The Macroeconomics of Private Equity Boyan Jovanovic and Peter L. Rousseau April 25, 208 Abstract We ask two questions about private equity. Why are returns to venture funds higher than those of buyout funds? And why does the investment of venture funds respond more strongly to the business cycle than that of buyout funds? To address them, we build a model in which venture and buyout play different roles in the private equity market. Venture brings in new capital whereas buyout largely reorganizes existing capital, and this can explain the stronger co-movement of venture investment with aggregate Tobin s Q. This stronger co-movement stems from a higher correlation of venture returns with aggregate consumption and therefore a higher premium than buyout, and a thicker right tail in the distribution of projects funded through venture magnifies that difference. The model embodies this logic and fits the aggregate time series of private equity investment and returns well. Introduction Private equity funds a growing share of real investment in the economy, averaging more than six percent of U.S. private investment since 200. We build a model consistent with that level, and with its cyclical properties. We fit both the level and the returns to private equity investment. Investment in both venture capital (VC) and buyout funds rise relative to other forms of investment as a function of aggregate Tobin s Q, but venture responds more strongly. Such VC intake is more Q-elastic than buyouts because buyout funds reorganize existing capital, which is costlier to acquire when Q rises. VC activity does not face this impediment, whereas ordinary investment faces it to an even greater extent than buyout. At the same time, the payoffs to both types of private equity activity rise with Q. Department of Economics, New York University, and Department of Economics, Vanderbilt University, respectively. We thank Sai Ma for help with the research, and the National Science Foundation and C.V. Starr Center for financial assistance.
3.5 3 log of intake ratio (right scale) 0.6 0.4 2.5 2.5 0.5 aggregate Q (left scale) 0.2 =0.0-0.2-0.4 log(intake ratio t ) = -0.268 + 0.64 Q t- (-.75) (2.06)r-0.6 995 2000 2005 200 205 0.5.5 2 2.5 3 3990 3.5 Aggregate Tobin's Q 2Figure : The relation between the log ratio of venture and buyout fund invstment to aggregate Tobin s Q, 987-206 The left panel of Fig. shows the response of the intake levels of VC and buyout funds as a percent of private domestic investment to fluctuations in Q from 987 to 206, and the scatterplot in the right panel shows the response of the ratio of venture to buyout intake with respect to Q. The ratio increases in Q because a high Q raises the costs of buyout investment more than it does for VC given that most bought-out capital is in place to begin with. Relative payoffs, in contrast, do not depend on Q because ventured or bought-out capital collects Q per unit in both cases when it is sold to the public. Only the costs differ. Opp (208) also models the implementation of heterogeneous ideas by VC funds, and in his model as in ours VC activity is procyclical, with funding standards declining during booms in the sense that the average quality of implemented ideas is countercyclical. He analyzes venture only, whereas we also analyze buyouts and the movementsintherelativeintakesshowninfig. aswellastheperformanceofventure and buyout funds. Fig. 2 shows that returns to each type of investment fund also differ over the cycle. Both outperform the S&P 500, and both co-move with it. Venture returns are higher and respond more strongly to returns on the S&P 500 stocks, and these properties arise in our model when the distribution of ideas has a thicker right tail for venture than for buyout. This is realistic if venture funds handle a larger fraction of radical innovations. Our model delivers premia over the S&P 500 return because private equity pays more when the cost of capital is high, but that s also the time when aggregate invest- See Appendix A for a description of all data and methods used in the empirical analysis. 2
