Irreversible Investment, Rental Rates and

Long-Run Cost

Robert D. Cairns∗

Department of Economics

McGill University

855 Sherbrooke St. W.

Montreal Canada H3A 2T7

email: robert.cairns@mcgill.ca

April 2008

∗I thank Leonard Cheung, Graham Davis, Kevin Fox, Chris Green, Daniel Leonard, Maxim

Sinitsyn, Alice Nakamura, Kwang Ng, Michel Poitevin, Bill Schworm and Dan Usher for comments,

and FCAR and SSHRCC for financial support.

1

Irreversible Investment, Rental Rates and Long-Run Cost

Abstract. Optimizing agents discount irreversible (or sunk) costs into their in-

vestment decisions. A technology involving sunk costs is represented as a present

value. A simple, dynamic analysis of the technology provides an expression for the

rental or recovery schedule of sunk cost over time and hence for long-run cost. If a

firm’s discounted profit is zero, unique rental payments are defined and long-run cost

can be represented as a unique, intertemporal flow. The rentals depend on demand

conditions as well as technical conditions. If discounted profit is positive, rental pay-

ments are not unique and long-run cost cannot be represented as a unique flow. In

general, the envelope property applies to a constrained total cost function, which is a

stock. The findings point toward generalizations of the expressions for capital rentals

and long-run costs in theoretical and empirical analysis.

Key words: irreversible investment, sunk cost, long-run cost, rental rates, capital

cost recovery, average cost, marginal cost

JEL Classifications: D20, L00, M40

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INTRODUCTION

It would be hard to overstate the economic significance of sunk cost. Any irre-

versible decision involves a sunk cost. Simple observation (e.g. Waverman 1989: 97)

as well as empirical estimates (e.g. Ramey and Shapiro 1998, Asplund 2000) suggest

that in many industries sunk costs are substantial. Sunk costs influence decisions

under uncertainty (Dixit and Pindyck 1994) and of firms with market power (Sutton

1991). They play a central role in issues of governance of the firm (Williamson 2005).

In a landmark study of the interaction of sunk cost and market size on industry

structure, Sutton (1991) envisages two types of sunk cost, exogenous sunk, or set-up,

cost and endogenous sunk cost such as advertising. His approach has been seminal

to further discussion (e.g. Ellickson 2006). McAfee, Mialon and Williams (2004),

Carlton (2004) and Schmalensee (2004) present perspectives on, among other things,

the effect of sunk costs in creating commitment and hence entry barriers. Martin’s

(2002) study of an infinitely lived project involves a point input of capital comprising

both an exogenous set-up cost and an endogenous capacity choice.

In Sutton’s model, the attribution or recovery of sunk costs over time is moot, as

there is only a single time period over which firms produce. Microeconomic theory

holds that in the long run a dynamic firm’s technology can be represented as a flow,

using a cost function which is dependent on output and input prices. Cost functions

are routinely used by economists in discussions ranging from informal evaluations to

policy analysis to rigorous theory. Attributed flows are used in productivity analysis,

in the valuation of returns to education, in index-number formulas, in models of in-

variant taxation and generally in the measurement of capital and of income. They are

fundamental to expressing and evaluating incentives for regulation and procurement

(Laffont and Tirole 1993). They affect the qualitative definition of nonrenewable-

resource rents and the interpretation of Hotelling’s rule (Cairns and Davis 2007).

3

In practice they are used in evaluating management performance because they are

considered to help in the measurement of profitability. They are used by banks and

other investors in the decision to allot capital to firms. They are fundamental to the

long-run, empirical analysis of industry.

Defining the long-run cost function requires a unique allocation of sunk cost to turn

it into a flow. The common allocation is the product of a rental price and the sunk

capital stock. For many types of sunk capital, however, there is no rental market

or observable rental price. Even if there is a rental market, as for some buildings,

vehicles or tools, by sinking costs firms reveal that they prefer not to use that market.

Observed rental prices may not be valid for a firm with sunk capital. Moreover, rental

markets do not eliminate the analytic problem of sunk capital; they simply displace

it. Capital is sunk in the hands of the owner.

The present paper examines the representation of a technology involving a sunk

investment. A central question is whether it is possible to disaggregate the sunk cost

into a flow per unit time as in a long-run cost function. Another is the relationship

of the long- and short-run cost functions, including the envelope property. The ideas

are introduced in the least complicated model possible, of capital as a point input.

The point of departure is the classic set of assumptions of linear investment cost,

zero profit, no capacity constraint and no physical deterioration. These assumptions

are relaxed in turn. The results do not depend on uncertainty. Even so they are

fundamental aspects of irreversibility.

Because they involve simple, small steps and are no deeper than discounting and

partial differentiation, the mathematical manipulations are transparent. As the small

steps are taken, the analysis veers from traditional theory. That the results are

counter to current understanding is clear from the universal acceptance of conven-

tional representations, e.g. in major graduate micro textbooks or studies of the

measurement of monopoly power.

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The technological definition of long-run cost is as a stock, not a flow. The envelope

property applies to a stock, a constrained, total cost function. Short-run cost is also

a stock but short-run total cost is not defined.

The allocation of sunk cost is a generalization of the concept of rental payment

or user cost. A rental schedule must be in place in the long run, before capital is

sunk, and be (expected to be) implemented to recover sunk costs as quasi-rents in

the short run (cf. Baumol 1971). If the present value of profits is zero, there is a

determinate way to define long-run cost as a flow. The allocation of the sunk cost

over time, however, is dependent on demand conditions, not on just the technology

as held in cost theory. If the present value of profits is positive there is, within limits,

indeterminacy in the rental schedule. Generalizations of cost theory are summarized

in a series of propositions that, among other things, delimit the economically valid

sets of rental schedules.

AVOIDABLE COST AND SUNK COST

At time t, let the vector of prices of variable factors xt be wt > 0. Suppose

that a profit-maximizing firm uses its sunk capital stock K ≥ 0 to produce output qt,

incurring avoidable cost c(qt, wt,K). Avoidable cost has the properties of neoclassical

variable cost, in which

c (q, w,K) , minx{w · x |q can be produced using x and K } ;

c (q, w, 0) = ∞ if q > 0; c (0, w,K) = 0; and c (q, w,K) is differentiable for positive

values of q and K. The signs of the partial derivatives are axiomatic in cost theory:

cq = ∂c/∂q ≥ 0 and cK = ∂c/∂K ≤ 0. Not all avoidable costs must be variable; there

may be a positive quasi-fixed cost, ψt, such as for heating or lighting.

