On Measuring Sunk Capital

Transcription

1 On Measurng Sunk Captal Robert D. Carns Department of Economcs and Creq McGll Unversty 855 Sherbrooke St. W. Montreal QC Canada H3A 2T7 November 3,

2 On Measurng Sunk Captal Abstract. For much of ts lfe, tangble captal goods are rreversbly commtted to nvestment projects. The paper examnes what t means for captal to be sunk. Value sunk s measured usng the optons avalable to managers of a project. External prces may or may not be relevant to an enterprse. Fxng the value of optons s the only role for external prces of the assets. Moreover, margnal prces are not used n valung sunk captal. The choce of real or nomnal prces s consdered n terms of measured benefts, costs, wealth and ncome of a project. Key words: captal value, rreversble nvestment, sunk captal, optons 2

3 1. INTRODUCTION The dea of sunk captal s ntegrated nto the statc, Marshallan theory of the frm. The assets that are sunk and the value sunk are defned. In much of economc analyss, however, sunk costs are consdered to be bygones and tend to be vewed as unmportant. In recent years, the theory of nvestment under certanty (e.g. Dxt and Pndyck 1994) has hghlghted the mportance of rreversble nvestment and brought the mplcatons of sunk captal back to promnence. The condtons derved tend to be nstantaneous condtons for makng an nvestment (or dsnvestment) as opposed to condtons that hold for the perods between changes n the sunk captal. A frm s vewed largely as a set of optons, and the condtons derved are for the exercse of an opton. Carns (forthcomng 2013) ndcates that rreversblty s also the vtal consderaton n the accountng for, and hence valuaton of, captal deprecaton and hence captal tself durng perods when an opton s not exercsed. When captal s sunk, t does not have a unque value. The present paper revsts the queston of sunk captal n a dynamc model. Sunk assets and the values sunk are defned. These determnatons form a foundaton for the valuaton of captal n place. 2. PROJECT VALUE AND DEPRECIATION Captal s organzed nto projects and projects nto frms. Doms (1996: 80, n. 7) notes that much nvestment s lumpy, not convex. The vew of a project n ths secton s consonant wth the rreversble nvestment of captal of multple types. The smplest case, of nputs at a sngle pont of tme, s consdered. The present paper departs from tradtonal analyses n ncorporatng non-marketed captal as well as marketed captal nto a frm s captal stock. Heren, captal s defned 3

4 n terms of realzatons of cash flows over tme. A surplus that accrues over tme can be captalzed. The source s vewed as a component of the captal of the project. Defnton 1. Any stock that contrbutes to cash flow over tme s a form of comprehensve captal. Let the vector of marketed captal goods at tme t be represented by the k t -vector K t and cost a total of M 0 at tme 0; the vector of non-marketed captal goods by the n t -vector N t ; and the project by the scalar P t.letthelengthoftheproject slfebe represented by T, the real nterest rate n perod t by r t and real, varable proft n perod t by m t. Furthermore, let the present value of the project be denoted by V.Att =0 T m t V (P 0 )= t=1 Π t s=1 (1 + r s ). (1) Havng no prce and ndeed no unts, the non-marketed assets are confounded n the project and can be valued only as a sngle, resdual asset. Ther contrbuton s specfc to the partcular project (cf. Olner 1996: 69). The value of the non-marketed assets s defned as a resdual, V (P 0 ) M 0. In a tradtonal analyss, Hulten and Wyckoff (1996: 11) consder the prce of a used machne to be both (a) the present value of the ncome accrung to the machne and (b) what a ratonal nvestor would pay for t. The value of the project as a whole at t, V (P t ),satsfes these condtons, snce the remanng net present value can be realzed n the captal market. The tangble captal goods nvested n a project are subject to (physcal) deteroraton. Deteroraton of non-marketed captal s not defned. Therefore, deteroraton of the project s not defned. Marketable components of the project are only occasonally, and not typcally, sold n a secondary market. Rather, t s the project tself (e.g. a mnng property) that can be sold. The project thus becomes a prmary object of economc analyss. 4

