Trying to Measure Sunk Capital

Transcription

1 Trying to Measure Sunk Capital Robert D. Cairns May 26, 2006 Abstract Standard analyses of the measurement of capital are based on several maintained assumptions. These assumptions are tantamount to assuming that projects are not unique, that projects are made up of a single type of capital good and that capital is not sunk. In reality, none of the assumptions is valid. When capital is sunk, the unit of economic analysis is the entire, realized project. Disaggregation of capital cost over time can be done in non-unique, but non-arbitrary, ways to find economic accounting costs. Depreciation is affected by the physical deterioration of capital, but is not the value of deterioration.

2 1. Introduction Any irreversible decision entails sinking a cost. In the measurement of capital, the fact that costs are sunk is of central importance. However, conventional analyses of the cost of capital do not fully recognize the restrictions imposed by irreversibility. Cairns (2005) explicitly considers the role of irreversibility in the measurement of production costs, which include the cost of capital, and finds instances in which conventional analyses can be considered incomplete. The present paper discusses the implications of the special assumptions of the conventional theory of the measurement of capital (e.g., Symposium on Economic Depreciation, Economic Inquiry 1996). The discussion is fundamentally an enquiry into what it means for capital to be sunk, or irreversibly committed, to a project. 2. Assumptions of Conventional Theory A main issue is the need to express capital in terms of a common numeraire. To make all forms of capital commensurate with each other and with output, capital is measured in value, not physical, units. Consequently, much analysis begins with an attempt to measure capital consumption (the physical deterioration of capital with time, use or obsolescence) in physical terms. Then the changes are valued and 2

3 the resulting money values are interpreted as the (financial) depreciation schedule of the capital. It is reasonable to assume that the physical deterioration can be measured, as there are engineering studies and also ubiquitous discussions of maintenance requirements for machines and buildings. But deterioration is only one factor that affects depreciation. Therefore, a section of the present paper is devoted to discussing the similarities and differences between physical deterioration and financial depreciation. This section helps to show why the financial depreciation of an asset cannot be identified with its physical deterioration, but depends on the pattern of net cash flows from the project. A second assumption is that the theoretical analysis can be conducted as if the capital of a project were a single asset and that discounted long-run profits, inclusive of sunk capital costs, are zero. Since the time of the economic decision is thetimeofinvestment,the originalvalue ofacapitalgoodcanbeidentified with its price at that time (usually the price of a new good). However, it is of central importance to recognize that there are many types of capital goods involved in any investment project. Moreover, capital comprehensively defined includes not only tangible goods with market prices, such as manufactured capital goods or a resource being extracted, but also non-priced, intangible goods such as the value of the investment opportunity, or the entrepreneurship involved in recognizing 3

4 the opportunity. The value of the non-priced goods at the point of investment is a residual, namely, the difference between the net present value of the project and the total cost of the part of the (comprehensively defined) capital that is mediated on markets. When the project is initiated, the various forms of capital are sunk in the project. Analytically, the formerly distinct capital goods cannot be distinguished once sunk (or to the degree that they are sunk). Rather, the very act of investment transforms them into a single, new capital good, namely a realized project, and the entire project becomes the unit of economic analysis. As a result, depreciation of the various assets is not unique, much less proportional to physical deterioration of the assets. Indeed, any intangible capital does not deteriorate, for it does not have a natural metric other than value. It can only depreciate. Another assumption is that changes in the values of tangible capital goods can be decomposed into a cross-sectional and a time-series component. The decomposition is typically accomplished through what is tantamount to a replication of the investment using older capital goods (cross section) or at a different time (time series). It is used for the measurement of the deterioration of the goods over a time period. This decomposition is conceivable, however, only if capital is not sunk. For sunk capital, the decomposition is a Gedankenexperiment that is 4