The simplest representation of sunk capital is as a point input.1 Suppose that a

1It is argued below that the main conclusions are general.

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firm invests a scalar stock of capital K at time t = 0. Let the cost of installing

this stock at time t = 0, using variable factors y with prices w0 > 0, be formally

represented using a neoclassical cost function,

Φ (K,w0) , miny{w0 · y | K can be installed using y} ,

where Φ (0, w0) = 0, limK↓0Φ (K,w0) ≥ 0 and ∂Φ (K,w0) /∂K > 0 for K > 0. The

choice of K is assumed to be endogenous; an exogenous cost can be represented if

limK↓0Φ (K,w0) > 0.2 The capital is assumed to be sunk, by which is meant specific

to the firm, having no value in any other use. The assumption abstracts from partially

sunk costs in order to focus attention on properties of sunk cost.3

A three-period (t = 0, 1, 2) model is a simple vehicle to present the ideas of this

paper. Let there be no physical deterioration of capital until time t = 2, when it

becomes useless. (Capital is like a light bulb or one-hoss shay in this simple model.)

The capital is used to produce outputs q1 and q2 in periods 1 and 2. The total cost

of producing these outputs, in constant prices and given a capital stock K, is

C (q1, q2, w0, w1, w2, K) = Φ (K,w0) +c (q1, w1,K)

1 + r+c (q2, w2,K)

(1 + r)2. (1)

The sunk cost Φ is added to the avoidable costs of producing the output vector but

is not expressed as a flow over periods 1 and 2.

In the long run the firm minimizes total cost of producing (q1, q2), including capital

cost. The investment decision is a choice from a menu of short runs, each element

corresponding to a different level of the capital stock. The effect of sinking cost is

recognized in this decision. Therefore, the long run cannot be analyzed independently

of the short run.2If all capital cost is exogenous (e.g. a taxi cab and license), there is no choice of an optimal

value of K. Analysis proceeds as below, but without the effect of this choice on other decisions.3The properties of Φ are broadly consistent with findings by Cooper and Haltiwanger (2006) for

adjustment costs of capital.

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Theminimization is through backward induction. First c (q2, w2, K), then c (q1, w1, K)+

c (q2, w2,K) / (1 + r) and finally C are minimized. Let optimal values be indicated

by circumflexes. If cK < 0, the optimal capital stock is the solution to

−cK(q̂1, w1, K̂)1 + r

+−cK(q̂2, w2, K̂)

(1 + r)2=

∂Φ³K̂, w0

´∂K

. (2)

Equation (2) states that the cost of the marginal unit of capital is offset by the

discounted savings in costs that it provides. Its solution expresses the optimal capital

stock K̂ as a function of the outputs and the input prices,4

K̂ = κ (q1, q2, w0, w1, w2) . (3)

Long-run cost is expressed by the RHS of equation (1) prior to the investment:

L (q1, q2, w0, w1, w2) = Φ (κ, w0) +c (q1, w1,κ)

1 + r+c (q2, w2,κ)

(1 + r)2. (4)

The functions L and κ have the same arguments.5

By construction, the constrained and unconstrained total-cost functions, C and L,

satisfy the regularity conditions of joint-cost functions. They can be used to represent

the aggregate technology of producing the output vector (q1, q2) in periods 1 and 2.

In the determination of the optimal output levels, the profit function, which is also

implied by the technology, is also relevant. At time t, let the firm face the linear

inverse demand pt (qt |αt ), with parameters (α1t,α2t, ...,αkt) = αt. The firm’s net

cash flow in period t is

Ft (qt, wt,ψt, K |αt ) = qtpt (qt |αt )− c (qt, wt, K) .4The conditions of the implicit function theorem must hold, but the assumptions of the model

are usually strong enough to satisfy them. It is possible to entertain the case that cK = 0 if there

is a capacity constraint. See below. The interest rate is viewed as a parameter of this problem.

Consistently with much of cost theory, effects of changes in that rate are not considered.5The need to incur the capital cost Φ as a step in the definition of long-run total cost L is

evocative of the “roundaboutness” of production using capital discussed in the nineteenth century.

Here, one technology is used purely to set up another technology for the production of final goods.

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The goal of the firm is to maximize its discounted cash flow or long-run profit, i.e. to

maxK,q1,q2

∙−Φ(K,w0) +

F11 + r

+F2

(1 + r)2

¸= Π

³K̂, q̂1, q̂2

´. (5)

Profit, also a function of (q1, q2, w0, w1, w2), must be non-negative, for the firm can

attain Π (0, 0, 0) = 0 by not investing. In the long run all cost is avoidable.

The short run is the state of affairs where the firm has sunk its capital stock (where

it has chosen a value of K). In the short run, the firm’s goal is to maximize the net

present value over the remainder of the path, given K. In period 1 the short-run

problem is to

maxq1,q2

∙F11 + r

+F2

(1 + r)2

¸= π1 (q̂1, q̂2 |K ) ; (6)

in period 2 it is to

maxq2

F21 + r

= π2 (q̂2 |K ) . (7)

In problems (6) and (7) the sunk capital K plays a role but the sunk cost Φ (K,w0)

does not. If not producing is optimal at time t then, for any q > 0, the net cash flow

Ft (q, wt,ψt,K |αt ) ≤ Ft (0, wt,ψt,K |αt ) = 0, or p ≤ c (q, w,K) /q. If it is optimal to

produce output q > 0, then the inequalities are reversed and p ≥ c (q, w,K) /q. In

the short run (given the capital stock), for the firm to find it worthwhile to produce,

price must be at least equal to short-run average avoidable cost. The firm expands

output in response to increases in demand, “beginning” from the point at which

p = minq c (q, w,K) /q such that p+ qp0 = cq (q, w,K) (with p0 = 0 for a price taker).

The first-order condition with respect to qt in all three problems (in long and short

runs) is that current marginal revenue be equal to marginal avoidable cost.

The discussion thus far can be summarized as follows.

Proposition 1 (i) Conditions applying specifically to the long run. (a) Discounted

profits are non-negative. (b) The long-run cost functions, constrained and uncon-

strained, are sufficient statistics to represent the technology in the long run. (c) Sunk

cost affects (only) the long-run decision.

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(ii) Long-run cost is a present value.

(iii) Conditions of internal optima in both the long and short runs. (a) Price is

no less than average avoidable cost. (b) Marginal revenue (price for a price taker) is

equal to marginal avoidable cost.

Part (i) confirms much of traditional analysis, but Part (ii) departs from it. Part

(iii) states that sunk cost is not allocated in the determination of long-run optimality

conditions.