5 Whle there s no measure of the deteroraton of a project, the analyss of the project s deprecaton has close parallels to the tradtonal one that nvolves a sngle captal good. It begns smlarly to that of, for example, Hulten and Wyckoff (1996). As n dualty theory, varable proft m t s a functon of prces and wages and the rreversbly chosen ( sunk ) captal stock. The conventon adopted s the usual one, that net cash flows or varable profts are receved at the end of a perod. If the project has a fnte lfetme T,letm t =0for t>t. For a gven project, for any t 0, the dscounted-varable-proft functon at t s At t +1, V (P t )= V (P t+1 ) = s=2 s=1 m t+s Π s k=1 (1 + r t+k). (2) m t+s Π s k=2 (1 + r t+k) = m t+s+1 (1+r t+1 ) s=2 Π s k=1 (1 + r t+k) = (1+r t+1 ) V (P t ) m t+1. (3) Hotellng (1925) and Samuelson (1937) provde the followng defnton for the deprecaton of (the aggregated, comprehensve captal of) a project. Defnton 2. Deprecaton of the project at tme t s the declne n ts value, t+1 = V (P t ) V (P t+1 )=m t+1 r t+1 V (P t ). (4) Defnton 3. Therateofdeprecatonsδ t+1 = t+1 /V (P t ). Deprecaton of the project s unque but t depends on the pattern of the net cash flows. These cash flows are specfc to the project and depend on economc condtons. Rearrangement of equaton (3) shows that the varable proft s equal to the sum of the rate of nterest and the rate of deprecaton appled to value: m t+1 = r t+1 V (P t )+[V (P t ) V (P t+1 )] = (r t+1 + δ t+1 ) V (P t ). (5) 5

6 A rental or user cost can be dentfed as (r t+1 + δ t+1 ) V (P t ). Therefore, the net cash flow at any tme s the user cost of the project, consdered to be a composte captal good. A further rearrangement yelds a no-arbtrage condton, that the return on the orgnal captal s equal to current proft mnus deprecaton, or the dvdend plus the change n captal value, r t+1 V (P t )=m t+1 +[V (P t+1 ) V (P t )]. (6) The followng apples to the aggregate captal, the project. Proposton 1 Therentalorusercostofaprojectsthesumoftheratesofnterest and deprecaton appled to the project s dscounted net cash flow. The return on the dscounted net cash flow satsfes the fundamental, no-arbtrage condton of equlbrum n asset markets. Deprecaton s endogenous to economc condtons. Snce the mathematcal propertes of varable proft m t or of the nvestment-cost functon have not been used, the results are robust to whether or not the assumptons of constant returns to scale or of convexty, etc. hold. Snce taxes are ncorporated nto varable proft, the results are also robust to the ncluson of any form of tax NUMERAIRE Dewert (2009) observes that the user cost s a beneft n terms of cash flow, what can be called a user beneft, andv (P t ) s the captalzed total beneft antcpated by the owner. Vewed retrospectvely (wth the understandng that sunk costs are bygones ), the full cash flow s ndeed a beneft to the project s owner n the current perod. 1 The captalzed values of taxes are socal assets but not prvate ones. 6

7 For example, the Hotellng rent of a canoncal exhaustble resource rses at the rate of nterest. Many resource economsts consder Hotellng rents to be costs; they are ndeed opportunty costs and are often called user costs. They are costs, however, only because n equlbrum the maxmum beneft s beng obtaned,as compared to the beneft avalable n other tme perods. Havng the use of the resource at the optmal tme (or any tme) s, consstently wth Dewert s reasonng, a beneft. On the other hand, deprecaton s the repayment of a cost. The antcpaton of that repayment s necessary f the project s to be ntated. Indeed, both recovery and nterest, δ t+1 V (P t ) and r t+1 V (P t ), are components of the requred repayments n the context of an antcpated nvestment. Vewed prospectvely, the rental comprses acostand abeneft: One may consder the deprecaton, δ t+1 V (P t ), to be recovery of captal cost and the nterest, r t+1 V (P t ),tobeanncomeorabeneft. The dscusson dentfes a source of ncome to be an nterest rate multpled by a captal value. Income and value (wealth) are lnked dynamcally: the captalzed value of any form of ncome s the value of an asset, nterest on whch s the ncome. It deprecates accordng to equaton (4). Frms nvest n nflatonary or deflatonary tmes as well as n tmes of stable prces. Suppose, then, that m t denotes varable proft n nomnal terms and r t the nomnal rate of nterest. Let the rate of nflaton be denoted by π t. The project s value n money terms s V (P t )= s=1 m t+s Π s k=1 (1 + r t+k) = s=1 Π s k=1 m t+s Π s k=1 (1 + π t+k) = (1 + r t+k ) (1 + π t+k ) s=1 m t+s Π s k=1 (1 + r t+k) = V (P t) : Results are, then, equvalent whether nomnal or real values are used. A case can be made for usng the numerare that s observed (for a preference for usng nomnal values) rather than real values calculated usng an estmated prce ndex. Proposton 2 The method of calculaton of captal value and deprecaton s robust 7