8.6 3.83 in 999 7 6 5 Venture (% GDPI) Buyout (% GDPI).4.2 Buyout funds 4 3 2 Aggregate Q 0.8 0.6 S&P 500 Venture funds 0 990 995 2000 2005 200 205 0.4 990 995 2000 2005 200 205 Figure 2: The intakes of venture and buyout funds with respect to gross private domestic investment and their respective returns, as related to Tobin s Q and the S&P 500, 987-206 ment is low and consumption is high. Returns to private equity funds must therefore be higher to compensate for the lower marginal utility of consumption. On the other hand, earnings of the S&P 500 firms depend primarily on TFP shocks, and there is no incentive to substitute toward consumption when TFP is high. Hence the premium for private equity. 2 A complementary explanation of the PE-return premium is the illiquidity of PE investment: there is an effective lock-up period ranging up to ten or twelve years. Sorensen, Wang and Yang (204) model the effects of these lock ups and find an annual premium of slightly more than one percent which we subtract from the returns that we target in the model, because our premia are driven entirely by the correlation of PE returns with aggregate consumption. Our discussion of PE is confined to PE funds. A broader measure of private equity include occupation-specific investments in physical and human capital such as are made by self-employed people. Hamilton (2000) and Moskowitz and Vissing- Jorgensen (2002) finds the risk-adjusted return to self employment insufficient to compensate for the foregone wage earnings, and that perhaps the non pecuniary benefits play a major role. Such benefitsarepresumablynottherewhenoneinvestsin PE funds. Vereshchagina and Hopenhayn (2009) point out, however, that the option of pulling out of a low-realized-return PE investment can also lower its equilibrium return. 2 Not pictured is the relation between returns and investment, but Kaplan and Stromberg (2009, Table 3, panel B) find that private equity commitments rise as a function of returns realized over thepreviousyear,andthisoccursinourmodelaswell. 3
Earlier work finds that the mode of business investment varies across firms and over the cycle; in particular, young firms contribute a larger fraction of investment when stock prices are high. Gompers et al. (2008) find that VC investment is positively related to Q. Jovanovic and Rousseau (204) show that a rise in aggregate Tobin s Q leads incumbents to reduce investment, thereby creating opportunities for investments by new firms. 2 Model We begin with a model of the real side of the economy in Sec. 2., and then add a private equity sector in Section 2.2. 2. An structure with two capital stocks Output,, is produced with physical capital,, andhumancapital,, as = () where is a shock. Output is consumed,, or invested in amounts and h in the two types of capital; the income identity reads Thelawsofmotionfor and are = + + h h (2) 0 = ( ) + and (3) 0 = ( ) + h (4) where a prime, 0, denotes a variable s the next-period value, and is the common rate of depreciation. P Households. Households are homogeneous with preferences are 0 =0 ( ) ª, with () = They own, which earns the competitive wage =( ) and households receive the aggregate wage bill =( ). Households own the firms, which pay out profits ( ) = as dividends back to the households. Optimal investment in requires that the cost of a unit of human capital equals the present value of its expected wage payments. Written recursively, this condition reads Z µ h = µ( ) 0 0 +( ) 0 0 h ( 0 ) (5) 4
where 0 is next period s output, and where the three aggregate shocks, ( h ) follow first-order Markov processes with the CDF of 0 given by ( 0 ). Firms. They own and optimal investment in requires that the cost of a unit of physical capital,, equals the present marginal value of expected dividends (discounted at the household s stochastic discount factor) of that unit. Expressed recursively, the condition reads Z µ 0 = µ 0 +( ) 0 ( 0 ) (6) 0 ³ This too is the price of a unit of, and the gross return on equity is E = 0 +( ) 0. 0 Let h (7) If is a constant, the model simplifies as and can be aggregated into a composite that is proportional to. Let = (8) and let be the scaled TFP shock Wesummarizetheresultsasfollows: = (9) Proposition If is constant, then is proportional to, output can be written as a function of alone, the goods cost of a unit of composite capital is = (0) = () ( + )= (2) and the income identity reduces to a function of,, and composite capital investment,, reads = + or in units of, where = is investment in physical capital. = + (3) 5