It is well known that for a static, multiproduct firm there are many possible ways

to allocate common costs (Baumol, Panzar and Willig 1982). One’s initial intuition

is that a firm with sunk costs is a special case of a multiproduct firm. For the

intertemporal model, current incremental cost can be defined, namely, the avoidable

cost, c (q, w,K). Similarly to fixed cost in a multiproduct firm, sunk capital cost,

which is a common cost to production in periods 1 and 2, is not incorporated into

incremental cost.

In a dynamic firm, however, there are links within the product set that do not

exist for a static multiproduct firm. There is a temporal ordering of the products.

There is a series of nested optimizations inherent in the grand optimization, as in

problems (5), (6) and (7). Links among the products of a firm producing through

time are surely the reason that Baumol et al. conclude that there is no contestable

equilibrium for some types of dynamic firm. Can the links be used to allocate capital

costs to define a long-run cost function as in traditional analyses?

ZERO-PROFIT EQUILIBRIUM

Let maximum profit be zero: Π³K̂, q̂1, q̂2

´= 0. This is a special condition in the

analysis of a firm, but a benchmark in microeconomics. A frequent assumption is

that investment cost is linear. This very important special case is discussed first, and

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then discussion turns to non-linear investment cost.

Linear Investment Cost

Let Φ(K,w0) = PKK,PK > 0 and ΦK (K,w0) = PK.6 Baumol (1971) carefully sets

out the traditional analysis of this important special case and occasionally relaxes

certain assumptions. His aim is to express long-run marginal cost as the sum of

marginal operating cost and an optimal rental payment for the use of capital.

By equation (2), an allocation of −K̂cK(q̂t, wt, K̂) to times t = 1, 2 exactly offsets

the investment cost PKK̂. At any time t, total (current) cost is the sum of avoidable

cost and allocated capital cost,

T` (q, w,K) = c(q, w,K)−KcK(q, w,K). (8)

The rental payment per unit of capital, −cK, is each unit’s contribution to reducing

avoidable cost.

In the long run, equation (2) defines the choice of the optimal capital stock as a

function of the other variables of the problem:

K̂ = κ` (q1, q2, w0, w1, w2) .

Long-run cost can be written using expressions for single-period total cost correspond-

ing to that in equation (8):

L` (q1, q2, w0, w1, w2) = PKκ` +c (q1, w1,κ`)

(1 + r)+c (q2, w2,κ`)

(1 + r)2

=2Xt=1

c (qt, wt,κ`)− κ`cK (qt, wt,κ`)

(1 + r)t, (9)

In contrast to traditional representations, the function κ` (q1, q2, w0, w1, w2) appears

6In this case the investment technology exhibits constant returns to scale and PK is a unit-cost

function: for some function γ, PK = γ (w0).

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in each term: single-period costs depend on the prices and outputs of all periods, not

just the current period; the cost function L` is a present value, a stock not a flow.

Since (maximized) profits are zero, discounted revenues equal discounted costs

(current and capital) so that

p1q11 + r

+p2q2

(1 + r)2=

2Xt=1

(c− κ`cK)t(1 + r)t

. (10)

If demand and cost conditions are stationary then output and total cost are the same

in both periods. Since discounted profits Π³K̂, q̂1, q̂2

´are zero, price is equal to

average cost in each period: pt = (c− κ`cK)t /qt.

However, it is not a criterion of the firm’s long-run decision that price be no less

than average cost at any time. If any demand or cost condition is not stationary,

the allocations of sunk capital cost and hence the functions c (q, w,K) are different

in the two time periods. Price may fall short of average cost at some times. A long-

run participation constraint, usually considered to be a short-run constraint, for not

shutting down in any period is that p ≥ c (q, w,K) /q.

These results for this special case can be summarized as follows.

Proposition 2 Let profits be zero and investment costs be linear.

(i) The allocation of sunk cost to form a flow total cost is unique. However, it

depends on demand conditions throughout the life of the project, not solely on tech-

nological conditions.

(ii) That price be no less than current avoidable cost is the appropriate long-run as

well as short-run participation constraint.

(iii) That price be no greater than average total cost is not a long-run constraint

but rather a defining property of average total cost.

Part (iii) does not point to a participation constraint but rather is a property of

average cost resulting from the assumption that profit is zero.

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This special case is prominent in investment theory because of its simplicity. Con-

stant returns to scale in the investment-cost function PKK allow, among other things,

the use of the partial derivative of avoidable cost, cK , to allocate sunk cost. Capi-

tal cost is not explicitly expressed as a rental payment, however. Doing so becomes

necessary when investment costs take a more general form.

Nonlinear Investment Cost

Now let ΦKK (K,w0) 6= 0. The investment cost Φ³K̂, w0

´figures in the participa-

tion constraint, Π³K̂, q̂1, q̂2

´≥ 0, while the marginal price ΦK

³K̂, w0

´appears in

optimality condition (2).7 This marginal price supports the decision to invest K̂, and

the condition can be solved to find, for some function κ, that K̂ = κ (q1, q2, w0, w1, w2).

Long-run cost is again a stock, expressed as a function of outputs and factor prices,

L (q1, q2, w0, w1, w2) = Φ (κ, w0) +c (q1, w1,κ)

1 + r+c (q2, w2,κ)

(1 + r)2. (11)

Also let the revenue function be represented by the more general form, Rt (qt |α0t ) , t =

1, 2, with parameters α0t, so that prices may be nonlinear as well. In this case, net

cash flow is given by Ft (qt, wt,ψt,K |α0t ) = Rt (qt |α0t ) − c (qt, wt, K). With profits

Π³K̂, q̂1, q̂2

´still assumed to be zero, a unique long-run allocation of capital cost over

time can be determined as follows. Let the remaining (more formally, undepreciated)

value of the project at time t be represented by V³K̂, t

´. Economic depreciation of

the project is defined to be negative the change in its value (Samuelson 1937):

−∆V³K̂, 0

´=

F11 + r

+F2

(1 + r)2− F21 + r

= F1−r∙F11 + r

+F2

(1 + r)2

¸= F1−rV

³K̂, 0

´;

(12)

−∆V³K̂, 1

´=

F21 + r

= F2 − rF21 + r

= F2 − rV³K̂, 1

´. (13)

7In a sense, when ΦKK (K,w0) 6= 0, the firm is not a price taker in the input market for capital.

Typically, ΦKK < 0. Although this cost is not convex, under conditions given by Cairns (1998) an

optimal level of investment can be found.