5 consistent with neither optimizing behaviour nor observability. For example, the analysis of optimal stopping indicates that investment at a time other than the optimal stopping point reduces value. If investors behave suboptimally (a possibility that prudence demands be admitted, given informational asymmetries and other market failures), the values at different stopping points are, in general, still different. Moreover, since economic conditions cannot be assumed stationary, in general they involve different quantities (in physical and value terms) of capital. The standard assumption of convexity does not appear to play a strong role in the problem of the measurement of capital. The project as a whole is the unit of economic analysis, and the same analysis can be carried out whether or not investment cost and production cost are convex. However, the assumption of constant returns to scale can lead to some misconceptions. Non-linearity implies that marginal analysis does not correspond with total analysis. Under constant returns a less general analysis can be carried out in unit rather than total terms. Furthermore, constant returns are sometimes justified by the assumption that an investment can be exactly replicated. Replication is only a small step from reversibility, and sometimes the distinction is not made. The analysis below abstracts from the important issue of inflation. 5

6 3. Replicability and Non-Replicability Mostprojectsareuniqueandirreversible. Itisnotpossibletocreateareplica of a unique project. Consonant with the theory of investment under uncertainty, one of the investor s choices in making an irreversible decision is when to make the investment. Even under certainty there is an optimal stopping point (Cairns and Davis 2006, Davis and Cairns 2006). The optimal stopping point is the strike time for which the net present value of the project is maximized. The maximal net present value is the value of the investment opportunity. Not all investments are made under perfect competition, and the value of the opportunity can be viewed as an entry barrier. For analytic purposes the net cash flow at a given time can be viewed as the user cost of the project, an abstract capital good, at that time. An old Marxian perspective on capital was that financial capital was transformed into physical capital, and returned financial capital of greater value. The transformation was represented as M C M 0, with M 0 M>0 representing the profit of the capitalist. 6

7 A more mainstream way of viewing the progression is that financial capital is expended to purchase physical capital. The physical capital, combined with non-priced capital, is congealed into a project. Thereafter, so long as the capital is sunk, in many cases until it is scrapped, the capital is not distinguishable analytically from the other capital inputs that make up the project. Let the vector of non-priced capital goods be represented by N, the vector of priced capital goods by C, andtheprojectbyp. The transformation can be seen as an investment and its resulting outputs at time t =0in which M 0 C 0 ; C 0 N 0 P 0 M 0, with the ultimate money value obtained being equal to the value of the project. In turn, because the value of intangible goods is a residual, the project value is the sum of the values of the comprehensive capital. Let the financial value of an asset be represented by using the function V.Then M 0 = V (P 0 )=V (C 0 )+V (N 0 )=M 0 + V (N 0 ). 7

8 Totheextentthataunitofphysicalcapitalgoodi, C i,isnotsunk,itremains with the project so long as its undeteriorated quantity, u i C i, contributes to the project, or so long as V (P t ) >V ³ P t \u i C i + V ³ u i C i. (3.1) A typical analysis assumes that V (P t )=V (P t \u i C i )+V (u i C i ).Ifequation (3.1) holds, the non-sunk capital is tantamount to being sunk in the project (the firm will not sell it) and can be analysed as if it were sunk. Sometimes assets are sold, but the selling of assets is not the most common observation over time. The identification of depreciation with deterioration is possible only if the capital good is not sunk. In a going concern, when capital is retired or sold, its value can be subtracted from the project value at that time. When new capital is added, it is simply added to the omelet called the project. The analyst simply needs to consider intertemporal valuation using an appropriate interest rate. Once one recognizes that there is an optimal stopping point, one must also recognize that the chosen investment cannot be replicated one period hence, nor in the present with one-year-old capital. The very notion of comparing the value of 8

9 an investment now with the value of investment in new capital one period hence, or of an investment now with capital goods a year old, is not consistent with optimizing or actual behaviour. Nor is the counterfactual investment observable, even in principle. An extreme example is that a mine cannot be developed with a year-old mineshaft. But the same sort of observation holds for an automobile factory, a school, etc. Therefore, depreciation is not decomposable into changes in new asset prices over the year and the difference between the productivities of old assets. Indeed, the assumption that the investment is replicable a year hence isnotfullyconsistentwiththefactthatitisirreversible. Moreover, the decomposition into cross-section and time series is not unique. Let the putative value of an investment with s-year-old capital in period t be represented by v s t. One could define cross-sectional and time-series components of depreciation to be x s t = v s+1 t v s t and z s t = v s+1 t+1 v s+1 t,ortobey s t = v s+1 t+1 v s t+1 and w s t+1 = v s t+1 v s t ; the two pairs are not necessarily the same. It costs no generality to assume that an investment is unique and to analyse the flows of real goods and finance that occur during the project s life. An implication is that an investment opportunity is inseparably combined with various forms of capital when the decision is made to strike by investing. These various capital goods sunk in the project no longer have independent existence; rather, 9