Proof. Equations (8) and (0) imply that ( ) 0 = 0 0 (4) 0 Substituting from (7) and (4) into (5) yields (6). Eq. (2) follows from (7) and (8). Finally, (2) reads = + + h h = + ( + h )= + ( + ) and division by yields (3). Return on equity. By () we have 0 0 = 0 and therefore the gross return on equity is E = (0 +( ) 0 ) (5) 2.2 Private equity and the implementation of ideas The arrival process for new ideas. Households get new ideas; their number is proportional to the aggregate stock of human capital,. The number of ideas implementable by VC funds is, and the number implementable by buyout funds is. Not all the ideas are implemented. The presence of is external to the households; a household does not take it into account when choosing its human capital investment which still satisfies (5). Formally, the production function () and the law of motion for in (4) remain the same, but in (3), is broadened as follows: = X (6) {} where represents investment in continuing projects, is venture-backed investment in new projects, and is investment mediated by buyout funds. In other words, all three methods create the same commodity physical capital, but their production functions differ.we nowdescribeeachinturn: () Continuing projects. c is created via existing projects and its total cost is c in units of consumption. () VC-backed projects. v denotes efficiency units of created by implementing new projects. A project uses as inputs a unit of the consumption good and an idea. As output it delivers units of capital ready for use in the next period. The quality of the project,, is known at the start. New projects are born each period, and their quality is distributed with a CDF v (). Households get ideas at the rate so that the unnormalized distribution of new ideas is v (). () Upgraded projects. b denotes efficiency units of created by buyout funds; units of existing physical capital canbeupgradedatacost per unit. When 6
upgraded, its efficiency changes from unity to. Idea qualities for upgrading are described by s drawn from the CDF b. The unnormalized distribution of new ideas is b (). As with the venture funds, is known before the fact. Contracting between agents and PE funds. Funds raise new subscriptions each year and close the following period. Each vintage of investors thus receives a return on their investments alone there is no mixing of dividends among investments of different vintages. Costs predate returns by a period. 3 We assume that the PE fund owners get all the rents from the projects they invest in. It takes one period for the projects to mature, and a fund s revenue from selling the project is 0 0. Capital 0 created though new projects is sold by venture funds to incumbent firms or floated through IPO at a price equal to the cost,, of making capital via the incumbents technology. Capital created through buyout activity results in an effectively larger capital stock in the hands of existing firms. 2.2. Venture Funds Let v be the minimum project quality accepted by a VC fund. We shall measure intake and payout relative to, and using (0), relative to the number of ideas is = The intake of VC funds. Each implemented idea costs one unit of consumption, and the total VC fund investment then is the same as the number of projects v # VC-backed projects = [ v ( v )] (7) VC fund payout. Since capital is delivered in time for next-period production, i.e., next period dividends are the closing revenue v ( 0 )=( 0 +( ) 0 ) v where Z v = v (8) v Gross return of VC funds. Realized returns are v v ( 0 ) v The VC fund s project-portfolio decision. A VC fund chooses v to maximize the expected utility of its investors. ½ Z µ ¾ max v 0 v ( 0 ) ( 0 ) v (9) 3 The typical fund lasts 0-2 years but our data will allow us to infer the year-to-year returns. 7
Proposition 2 The minimal accepted quality of VC-backed projects is v = (20) Proof. Since R 0 v = R v R ³ 0 v 0 +( ) 0 0 (6) reduces the VC s maximization problem to µ Z v () max v v v and (20) follows. 2.2.2 Buyout funds v = R v v, (2) Let b be the minimum project quality accepted by a VC fund. The intake of Buyout funds. Each implemented idea costs units of consumption and one unit of physical capital the price of which is. The total buyout fund investment then is b ( + ) b ( b ) (22) Buyout fund payout. Next period dividends are the closing revenue b ( 0 )= µ 0 +( ) 0 0 where Z b = b (23) b Gross return of buyout funds. Realized returns are b b ( 0 ) b The buyout fund s project-portfolio decision. The fund chooses b to maximize the expected utility of its investors, Z µ 0 b ( 0 ) ( 0 ) b and using the same logic as that behind the proof of (20), we get µ Z b () max b b (24) b b 8