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Since profit is zero, the undepreciated value of the sunk capital can be identi-

fied with the undepreciated value of the project. Baumol et al. (1982: 384ff.) call

rV³K̂, t

´−∆V

³K̂, t

´, interest plus depreciation at time t, the payment to capital.

The distinction between economic and accounting costs is evident here. Long-run

economic cost includes rV −∆V , the full payment to capital or the economic recov-

ery of capital, while accounting cost includes only depreciation, −∆V , the accounting

recovery of capital.

Payments having present value Φ³K̂, w0

´must be recoverable if the firm is to

invest K̂ in the long run. They constitute an allocation of capital cost over time.

Since discounted profits are zero, the payment to capital is equal to the net cash flow,

Ft. In the context of equation (10), the allocation of capital cost is (pq − c)t rather

than −K (cK)t. In general, the discounted sum of the latter values is not the cost of

capital, Φ (K,w0).

Given the identification of the value of the capital stock with the value of the

project, total cost at time t, including sunk capital cost, can be defined to be

Tt (q̂1, q̂2, w0, w1, w2) = c³q̂t, wt, K̂

´+ rVt −∆Vt.

As in the traditional equilibrium of the firm, marginal revenue (price for a price taker)

equals marginal cost and price equals average cost (Tt/q̂t).

It may be that Ft < rVt in some situations. In this case, ∆Vt > 0: the project

appreciates. Appreciation is defined by ∆Vt = rVt − Ft. For example, if Ft = 0,

∆Vt = rVt.

Let the rate of depreciation at time t be represented by δt = −∆Vt/Vt. (If there

is appreciation, the rate of depreciation is negative.) The rate of depreciation is

intrinsic to the equilibrium of this particular firm. Total cost is the sum of avoidable

cost c³q̂, w, K̂

´and a rental rate (r + δ) applied to the undepreciated capital stock,

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as identified with the project value V :

Tt (q̂1, q̂2, w0, w1, w2) = c³q̂t, wt, K̂

´+ (r + δt)Vt

In the long run, the rental payment (r + δt)Vt is a part of the opportunity cost of

capital.

The definition implements the common allocation mentioned in the introduction.

The allocation is obscured in the analysis of a linear investment cost but is implicit:

in equation (8),

2Xt=1

ptqt − c (qt, wt,κl)(1 + r)t

= V (0,κl) = PKκl =2Xt=1

−κlcK (qt, wt,κl)(1 + r)t

and

p2q2 − c (q2, w2,κl)1 + r

= V (1,κl) =−cK (q2, w2,κl)

1 + r.

When investment cost is linear, (r + δt)Vt = −cK (qt, wt,κl).

When investment cost is not linear, however, the partial derivative cannot be substi-

tuted for (r + δt)Vt. In general, the rental payment to capital at any time, (r + δ)V ,

depends on the pattern of output and hence demand at other times; the components

rV and δV , and their sum, may be irregular through time. Of the components of

cost, only the avoidable cost, c³q̂, w, K̂

´, depends purely on the technology and input

prices.

Equations (6) and (7) imply that short-run cost, too, is a present value rather than

a flow per period. In period 1, short-run cost is

S1 (q1, q2, w1, w2,K) =c (q1, w1,K)

1 + r+c (q2, w2,K)

(1 + r)2, (14)

and in period 2, S2 (q2, w2,K) = c (q2, w2, K) / (1 + r).8 It is possible to imagine qt8Strictly speaking, the single-period costs c (q, w,K) are not the short-run costs that influence

the firm’s decisions unless, as frequently assumed, costs are independent through time. The single-

period costs are not independent if, for example, capital deteriorates through use. Analysis of

deterioration through use appears most commonly in models of natural capital, e.g. an exhaustible

resource subject to a stock effect. See below.

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to vary for a fixed value of K. But if levels of output vary, what of the allocation of

Φ (K,w0)? For making decisions it is not relevant to allocate the sunk cost to form a

short-run total cost.

For example, Scherer (2001) analyzes the flow of avoidable cost in printing sheet

music, c (q, w,K) in our notation. A quasi-fixed set-up cost (our ψ) incurred for

printing-runs of individual pieces of music, is included. Excluded from the cost func-

tion are general overhead costs including, presumably, the sunk values of machine

capital and the industrial premises. Usually, the average total cost mentioned in

Scherer’s title includes the allocated cost of capital (the fixed cost), in both the short

and long runs, but he does not allocate the sunk cost. For yardstick comparisons,

however, it may be useful in evaluating future investments to estimate a short-run

total cost using undepreciated capital values.

Proposition 3 (i) Only avoidable cost is solely dependent on the technology.

(ii) Short-run cost and long-run cost are properly viewed as stocks.

With zero profits, equality of the value of the capital stock and the value of the

project is implied. A unique depreciation schedule of capital is defined. If one iden-

tifies the value of the capital stock as the value of the project, viewed as a unit as in

many treatments, the following can be stated.

Proposition 4 Unitary Determinacy. Let the entire capital stock be viewed as a

single unit. If discounted profits are zero, the long-run participation constraint allows

a unique definition of total, single-period costs by identifying the value of the capital

stock as the value of the project. A component of the cost is a unique rental payment

or user cost for the capital at each time.

Unitary determinacy applies to the depreciation of the total capital of an enterprise

with zero profit.

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In micro theory all costs, including capital costs, in the long run are allocated to

the time periods of production. Long-run cost is a flow. The long-run cost function is

invariant through time and depends on only the level of output and factor prices. It

is an envelope of short-run total cost functions and is not a function of output price.

When there is sunk capital, however, a long-run cost function is a present value

over the life of the sunk capital as in equation (11); it is a stock. Long-run cost

L (q1, q2, w0, w1, w2) is the envelope of a family of joint-cost functions in which partic-

ular values of K are substituted for the function κ, as in equation (1). (See Appendix

1.) From equation (4), the long-run marginal cost of producing qt is

dLdqt

=∂L

∂κ

∂κ

∂qt+

∂L

∂qt=

∂L

∂qt=

1

(1 + r)t∂c (qt, wt,κ)

∂qt=dCdqt

|κ .

Proposition 5 Envelope Property. A “total” envelope property applies to the con-

strained long-run cost function. Long-run marginal cost is equal to discounted mar-

ginal avoidable cost and does not include sunk capital cost.

No envelope property applies to average cost as in traditional cost theory.