10 like an omelet, a new capital good, a realized project, emerges from the combination. Strictly, this is the only capital entity amenable to analysis through time. Depreciation is measured simply as v t t v t+1 t+1 (Samuelson 1937). 4. Deterioration and Depreciation Usually, economic theory finds a strong link between the physical deterioration of capital through time and the depreciation of the project, which is the change in its value to the firm. Frequently, the physical deterioration of an asset is considered to be capital consumption, and the value of that deterioration is then derived as the depreciation of the asset. It is argued above, however, that once a capital good is sunk it no longer has an economic existence independent of the other capital goods, of which one is the investment opportunity, that constitute the project. Value is defined by the net present value of production from the project, and the various capital goods contribute that value in concert. Deterioration of the component capital goods is only one influence on depreciation. It is worthwhile to point out the mathematical similarities of deterioration and depreciation in order to understand the tendency to link them. Over the life of a capital good, the deterioration sums to the original quantity of capital invested. Let the undeteriorated (net remaining) quantity of physical capital equivalent at 10

11 time t be represented by u t and let the rate of deterioration in period t 0 be represented by d t.also,define d 1 =0(the rate of deterioration before investment is nil). Then u 0 =1and lim t u t =0.Also,0 d t 1, sothatu t+1 u t.in fact, u t = u 0 (1 d 0 )(1 d 1 )... (1 d t 1 ). Total deterioration over the infinite future 1 is d t u t = = d t u 0 (1 d 0 )(1 d 1 )... (1 d t 1 ) (1 d 0 )(1 d 1 )... (1 d t 1 ) u 0 (1 d 0 )(1 d 1 )... (1 d t ) u 0 = (1 d 1 ) u 0 = u 0, since d 1 =0. Total deterioration over all future time periods, is equal to the original stock. This is also a property of depreciation: undiscounted depreciation sums to the original sum invested: ³ v t t v t+1 = v 0 0 v = v While the symbol for infinity is used, in practice the sum ends at some finite time. From then all values are zero. 11

12 Furthermore, both deterioration and depreciation of a given capital good may be at different rates for different projects (different uses of capital, such as trucks, for different purposes). Both may be at different rates for two capital goods within the same project. However, depreciation is evaluated in terms of numeraire, not of physical units of capital. When the tangible capital goods are invested in the project, they are subject to deterioration, but deterioration of the intangible capital (the value of the opportunity or entrepreneurship, for example) or of the project itself is not defined. Depreciation of the project can be defined, however. Let the value of the project, and hence of its comprehensive capital, including entrepreneurship and so on, at time t be represented by V t, and let current variable profit (total, not unit) be represented by π t.(inturn,π t may be a function of prices and wages and the irreversibly chosen capital stock.) The convention adopted is that net cash flowsorvariableprofits are received at the end of a period. For a given project with sunk capital, V 0 = V 1 = t=1 π t t+1 and (1 + r) π t (1 + r) t =(1+r) X t=1 π t (1 + r) t+1 =(1+r) V 0 π 0. 12

13 The differences between the arithmetic expansions of V 0 and V 1 are the discount factors, since looking forward from period 1 there is one period fewer of discounting for each included term, and the number of terms. Depreciation of the project (of its congealed capital stock) in period 1 is defined to be 0 = V 0 V 1 = π 0 rv 0. (4.1) The rate of depreciation is defined to be δ 0 = 0 /V 0. By a rearrangement of equation (4.1), the dividend from the project in period zero (its net cash flow π 0 ) is equal to the sum of the rate of interest and the rate of depreciation applied to value in period zero, π 0 = rv 0 +(V 0 V 1 )= µ r + 0 V 0 V 0 =(r + δ 0 ) V 0. The user cost of the project can be identified as (r + δ 0 ) V 0. By a further rearrangement, the return on the original capital is equal to current profit minus depreciation, or the dividend plus the capital gain, rv 0 = π 0 +(V 1 V 0 ). (4.2) 13

