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: [email protected]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
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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.
6
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.
7
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.
8
(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
9
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).
10
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.
11
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.
12
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,
13
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.
14
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.
15
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
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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