POSITIVE PROFITS

The possibility of positive profits is the main theme of Industrial Organization. Let

Π³K̂, q̂1, q̂2

´> 0. The source of profit–some artificial or natural entry barrier–is

an attribute of the firm which can be capitalized to define a generalized capital good,

a virtual asset. A concrete way of visualizing this asset is as a mineral deposit, the

market value of which is its discounted rent. In this case there are two types of

capital, natural and manufactured.

If profit is zero, the (undepreciated) value of the manufactured capital is identified

with the (undepreciated) value of the project in the short run; consequently, payments

to capital are uniquely pinned down as the net cash flows over the life of the firm. If

16

profitΠ³K̂, q̂1, q̂2

´is positive, however, capital value cannot be identified with project

value as was done in deriving equations (12) and (13); the payment to capital, or the

long-run allocation of capital cost over time, cannot be equal to the net cash flow.

Rather, to define long-run cost as a flow, an appropriate schedule of rental payments

to manufactured capital, or generalized user costs, (u1, u2), must be devised such that

u1(1 + r)

+u2

(1 + r)2= Φ

³K̂´. (15)

The anticipation of payment of, and hence the definition of, a schedule like (u1, u2) is

necessary for the investor to be willing to invest the value Φ³K̂´. An agreement (in

essence a contract) for these payments to be recovered is a participation constraint.

The vector of rental payments constitutes the (long-run) opportunity cost of capital.

It makes no sense to hold that a loss is incurred in period t when that period is

contributing to variable profit, and hence contributing to the objective (5). This

observation follows from the long-run participation constraint, Π³K̂, q̂1, q̂2

´≥ 0 and

the definition of total costs, T , whenΠ³K̂, q̂1, q̂2

´= 0. Nor does it make sense to hold

that the contribution of period t to the objective is greater than its net cash flow. This

observation follows from the current participation constraint, Rt (q̂t) ≥ c³q̂t, wt, K̂

´.

Therefore,

0 ≤ ut ≤ Ft = Rt (q̂t |α0t )− c³q̂t, wt, K̂

´. (16)

Conditions (15) and (16) are the only economic conditions that must be obeyed

by a schedule of rental payments.9 The rental schedule establishes a sequence of

(undepreciated) capital values equal to the present values of the remaining payments.

The schedule thereby imposes a depreciation schedule and vice versa. The marginal

conditions hold (determine output levels, etc.) but they are not employed in defining

the schedule. The undepreciated values of the manufactured asset at times t = 0, 1, 2

9In general, assets may appreciate and payments in any given time period may be zero.

17

are

V (0, K) =u1

(1 + r)+

u2

(1 + r)2,

V (1,K) = u2/ (1 + r) and V (2, K) = 0. Depreciation in period 1 is

δ0V (0, K) = −∆V (0,K) =∙u11 + r

+u2

(1 + r)2

¸− u21 + r

=u11 + r

− ru2

(1 + r)2,

so that

δ0 =−∆V (0,K)V (0,K)

=(1 + r)u1 − ru2(1 + r)u1 + u2

. (17)

In period 2, δ1V (1,K) = u2/ (1 + r) = V (1,K), so that δ1 = 1. Total undiscounted

depreciation is equal to the cost of capital:∙u1

(1 + r)− ru2

(1 + r)2

¸+

u21 + r

=u1

(1 + r)+

u2

(1 + r)2= Φ

³K̂, w0

´.

Moreover,

ut = (r + δt)V (t,K) .

In each period the payment to capital is a rental rate, r+ δ, applied to the undepre-

ciated value of capital.

Depreciation at time t = 1 can also be written

u11 + r

+u2

(1 + r)2− u21 + r

= u1 − r∙u11 + r

+u2

(1 + r)2

¸= u1 − rV (0,K) . (18)

Equation (18) is a rearrangement of the fundamental asset-market-equilibrium con-

dition, which states that the dividend, u, plus the capital gain (the negative of de-

preciation) is equal to the return on the asset. At time t = 2, depreciation satisfies

that condition as well:u21 + r

= u2 − rµu21 + r

¶.

When Π³K̂, q̂1, q̂2

´> 0, a rental schedule satisfying conditions (15) and (16) is not

unique. Any admissible rental schedule is a vectorial representation of the long-run

user cost of capital, Φ³K̂´. The series of payments is “...one of [the] intertemporal

18

patterns of prices which will yield one of the income streams adequate to compensate

investors” sought by Baumol (1971: 640).10

Given any rental schedule, let the payment schedule to the virtual asset be θt =

Ft − ut. That schedule assures that

θ11 + r

+θ2

(1 + r)2= Π

³K̂, q̂1, q̂2

´.

The undepreciated values of the virtual asset are as above with θt replacing ut. Alter-

natively, payments to the virtual asset, θt ∈ [0, Ft], with discounted sumΠ³K̂, q̂1, q̂2

´,

can be posited. The payment to capital is then defined as ut = Ft − θt. Within the

limits specified in conditions (15) and (16), the payment schedule (θ1, θ2) and its as-

sociated depreciation schedule are not unique. Profit at time t is θt and total cost is

c³q̂t, wt, K̂

´+ ut.

Therefore, any rental schedule satisfies all relationships stipulated by capital theory.

Any schedule that satisfies conditions (15) and (16) is an admissible allocation of the

cost of sunk capital Φ³K̂, w0

´to periods 1 and 2. When profits are positive, allocated

cost, and hence single-period (flow) total cost, profit and economic depreciation, are

defined up to a rental schedule. The pattern of economic depreciation is derived from

the schedule and may be irregular. Even if profits are zero, when there are two or

more sunk assets, unique rental schedules cannot be attributed to them individually.

Rather, within the limits discussed above, there is a choice among rental schedules.

Proposition 6 Componential Indeterminacy. If there are more than one type of

comprehensive capital (including a form of intangible capital or a source of profit,

rent or quasi-rent), long-run, economic rental schedules and their implied economic

depreciation schedules are defined but are not unique. Such schedules apply to all

10Compare Triplett’s (1996) discussion, in attempting to put structure on the problem of depreci-

ation, of the vectorial representation of deterioration of capital and its implication for capital “used

up in production” and depreciation.

19

forms of comprehensive capital.

Unique allocations of sunk cost can be obtained, for example, through co-operative

game theory (e.g. the Shapley value). In the present context, however, there is a

single decision maker and hence no reason for any such equilibrium to take a priv-

iledged position over, say, paying all assets, tangible and intangible, virtual or not,

proportionally to net cash flows. Rental payments for particular assets can be devised

such that, for example, the value of manufactured capital is recovered early in the

life of the project. The effect of the choice of payment schedule on the present values

of different assets is neutral.