14 Equation (4.2) is an expression of the fundamental equation of equilibrium in asset markets. Suppose now, as is often assumed, that there is a single capital good and that its purchase price is equal to the present value of variable profits, so that total profit iszero. 2 Is there a relationship between the rate of deterioration, d t,and the rate of depreciation, δ t? One can put V 0 = π 0 1+r + X π t 1 (1 + r) t and t=2 V 1 = π 1 1+r + X π t t+1,sothat (1 + r) t=2 0 = π 0 π 1 1+r + X = t=2 π t π t+1 (1 + r) t+1. π t 1 π t (1 + r) t Let v t = π t /π 0 be the profit ratio in period t 0. That is to say, the profit ratio incorporates everything affecting profitability, i.e., the productivity of undepreciated capital in producing variable profit or net cash flow. The current 2 In standard analysis the single capital good is a tangible, priced good such as a machine or building. In the present paper it is the project. The project is not deteriorated. 14

15 cash flow at time t, π t, incorporates the values of the physical deterioration of the capital as well as of any other changes in economic conditions, into a single measure. Let γ t represent the change in productivity of capital in period t, sothat 0 = π 0 (υ t υ t+1 ) (1 + r) t+1 = π 0 γ t (1 + r) t+1. Depreciation is equal to the discounted sum of the changes in productivity of comprehensive capital, or its period-to-period deterioration, evaluated at the productivity in period zero. Unlike deterioration d t, however, the productivity series γ t need not be a declining series and indeed may exceed unity at certain points. All that is required is that the infinite sum converge 3. Depreciation need not be positive at any given point; the project may appreciate, even though physical capital can only deteriorate. In general, depreciation is a complicated, discounted sum of rates of change, andthoseratesofchangedependoneconomicconditions. Theconditionsare confounded within the variable-profit function. The effects of the various forms 3 Convergence is guaranteed for a project with at finite life of a project. A firm may go on forever, with mergers, divestitures, buy-outs and so on as well as investments, sales and retirements of assets. 15

16 of capital cannot be assessed separately. The rate of depreciation is δ 0 = 0 V 0 = X X π 0 γ t (1 + r) t+1 π 0 υ t = (1 + r) t+1 X X υ t υ t+1 (1 + r) t υ t. (1 + r) t The rate of deterioration, even in a model of a single capital good cannot be determined from observations of variable profit (netcashflows). Moreover, the rate of deterioration is specific to the project and so comparisons among projects are not relevant. The only way to obtain an arithmetically simple representation of the rate of depreciation is to assume that υ t υ t+1 =(1 k) υ t for some constant 1 k, so that υ t+1 /υ t = k. This assumption gives rise to the common, special form of exponential deterioration at rate k, υ t υ t+1 =(1 k) t (1 k) t+1 = k (1 k) t. Under this assumption, 0 = π t 0 (1 k) t k (1 + r) t+1. The sum converges if and only if k (0, 1], butifk =1the good lasts only one period and so does not qualify as a capital good. Exponential depreciation does 16

17 not allow for improvements, e.g. pre-building infrastructure such as pipelines. If 0 <k<1, V 0 = π t (1 + r) t+1 = X π t 0 (1 k) t (1 + r) t+1, so that δ 0 = 0 V 0 = k. The rate of depreciation δ 0 at time zero (and in general at time t) isequalto the rate of deterioration of economic conditions, k. If physical deterioration of capital is the only source of change in conditions, then the rate of depreciation is equal to the rate of physical deterioration. In this example, the rate k is constant. Equality of the rates of depreciation and deterioration depends on stationarity of economic conditions. All of this analysis was carried out for the whole project using total and not unit values. The properties of π t or of the investment-cost function were immaterial. The results are robust to the holding or not holding of the common assumptions of constant returns to scale or of convexity. In general, the measurement of committed capital is specific to its use and is confounded with properties of that use. 17