and the optimal buyout cutoff b = + (25) To summarize: Firms live forever and are publicly owned with share prices in (6). Funds live for one period and are not publicly tradeable. The timing within the period is as follows: () () and the profile of s are realized at the start of the period; () Production, and idea-implementation occur, determining v b, c and Γ 0 ; () Theequitymarketopens;firms invest, converting goods into and buying up the PE-mediated capital at the price. PE funds distribute their profits to households and consumption takes place. 2.3 Equilibrium and its properties Proposition continues to hold with the two forms of private equity except the accounting equations change slightly. First, the law of motion for in (3) changes to 0 = + (26) where = c + v + b Second, in our new notation, the RHS of the income identity (3) reads = +,the presence of reflecting the investment of goods into, which itself is proportional to The two PE funds, however, generate at a cost lower than and the difference is reflected in their profits v + b whichareaddedtothelhsof(3)alongwith + b of the buyout funds costs, which are transfer payments. Thus the income identity becomes + v + b + + b = + (27) With both and dropping out of the equations, the state of the economy is =(). Using(),were-write(6)as Z µ ( 0 ) = () Γ () ( 0 +( ) 0 ) ( 0 ) (28) The functions ( ) are defined in terms of primitives, and the remaining unknowns are () c (), andγ () These functions solve Eqs. (27), (26) and (28) for all. Existence and characterization. If Z µ + () (29) then Appendix B proves the following result: 9
Proposition 3 There exist a constant 0 such that + P h ³ i ()+ + +( ) = (30) + ³ + P h ³ i ()+ + ( ) = (3) + 2.4 Implementation Since venture deals with implementation of ideas that need not need to relate as much to technologies that are embodies in capital already in place, we argue that it, more than buyout, deals with ideas drawn from a distribution with a thicker right tail. This notion is easiest to model cleanly if we from the outset use Pareto distributions. Better fits could no doubt be obtained if we were to assume only tail properties but otherwise kept the two distributions general in form, but this would interfere with the brevity of the exposition. Distributions of ideas. We assume that the distributions are Pareto: i () = ³ 0 { }, where b v so that the means exist and so that the venture distribution has a thicker tail. The intakes. Then the intakes are v ( v ) v 0 v and µ b + b ( b )= ( + ) = b 0b 0b ( + ) b b The intakes are both increasing in as Fig. indicates. The intake ratio v b = v ( + ) 0 v ³ + 0b b v b ( + ) b = v b ( + ) b (32) is also increasing in if v (33) b v an inequality that holds at the estimates reported in Table 3. As 0, v v (34) b 0
which is a log-linear version of the regression reported in the right-hand panel of Fig.. Returns. Gross returns to private equity are payouts divided by the intake: for { }. The return to VC is therefore R v =( 0 +( ) 0 ) v v v = 0 +( ) 0 v v = v v E (35) which is increasing in end-of-period q and decreasing in start-of-period q. The return to buyout funds is R b = 0 +( ) 0 ( + ) b b b = 0 +( ) 0 ( + ) b + b = b b E (36) with similar implications for q-elasticity. The return ratio, therefore, does not depend on : v = T as b b T v (37) v v b b The intuition for the result in eq. (37) is the thicker right tail for venture means that more of its projects are drawn in when rises,and therefore venture payout us more correlated with and, hence, with aggregate consumption. The profit for VC funds is µ v µ v v v () = 0,v v v where the last equality follows from (20). Buyout profit is µ b µ b b b () = 0,b b b ( + ) = = b 0,b b ( + ) b µ b where the last line follows from (25). 2.4. Regression tests µ v µ v v = 0,v v = v v 0,v v (38) = b b 0,b ( + ) b µ b µ + b ( + ) 0,b b b (39) Recall that Kaplan and Stromberg (2009) measure returns over the previous year and find in Table 3 that they relate positively to current period commitments. Our model predicts this; commitments rise with in andsodoreturnsoverthepreviousperiod see Eqs. (35) and (36). Gompers et. al.. alsofind that an increase in initial public offering (IPO) valuations leads venture capital firms to raise more funds, an effect that is particularly strong among younger venture firms (Kaplan and Schoar, 2005).