8 to (a) changes n the prce level, (b) the mathematcal propertes of current varable cost, (c) taxaton, (d) the retrement, deteroraton, agng, obsolescence or replacement cost of tangble captal, etc. 4. SUNK ASSETS AND SUNK VALUE At any tme t < T, a partcular marketed, tangble captal good has deterorated through use n the project. It remans wth the project at tme t so long as ts productve capacty n the project, represented by u tk0,u t 1, contrbutes to the project more than ts current market value net of transactons cost, denoted by p t (Kt), wth Kt not necessarly equal to u tk0: V (P t ) >V P t \u tk0 + p t K t. (7) Sometmes, used captal goods are sold. Far more often they are not sold. Projects reman n operaton because ther assets, tangble and ntangble, marketed and nonmarketed, are better deployed n the project than elsewhere, as n equaton (7). So long as a captal good remans wth ts project, n many cases untl t s scrapped, ts contrbuton comes not solely on ts own account but as a result of complementarty wth other assets. The value of a sunk asset satsfes condton (7). Defnton 4. Captal s rreversbly nvested or sunk so long as ts ncremental contrbuton to ts current project exceeds ts market value net of transactons costs. In tradtonal captal theory, the relatve prces of assets of dfferent vntages are consdered to be proportonal to ther relatve productvtes n the project (e.g., Jorgenson 1996: 29). In contrast to equaton (7), the followng condtons are mpled: Kt = u tk0 and (8) V (P t ) = V P t \u tk0 + p t K t. (9) 8

9 The dentfcaton of deprecaton wth physcal deteroraton of a tangble, marketable captal good s possble only f condtons (8) and (9) hold. These condtons mply that the captal good s not sunk n that t can be economcally dsengaged from the project at any tme and ts full value to the project realzed n the captal market. In terms of the producton functon F (K, L), a non-sunk asset, whch satsfes condtons (8) and (9), has F/ K = r + δ. The defnton of the remanng captal as u tk t, however, remnds one that the stock of captal good has lnks to the other perods through the factors u t. If condton (7) holds, the value of the good n the second-hand market, p t (K t), snotrealzed. The asset s not sold, s not removed from the project. The productvty of a sunk asset s lnked to the other perods of ts lfe. An asset s ether sunk or not, sold or not. Once a partcular unt of an asset s sunk t acqures some propertes of a non-marketed asset, even f t can be traded n a second-hand market. Usually, the value sunk n a project s not specfed. Defnton 5. Value sunk n a project s the dfference between the value of the project and what can be realzed by sellng some or all of ts marketable consttuents. For a sngle type of captal, the value sunk s V (P t ) V P t \u 0K 0 + p t K t. (10) The values of smlar assets n a project are not necessarly equal to that of the margnal unt of that type of asset, nor, f sold, are necessarly sold at once. dfferent value s mpled f not all of the unts of a gven type of captal are beng consdered for possble sale. Some molecules of water evaporate from a lake. Others do not. There are a multtude of such values, then, dependng on what comparson s made. Also, through complementartes, non-marketed captal contrbutes to value sunk. If the project s magned as beng possbly completely wound up, the value sunk s V (P t ) V (P t \ u 1 t Kt 1,..., u kt t Kt kt )+ kt =1 p t (Kt). 9 A

10 Dxt and Pndyck (1994: 211) vew a frm s value as beng largely the value of a set of optons. The external (second-hand-market) prce, p t (Kt), s the prce of an opton to sell, as s mpled by equatons (7) through (10). If a transacton s the best opton, as n equaton (9), t s acted upon. Remark 1 The value of the assets sunk n a project s the value of an opton to contnue wth the project rather than to sell some or all of the marketable captal. If the best opton s to keep the asset n the project (to contnue), the external prce, the prce of an neffcent opton, s not an approprate crteron for acton. For purposes of valuaton, t s not necessary to take the outsde opportunty nto account. As wth a non-bndng constrant, whose shadow value s zero, the value of the outsde opportunty may as well be zero. The best opton s chosen and that opton provdes the realzed cash flow. One of the optons may be to rent out the captal for a perod of tme less than ts remanng lfe. If the rent flow at any tme t s less than m t, one mght suggest that the rent puts a floor on the user cost of captal at t. But the level of and the opportunty of obtanng that rent does not affect the realzed cash flow from the project at any tme (ncludng t) and does not affect the choce of the captal stock. Agan, the value of the opportunty to rent the asset out s the value of a sub-optmal opton and should not affectwhatsdeemedtobetherangeofpossblevaluesofthe asset s current rental, and hence not affect ts mpled deprecaton. Proposton 3 The value of an outsde opton s not used n valung sunk captal. In tradtonal captal theory, captal s valued at margnal values. Snce for a functon f, df (x) = f(x) dx, margnal values are addtve. However, only f Euler s (very strong) condton, that f (x) = f(x) x, holds s value f (x) addtve or gven by margnal prces. Furthermore, a margnal analyss does not gve a complete analyss 10

