imputed marginal value. This is the ﬁrst-order neces-
sary and suﬃcient condition for a proﬁt-maximizing choice of the investment k: together
with a choice of output y that maximizes the short-run proﬁt (SRP), such a choice of k
maximizes the long-run proﬁt (LRP), and thus turns the short-run equilibrium into the
long-run equilibrium.
5In terms of the subdiﬀerential, ∂C, of the long-run cost (5) as a function of output, the ﬁxed-point
problem is to ﬁnd a function p such that p ∈ ∂CLR (D(p)),w h e r eD(p)(t)=Dt (p(t)) if demands are
cross-price independent.
4When the producer is a public utility, competitive proﬁt maximization takes usually
the form of marginal-cost pricing. In this context, the equality r = ∂ΠSR/∂k,o rr
= ∇kΠSR when there is more than one type of capacity, guarantees that an SRMC price
system is actually an LRMC. The result applies to any convex technology–even when
the costs are nondiﬀerentiable, and the marginal cost has to be deﬁned by using the
subdiﬀerential as a generalized, multi-valued derivative. This is so in the above example
of capacity pricing, since the long-run cost
CLR (y(·),r,w)=w
Z T
0
y(t)dt + r sup
t∈[0,T]
y(t) (5)
is nondiﬀerentiable if the output y has multiple peaks: indeed, for every γ satisfying (4),
the function p = w+rγ represents a subgradient of CLR with respect to y. The function
γ is the density of a nonunique distribution of the total capacity charge (r)o v e rt h e
multiple peaks. And multiple peaks are more of a rule than an exception in equilibrium
(note the peak output plateau in Figure 2d here, and see [3] for an extension to the case
of cross-price dependent demands). Similarly, the short-run cost
CSR (y(·),k,w)=
½
w
R T
0 y(t)dt if 0 ≤ y ≤ k
+∞ otherwise
(6)
is nondiﬀerentiable if supt y(t)=k.A tat i m et when y(t)=k, an instantaneous SRMC
is the sum of the unit operating cost (w) and an indeterminate capacity premium. In
Figure 2a, the nondiﬀerentiability shows in the (inﬁnite) vertical interval [w,+∞) that
represents the multi-valued instantaneous SRMC at y = k.6 In Figure 2c, it shows as a
kink, at y = k, in the graph of the instantaneous cost function
cSR (y)=
½
wy if 0 ≤ y ≤ k
+∞ otherwise (7)
(which gives CSR (y) as
R T
0 cSR (y(t))dt,s ot h a taT O Up r i c ep is an SRMC at y if
and only if p(t) is an instantaneous SRMC at y(t) for each t). With this technology,
CSR is therefore nondiﬀerentiable whenever k is the cost-minimizing capital input for
the required output y: cost-optimality of k means merely that it provides just enough
capacity, i.e., that k =S u p ( y). This condition, being quite unrelated to the input
prices r and w, obviously cannot ensure that an SRMC price system is an LRMC. To
guarantee this, one must strengthen it to the condition that r =
R T
0 (p − w)+dt in this
example or, generally, that r = ∇kΠSR (or that r belongs to the supergradient set b ∂kΠSR,
should ΠSR be nondiﬀerentiable in k).7 The capital’s cost-optimality would suﬃce for
6The SRMC and the short-run supply correspondences are inverse to each other, i.e., have the same
graph: in Figure 2a, the broken line is both the supply curve and the SRMC curve.
7This condition (r = ∇kΠSR) is stronger than cost-optimality of the ﬁxed inputs when p is an SRMC.
5t h eS R M Ct ob et h eL R M Ci ft h ec o s t sw e r ed i ﬀe r e n t i a b l e ;t h i si st h eW o n g - V i n e r
Envelope Theorem. The preceding remarks show how to reformulate it to free it from
diﬀerentiability assumptions. This is detailed in [6] and [8].
Figure 2: Short-run approach to long-run equilibrium of supply and (cross-price indepen-
dent) demand for thermally generated electricity: (a) determination of the SR equilibrium
price and output for each instant t, given a capacity k; (b) and (d) trajectories of the
SR price and output; (c) the SR cost curve. When k is such that the shaded area in (b)
equals r, the SR equilibrium is the LR equilibrium.
6References
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ter 4), ed. by J. R. Nelson. Engelwood Cliﬀs, NJ: Prentice Hall. (A translation of
“La tariﬁcation des demandes en pointe: application de la théorie de la vente au
cout marginal”, Revue Général de l’Electricité, 58 (1949), 321—340.)
[2] Drèze, J. H. (1964): “Some postwar contributions of French economists to theory
and public policy”, American Economic Review, 54 (supplement, June 1964), 1—64.
[3] Horsley, A., and A. J. Wrobel (2002): “Boiteux’s solution to the shifting-peak prob-
lem and the equilibrium price density in continuous time”, Economic Theory,2 0 ,
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[4] Horsley, A., and A. J. Wrobel (2002): “Eﬃciency rents of pumped-storage plants and
their uses for operation and investment decisions”, Journal of Economic Dynamics
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[5] Horsley, A., and A. J. Wrobel (2005): “Continuity of the equilibrium price density
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http://www.springerlink.com/index/10.1007/s00199-004-0568-3
[6] Horsley, A., and A. J. Wrobel (2005): “The short-run approach to long-run equi-
librium: a general theory with applications”, CDAM Research Report LSE-CDAM-
2005-02.
http://www.cdam.lse.ac.uk/Reports/reports2005.html
[7] Horsley, A., and A. J. Wrobel (2005): “Proﬁt-maximizing operation and valuation
of hydroelectric plant: a new solution to the Koopmans problem”, CDAM Research
Report LSE-CDAM-2005-03.
[8] Horsley, A., and A. J. Wrobel (2005): “A reformulation of the Wong-Viner Enve-
lope Theorem for subdiﬀerentiable functions”, CDAM Research Report LSE-CDAM-
2005-05.
http://www.cdam.lse.ac.uk/Reports/reports2005.html
[9] Tiel, J. van (1984): Convex analysis. Chichester-New York-Brisbane: Wiley.
7

Boiteux’s solution to the shifting-peak problem and the equilibrium price density in continuous time”,

Continuity of the equilibrium price density and its uses in peak-load pricing”,

Wrobel (2002): “Eﬃciency rents of pumped-storage plants and their uses for operation and investment decisions”,

Wrobel (2005): “A reformulation of the Wong-Viner Envelope Theorem for subdiﬀerentiable functions”,

A Practical Short-run Approach to Market Equilibrium

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A Practical Short-run Approach to Market Equilibrium

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