Again, the long-run cost function that can used in long-run decision making is

expressed as a present value. The only well-defined single-period (flow) cost concepts

are avoidable cost c³q̂, w, K̂

´, marginal avoidable cost cq

³q̂, w, K̂

´, and average

avoidable cost c³q̂, w, K̂

´/q̂. None involves sunk costs. Since profit is defined net of

all costs, profit is also a present value and is not related to single-period costs and

revenues. Componential Indeterminacy implies the following when there is an entry

barrier.

Corollary 7 (i) Profit (the return to some intangible asset such as entrepreneurship

or organization) is a stock, not a flow. (ii) If present value is positive, the rentals over

time are not unique. (iii) Since depreciation is subject to componential indeterminacy

and is not defined at the margin, there is no optimal depreciation schedule.

CONSTRAINED CAPACITY

Suppose that capacity constrains production, so that qt ≤ K, t = 1, 2. In the

Lagrangian of both the long-run and short-run maximization problems there is a

term, vt (K − qt), which is zero by complementary slackness, so that vt > 0 only if

qt = K. The optimality condition for the choice of K, corresponding to condition

20

(2), is

v1 − cK(q̂1, w1, K̂)1 + r

+v2 − cK

³q̂2, w2, K̂

´(1 + r)2

= ΦK(K̂, w0). (19)

The first-order condition for the choice of qt, in both the long- and the short-run

problems, is

R0t

³q̂t

¯̄̄α0t

´= cq(q̂t, wt, K̂) + vt. (20)

When output is at capacity, marginal revenue is equal to marginal cost only if mar-

ginal cost is (re)defined to include the shadow value of capacity, vt.

The marginal scarcity rent, v, is paid in both the long and short runs. The

scarcity rent, however, is not the rental payment to capital, even at the margin,

if cK (q, w,K) 6= 0. For example, let Π³K̂, q̂1, q̂2

´= 0 and investment cost be PKK.

Since ΦK (K,w0) = PK , the rental payment is ut = K̂ (v − cK) |t . There is no tech-

nological definition of the rental payment: a different path of price, varied so that the

optimal capital stock remained K̂, would give rise to a different path of v, and hence

to a different path of v− cK(q̂, w, K̂). In general, the magnitude of vt depends on the

pattern of demand, rather than solely on the technology. There is no technological

link between the left- and right-hand sides of equation (19). It may be possible to

assign an equal cost of capital to each time period by equal amortization, but that

assignment has no economic content in the sense of aiding long- or short-run decisions.

The discussion can be summarized as follows.

Proposition 8 (i) If production is constrained by capacity, the shadow value of ca-

pacity is a part of marginal cost in both the short and long runs.

(ii) Marginal cost is defined with respect to avoidable cost only.

SOME GENERALIZATIONS

At a cost of increased notation (which cannot be uniquely allocated to the different

subsections!), the present discussion can be generalized in a natural way to incorporate

21

partially sunk costs, investments over several time periods, more general demand

conditions, varying interest rates, continuous time, various forms of taxation and

so on. For a going concern investments, including maintenance, and returns are

interwoven through time. The analysis becomes intricate but remains conceptually

analogous (Baumol 1971).

Physical Deterioration

Of special significance in economic theory is the physical deterioration of capital

through time or use. It is allowed above that parameters of demand α0t may vary; the

variation may result from a decline in quality of the product because of deterioration of

capital. Nevertheless, deterioration is usually viewed as a change in the productivity

of the capital stock which leads to changes in current costs.

One can depart from the simplicity of the model by writing ct (qt, wt,K) as the

current cost in period t. Physical deterioration, dt, can be encompassed in the analysis

by redefining the avoidable-cost function as

ct (qt, wt,K) , c (qt, wt, (1− dt)K) , d1 = 0, 0 ≤ d2 (q1) < 1.

Deterioration through use is incorporated in the function d2 (q1). For time deterio-

ration, (a part of) d2 (q1) is constant. If this avoidable-cost function is substituted

for c (qt, wt,K) in the analysis above, then, for pure time deterioration, all equations

are as presented except for adding the appropriate subscript (so that, for example,

ct,K , ∂ct/∂K). Conclusions are similar for use deterioration, and have been studied

extensively in nonrenewable-resource economics (Levhari and Livitatan 1977; Solow

and Wan 1977; Davis and Moore 1998). The marginal cost of producing q1 is

dCdq1

=1

1 + r

∂c1∂q1

+1

(1 + r)2

µ−∂c2∂K

∂d2∂q1

¶,

and the long-run marginal cost of producing q1 has a similar expression. A discussion

22

of use deterioration for a project with a general life of T periods is presented in

Appendix 2.

The optimal values of qt and K are, of course, affected. The forms of the equations,

however, are analogous. Moreover, the thrust of the findings is not changed. Even in

the simple, conventional case (one capital good, zero profit, linear investment cost),

economic depreciation is linked to engineering deterioration only through the effects

of the latter on the optimized values of capital and output. In general, deterioration

bears no necessary arithmetic relation to depreciation, much less proportionality. In

particular, it remains true that, if there are two or more different capital assets–

including one or more virtual assets and hence if there is economic profit–there is

not a unique rental or depreciation schedule for any asset. Rather, the discussion

above provides restrictions on the economically admissible schedules.

As was observed in the preceding section, depreciation is positive in the final period

but may be negative in an earlier period. In the two-period model of that section,

δ0 < 0 if V (0,K) < V (1,K), or by equation (17) if (1 + r)u1− ru2 < 0. In this case

the asset appreciates in value. An asset may appreciate in value even if it deteriorates

physically.

Proposition 9 If a capital good deteriorates physically, its rate of depreciation bears

no necessary relation to its rate of deterioration. An asset may appreciate in value

as it deteriorates.

The results for the dynamic firm discussed in this paper show it to be similar in

certain respects to the static multiproduct firm discussed by Baumol et al. (1982)

and dissimilar in others.

Proposition 10 (i) As in the multi-product firm, for a dynamic firm with sunk cost,

long-run cost is a function of all outputs through time. The allocation of common

costs is ambiguous. Demand conditions delimit the permissible allocations of cost.

23

(ii) Unlike in the multi-product firm, cost depends on demand if profit is zero. One

can define an allocation of the common costs that is consistent with capital theory but

is not unique if there is more than one form of comprehensive capital.