18 5. The Disaggregation of Variable Profit If (discounted) profit is positive, then that profit arises because of the existence of some virtual asset, usually considered an entry barrier. That virtual asset can be treated like other assets. There may be several different types of capital, including manufactured capital, natural capital, entrepreneurship and an entry barrier. The first two have an obvious price at the time of investment but not afterward. The last two may be confounded at the beginning of the project because there may notbemarketpricesforthem. Total profit is the discounted variable profit minus the cost of marketed forms of capital. It is the investment of the non-marketed capital. Total profit can be depreciated as above. It is also possible to attribute user costs to each of the other forms of capital. Baumol, Panzar and Willig (1982) call the user cost, the payment to capital. The economic conditions that must be satisfied by the user cost of each form of capital are: 1. User cost, κ i, of capital good i must be non-negative and the sum of the user costs is the variable profit: nx i=1 κ it = π t. (5.1) 18

19 2. The discounted total user cost of each capital good C i is equal to the cost of the good, Φ i (C i ): κ ³ it t+1 = Φi C i. (5.2) (1 + r) Any schedule of user cost gives rise to a depreciation schedule, equal to the change in discounted user cost (and vice versa). As of any time s>0, the undepreciated value of good i is V is = t=s κ it (1 + r) t s+1, and so (1 + r) V is = V i,s+1 + κ is. Given the schedule κ is, depreciation of good i at time s is V is V i,s+1,and κ is = rv is +(V is V i,s+1 )= µ r + V is V i,s+1 V is V is =(r + δ is ) V is. These conditions for user cost satisfy the requirements of both capital theory and economic depreciation. The undiscounted sum of depreciation is also equal to the original cost of capital. A requirement of economic accounting is that depreciation sum to the original value. Requirements of capital theory are that (i) the capitalmarket equilibrium condition hold and (ii) the discounted payments to capital 19

20 sum to the original value. If there is more than one form of capital (including non-marketed forms) then the user cost and depreciation schedules are not unique. That is to say, the analyst has leeway in choosing the schedules κ is within the constraints given by the economic conditions (5.1) and (5.2). Indeed, if an asset is sunk, then it is not necessarily true that the period over which it is depreciated needs to be the same as its economic life. Making the two periods the same, however, seems aesthetic. 6. Conclusion Conventional analysis in capital theory has a number of simplifying assumptions that need to be modified if investments are irreversible. An irreversible investment transforms capital goods which may retain their physical forms into a project. the project has a unique rate of depreciation but its rate of deterioration is not defined. Depreciation is the rate of change of the value of the project. User cost is interest plus a rate of depreciation applied to the value of the project. Deterioration of capital goods is among the influences that affect productivity (variable profit) but is not the only consideration in the depreciation of sunk capital. Rather, the measure of capital depends directly on projected levels of profit. Total variable profit (netcashflow) can be attributed to particular capital goods 20

21 as a user cost. Viewed from the long run, before the commitment of a particular capital good to the project, the schedule of cost attributed to the good satisfies two mathematical conditions. In general the attribution is not unique. The nonuniqueness has implications for measurement of capital and the application in index numbers and elsewhere. References [1] Baumol, William J., John C. Panzar and Robert D. Willig (1982), Contestable Markets and the Theory of Industry Structure, Harcourt, Brace, Jovanovich, New York. [2] Cairns, Robert D. (2005), Sunk Costs and Cost Functions, mimeo. [3] Cairns, Robert D. and Graham A. Davis (2006), Strike when the Force is With You: Optimal Stopping and Resource Equilibria, forthcoming, American Journal of Agricultural Economics [4] Davis, Graham A. and Robert D. Cairns (2006), Optimal Stopping under Certainty and Uncertainty, mimeo. [5] Economic Inquiry (1996), Symposium on Economic Depreciation. 21

22 [6] Samuelson, Paul A. (1937), Some Aspects of the Pure Theory of Capital, Quarterly Journal of Economics 52,May,