11 of captal: Non-marketed captal does not have physcal unts and does not have a margnal value. 2 When an rreversble nvestment s beng contemplated, even under certanty, there s an optmal strke or stoppng tme for whch the net present value of the nvestment opton or opportunty s maxmzed (Carns and Davs 2007). A margnal condton s also a form of stoppng condton, expressed n terms of a quantty rather than of tme. It sgnals how much to nvest (when to stop ). Only the margnal unt bought or sold can be valued at the margnal prce. The market prce s possbly a lower bound on the value of the tem to the owner but not f the owner s consderng sellng the asset and s watng for the optmal tme. Then the value to the owner ncludes an opton premum. Proposton 4 Margnal values or margnal shadow prces are nputs to stoppng condtons n quantty space. They are relevant to tangble captal only. They are not nputs to the measurement of captal value. Three propertes that mltate aganst the use of margnal or shadow prces to value captal are (1) non-constant returns to scale and non-convexty, (2) non-optmalty and (3) the mportance of non-marketed captal. Even though margnal values are defned for the marketed captal, they cannot be used to measure the value of captal. The prmary captal entty s the project. Its value s well defned. Measures of ts components requre economc assessments of ther contrbutons to dscounted net cash flow. In realty, frms also nvest and dvest captal, reft, merge and so on. They are made up of a portfolo of projects and bundled together usng yet more non-marketed 2 The ncremental value of asset (attme0)sv P 0 \K0 j = M j 0. However, ths value s equal to the sum of (a) the M j 0 and (b) the non-marketed resdual. Snce each ncremental value ncludes non-marketed captal value, they do not sum to total value, V (P 0 ). 11

12 captal. The value of a frm at any date s ts net present value plus the values of the net nvestments n comprehensve captal made at that date. Values are updated recursvely usng a perpetual nventory smlarly to values defnednthenatonal accounts. When new captal (ncludng non-marketed captal as a resdual) s added, t s coalesced nto the project or frm, so that condton (7) holds. The analyst smply adds ts value to that of the project. When captal s retred or sold ts market value s subtracted from the project value at that tme. Under certanty, that value s foreseen n the rental schedule. Some forms of non-marketed captal may be common to more than one project. 5. CONCLUSION In much of captal theory, the value of captal s measured through the equalty of the margnal value to the frm and the external, market value. The paper has argued (a) that sunk costs are mportant for the contnung valuaton of the frm and (b) that there are mportant forms of non-marketed captal n any project or frm. Margnal values are not approprate for valuaton. Moreover, the opportunty to sell a marketable asset s vewed as a real opton. Even f external market prces exst for some forms of such captal, they have a specfc role n determnng whether an asset s sold or not but not n valuaton of sunk captal. If the asset s not sold (remans sunk) then, lke non-marketed captal, t has only a derved prce, whch does not affect optmal decsons n the project. The paper also argues for cauton n the choce of numerare. Use of observed, nomnal prces may be preferable to estmated, real prces for valuaton. 12

13 REFERENCES [1] Abraham, Katherne G. (2005), What We Don t Know Could Hurt Us: Some Reflectons on the Measurement of Economc Actvty, Journal of Economc Perspectves 19, 3, [2] Carns, Robert D. (forthcomng 2013), The Fundamental Problem of Accountng, Canadan Journal of Economcs. [3] Carns, Robert D. and Graham A. Davs (2007), Strke when the Force s Wth You: Optmal Stoppng and Resource Equlbra, Amercan Journal of Agrcultural Economcs 89,2, [4] Corrado, C., J. Haltwanger and D. Schel, eds. (2005), Measurng Captal n the New Economy, Chcago: Unversty of Chcago Press. [5] Dewert, W. Erwn (2006), The Measurement of Income, CAER Workng Paper 2006/11, Faculty of Commerce and Economcs, Unversty of New South Wales. [6] Dewert, W. Erwn (2009), The Aggregaton of Captal over Vntages n a Model of Emboded Techncal Progress, Journal of Productvty Analyss 32: [7] Dxt, Avnash K. and Robert S. Pndyck (1994), Investment under Uncertanty, Prnceton Unversty Press. [8] Doms, Mark E. (1996), Estmatng Captal Effcency Schedules wthn Producton Functons, Economc Inqury XXXIV, [9] Jorgenson, Dale W. (1996), Emprcal Studes of Deprecaton, Economc Inqury XXXIV, [10] Olner, Stephen D. (1996), New Evdence on the Retrement and Deprecaton of Machne Tools, Economc Inqury XXXIV,