Uncertainty

Recent analyses of irreversible investment have given a central place to conditions

of uncertainty (e.g. Dixit and Pindyck 1994). Consonant with microeconomic theory,

attention herein is confined to the limiting case (as any variance goes to zero) of

certainty. The approach is simple and transparent in laying out the basic ideas. Even

so, uncertainty is obviously a very important consideration when an investor commits

assets. A major difference is that realized cash flows in future periods are different

(usually with probability one) from the expected cash flows projected in the original

decision. Whatever the rental payment schedule, realized quasi-rents (in the short

run) may not be the same as the projected payments (in the long run) to capital

assets. Only under uncertainty is there a meaningful distinction between the two.

Usually, for manufactured capital the payment schedule is followed to the extent

possible, and deviations are attributed to some residual that is in fact the return to

some intangible item such as entrepreneurship or organization. It may not be possible

to identify the effects of mistakes (decisions that are inept at the time made), which

are confounded with randomness of economic events. Componential indeterminacy

remains.

Remark 1 The results of the present paper would be rendered more complicated by

generalizing the model to incorporate uncertainty, but it is clear that none would be

negated in the sense of enabling stocks to be converted uniquely into flows.

The present analysis has assumed a single decision maker. If, under uncertainty,

there are different owners of assets (e.g. a government receiving a royalty from a

24

mineral deposit) who have different discount rates, timing of payment schedules may

be important. Ceteris paribus, present value is maximized if payback is in order of

discount rates. Other things may not be equal if there are asymmetries of information.

DISCUSSION

For a dynamic firm there are unique definitions, and graphs, of avoidable cost

c (q, w,K), of average avoidable cost c (q, w,K) /q, and of marginal avoidable cost

cq (q, w,K). Average variable cost, [c (q, w,K)− ψ] /q, is also defined but has little

economic significance. The only well-defined flow concept of surplus is net cash flow.

Profit is not a flow but rather is a stock, a present value of the net cash flows.

If discounted profits are zero, a schedule of rental payments to capital can be natu-

rally and uniquely defined and then used to construct flow total cost. The definition,

however, is not a technological one in the sense of cost theory. Rather, the payments

(anticipated and required when the long-run decision is made) depend on the pattern

of demand over the life of the project. Long-run cost is a present value dependent on

outputs and on input prices throughout the project. It is not a flow.

If discounted profits are positive, the long-run allocation or recovery of sunk capital

(irreversible investment) cost as a flow is indeterminate and requires a choice of rental

schedule. Conditions (15) and (16) are the economic constraints on rental schedules.

Because the schedule is not unique, there is not a unique total-cost function for any

given time period. A consistently allocated cost of capital need bear no relation to

the shadow value of capital or its physical deterioration. Total cost remains fixed

through time, as in Marshallian analysis, cost theory and econometric practice, only

if a specific rental schedule can be and is chosen to make it so.

In rate-of-return regulation, regulators use a “future test year” in determining the

levels of unit cost to apportion the cost of service over time. The unit costs must

be determined using long-run anticipations. The practice amounts to assuming that

25

conditions will remain stationary until the next hearing. Under this assumption,

static diagrams depicting natural monopoly, with the second-best price being equal

to “average” cost, incorporate allocations of sunk cost that are not unique and, if

discounted profit is indeed zero, depend circularly on the path of regulated prices.

Since average total cost is uniquely defined only at the equilibrium point where

profit is zero, the analysis of Chamberlinian monopolistic competition is not pertinent

when there are sunk costs. The only case in which total cost is both well defined and

purely technological is that of zero sunk cost, as assumed by Baumol et al. (1982)

in the bulk of their book (outside ch. 13). In this case, long-run cost collapses to

current avoidable cost. This finding applies a fortiori to incremental cost and average

incremental cost in a multiproduct firm.

Traditional micro theory neglects the links among periods that are implied when

costs are sunk, i.e., when there is a distinction between long and short runs. Con-

ventional cost functions are (by design) atemporal rather than dynamic; they are an

imprecise way to model technology or to formulate empirical hypotheses.

In the 1980s, the properties of long-run cost in the telecommunications industry

were the subject of many empirical studies. In reviewing several of these studies

Waverman (1989) despaired at their inability to obtain believable, reproducible esti-

mates of such key statistics as the degree of economies of scale. He observed, among

other things, that the studies were subject to a form of aggregation bias. His point

was valid. However, all the studies used long-run cost functions, the arguments of

which were current outputs and current prices of factors including sunk capital. The

problems arose in spite of the fact that the firms were subject to rate-of-return regu-

lation, which was essentially an attempt to implement unique, economic rental prices

for capital. The implementation was hotly debated in rate hearings using sophisti-

cated theoretical and empirical arguments. According to the findings of the present

paper, the specification of costs was not general and may have been a fundamental

26

yet empirically unverifiable reason for the disappointing results.

If variable inputs have linear prices, then for various purposes Shephard’s lemma

can be applied to the avoidable cost, c, to the short-run cost, S, or to the constrained

or unconstrained long-run cost function, C or L. If capital has a linear price (if

Φ (w0, K) = PKK) then ∂L/PK = K. Cost functions retain their advantage over

production functions for representing technologies for empirical analysis. Still, be-

cause of componential indeterminacy, sunk costs make economic measurement lumpy

and clumsy rather than smooth and easy. Shephard’s lemma cannot be applied to

total cost, T = c+ (r + δ)V .

CONCLUSION

According to microeconomic theory, for a dynamic firm long-run cost and profit are

flows that depend solely on current variables. They are implicitly reversible. But the

long run in reality is not an idealized state of reversible decisions; it is the opportunity

to make a choice among irreversible future scenarios called short runs.

It is worthy of stress that, if there is no sunk cost, there is no distinction between

the long run and the short run (cf. Baumol et al. 1982). A single project with a finite

life has been studied herein. An alternative is to study a steady state as being the

ultimate long run. In the steady state, the properties of long-run cost may be masked

if sunk costs are attributed to current production rather than the future production

to which they contribute.

The analysis of the present paper confirms that sunk costs matter in the long run

but do not “count,” in the sense of influencing decisions, in the short run (once they

are sunk). The short-run cost function, written St above, is the discounted sum of

future avoidable costs, c (q, w,K); the sunk capital is not treated as incurring a cost.

The long-run allocations of capital cost for the future periods are not incorporated.

The long-run cost, which is a stock, is the envelope of a family of constrained func-

27

tions, which are stocks. Written C above, they involve parametric levels of installed

capital as an argument. There is not an envelope property for flows of cost.

Irreversibility entails analytical links among time periods. Allocating sunk cost in

the long run is subtle exercise for which an explicit intertemporal analysis provides

insights. A firm’s total cost at any time can be defined through the adoption of an

appropriately defined rental schedule.

If discounted profits are nil, a unique rental schedule can be derived. That schedule

depends not only on technology but also on the pattern of demand. The Marshallian

diagrams that are valid are those which depict marginal and average avoidable costs

(for the multiproduct firm, marginal and average avoidable incremental costs).

If profits are positive, the rental schedule is indeterminate and can, within limits, be

chosen. Current total cost is defined up to the choice of rental schedule. Rental prices

of sunk capital goods are not unique. Although the rental-price schedules are long-run

opportunity costs, they are not short-run opportunity costs. Since no decision is made

about deploying the capital at the time, no unique price emerges. It is economically

legitimate to select from among the valid rental schedules, or equivalently from among

their implied depreciation schedules.

Still, it is of economic significance to delineate the set of admissible rental payments

for the recovery of capital. They are the set of deferred payments required to induce

an economic actor to make resources available to a project, such as a basis for royalty

payments. An admissible payment to capital, a generalization of the concept of user

cost, is defined by two mathematical conditions. It is composed of an interest rate

(exogenous) plus a depreciation rate (endogenous to the chosen payment schedule)

applied to an undepreciated value (endogenous to the chosen payment schedule).

Market power and monopoly profit are frequently measured with reference to the

difference between price and marginal cost as it appears in Lerner’s index. There has

been considerable discussion of whether marginal cost applies to the long or the short

28

run. When costs are sunk, rational single-period decisions are based on marginal

avoidable costs. But the relationship of price to marginal avoidable cost is not an

indicator of profit, which is a long-run concept. Since long-run marginal cost is equal

to discounted marginal avoidable cost, and does not include capital cost, Lerner’s

index of monopoly power is not a useful long-run index. In the short run, price net of

marginal avoidable cost, p− cq (q, w,K), is no measure of monopoly profit per unit,

and Lerner’s index, [p− cq (q, w,K)] /p, is no measure of monopoly power in, for

example, antitrust analysis or utility pricing or any capital-intensive industry. Nor is

it for purposes of regulation or for the evaluation of mergers or of a firm’s performance.

Nor is the difference between price and average cost a measure of profit, since, if there

is profit, average cost is not uniquely defined.

Even given a measure of engineering deterioration, depreciation may not be unique,

let alone observable. Empirical analysis and resulting statistics based on period-by-

period, “long-run” estimates may lead to misinterpretations unless there is explicit

recognition that allocations of sunk capital costs, or rental schedules, are not unique.

The present paper has expanded theoretic perceptions of cost. If there is no sunk

cost, the long and short runs collapse into one another. If there is a sunk cost, the long-

run cost function is more general than described in micro theory and used in empirical

investigation. In many cases, traditional theory may be quite serviceable; it has, after

all, been in use for well over a century. Some atemporal allocations of sunk cost have

proved useful. For a sunk point input, the discussion herein has provided a beginning

of a more general, dynamic treatment by defining appropriate, economic rental rates

and by defining functions to which the envelope property applies. Among possible

extensions to the analysis is the treatment of firms with continuing investments and

retirements.

29

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APPENDIX 1: THE LONG-RUN COST FUNCTION AND THE

ENVELOPE PROPERTY.

minKC (q1, q2,K,w1, w2)

= minK

µΦ (K,w0) +

c (q1,w1,K)

1 + r+c (q2,w2,K)

(1 + r)2

¶= min

K

½miny0[w0 · y0 |F (y0) ≥ K ]

¾+miny1,y2

∙µw1 · y11 + r

+w2 · y2(1 + r)2

¶|ft (yt, K) ≥ qt, t = 1, 2

¸= min

K

½miny0,y1,y2

µw0 · y0 +

w1 · y11 + r

+w2 · y2(1 + r)2

¶|F (y0) ≥ K, ft (yt, K) ≥ qt, t = 1, 2

¾= min

y0,y1,y2minK

½µw0 · y0 +

w1 · y11 + r

+w2 · y2(1 + r)2

¶|F (y0) ≥ K, ft (yt, K) ≥ qt, t = 1, 2

¾= min

y0,y1,y2

½µw0 · y0 +

w1 · y11 + r

+w2 · y2(1 + r)2

¶|F (y0) ≥ K, ft (yt,K) ≥ qt, t = 1, 2

¾= L (q1, q2, w0, w1, w2)

At the margin,

dL

dqt=

∂L

∂κ

∂κ

∂qt+

∂c |t∂qt

1

(1 + r)t=

∂c |t∂qt

1

(1 + r)t=dC

dqt.

32

APPENDIX 2. DETERIORATION THROUGH USE

Suppose that the firm’s avoidable cost function at date t is ct (qt, wt, [1− dt (q1, ..., qt−1)]K).

Then

C = Φ (K,w0) +TXt=1

ct (qt, wt, [1− dt (q1, ..., qt−1)]K)(1 + r)t

.

Then the marginal cost of producing qs is

∂C

∂qs=

1

(1 + r)s∂cs∂qs

+TX

t=s+1

1

(1 + r)t∂ct∂K

µ−∂dt∂qs

¶.

The second term is a deterioration effect, and is analogous to the stock effect in the

economics of non-renewable resources. The optimal choice of the capital stock is

affected in the following way:

∂C

∂K=

∂Φ

∂K+

TXt=1

∂ct∂K

(1− dt) ,

so that ∂Φ/∂K =P(−∂ct/∂K) (1− dt) replaces equation (2) in the text. In the

usual case in which ∂2Φ/∂K2 < 0 (increasing returns), more capital is installed.

In the long run, the capital stock is a function κ (..., q1, ..., qT , ...):

L = Φ (κ, w0) +TXt=1

ct (qt, wt, [1− dt (q1, ..., qt−1)]κ)(1 + r)t

.

Long-run marginal cost of producing qs is

dLdqs

=

"∂Φ

∂κ+

TXt=1

∂ct∂κ(1− dt)

#∂κ

∂qs+

1

(1 + r)s∂cs∂qs

+TX

t=s+1

1

(1 + r)t∂ct∂K

µ−∂dt∂qs

¶

=1

(1 + r)s∂cs∂qs

+TX

t=s+1

1

(1 + r)t∂ct∂K

µ−∂dt∂qs

¶.

The second line is an envelope result: because of the optimal choice of the capital

stock, the term in square brackets in the first line is zero.

If there is such deterioration, of course, there may be re-investment or maintenance

expenditures at some points. There would be more complicated expressions, but the

idea remains.

33