We pause to verify the implications of the model for observed intakes and returns toprivateequity.tablereportstheresultsoftimeseriesregressionsonthelogsof venture and buyout intakes, as well as their log ratio, with respect to aggregate Tobin s and the productivity shock. We describe the data sources and methods in Appendix A. The table shows that the log intakes of both venture and buyout funds are positively related at the one percent level to aggregate (i.e., measured at the start of the period), but that venture intakes are more responsive to. The regressions in the right panel show that the log ratio of the intakes is also positively related to at the one percent level when we include a time trend and at the five percent level without a time trend. Table Private Equity Intake Regressions, 987-206 ln( ) ln( ) ln( / ) Q.4 0.94.248 0.69 0.64 0.250 (7.23) (6.57) (5.47) (4.23) (2.07) (2.87) trend 0.060 0.07-0.0 (6.52) (6.77) (-.98) constant 7.763 7.640 8.030 7.884-0.268-0.244 (20.66) (3.94) (8.28) (28.85) (.77) (.68) 2 0.65 0.865 0.57 0.82 0.33 0.242 No. of observations 30 30 30 30 30 30 Note. Estimation is by OLS, with T-statistics in parentheses. Q is aggregate Tobin s Tobin s Q measured at the start of year. Thevariables and are in millions of constant 2000 dollars. Table 2 reports time series regressions for the returns to the two funds and their ratio on aggregate and the productivity shock. Returns to both venture and buyout funds are related negatively to start-of-period and positively to end-ofperiod as the model predicts, and typically at the five percent level or less, and returns to venture are more -elastic than buyout returns The regressions in the right panel, while consistent with the model in showing no significant relation between the ratio R /R and, indicate that the relation with is positive and statistically significant. 2
Table 2 Private Equity Fund Return Regressions, 987-206 Dependent variable: fee and inflation-adjusted returns R R R /R Q -0.54-0.53-0.254-0.256-0.2-0.222 (-2.5) (-2.25) (-5.26) (-5.3) (-.0) (-.5) Q.04 0.966 0.32 0.304 0.557 0.527 (.0) (3.92) (6.24) (6.03) (2.79) (2.62) z 4.258 0.67 2.708 (.34) (.03) (.05) constant 0.22 -.30 0.996 0.756 0.38-0.586 (0.76) (-.2) (7.6) (3.7) (.69) (-0.62) 2 0.443 0.479 0.59 0.607 0.34 0.342 No. of observations 30 30 30 30 30 30 Note. Estimation is by OLS, with T-statistics in parentheses. Q is aggregate Tobin s Q measured at the end of the year. 2.4.2 Simulations of the model In this section we simulate the model and evaluate its performance with respect to the available time series for private equity returns, intakes and other variables of interest. To do this, we assign values to,,, and and then choose,,,,, 0,b, and 0,v to target the means of,, v, b,,,and S&P. 4 The National Income and Product Accounts provide us with k, the ratio of output to physical capital. From it, we compute using Eqs. (7)-(9) as follows: = k + Gross domestic product is then defined as µ = = k + 4 We take the S&P 500 return, denoted by S&P, as our observed measure of the return on equity in (5) 3
such that the true output and output-capital ratio are higher after bringing human capital into the measurement. 5 Table3reportsthevaluesweassignforparametersfor,,, and, alongwith those we estimate in the course of simulating the model. Table 3. Parameter settings for simulation. b v 0,b 0,v 025 095 5 00 567 00 03028 00098 00329 02000 0250 Table 4 reports the means of the various series of interest as simulated and in the data for 987-206. We subtract.05 percent from buyout and venture returns, which is liquidity premium reported in Sorensen, Wang and Wang (204) for these types of funds. 6 This value was also subtracted from the data in the series shown in the top two panels in Fig. 3. The model matches most of the means in the data well. It overestimates the mean return to venture funds by about four percentage points, however, and our estimate of venture intake is only one-third the value in the data. Table 4. Fit of model to observed means Target? v 0024 0008 Y b 0023 002 Y mean 69 20 Y mean 08 08 Y mean S&P 088 089 Y 0300 027 Y 020 054 Y 0050 0064 N Figure 3 shows the intakes of venture and buyout funds as percentages of gross private domestic investment in the data and in the model. The series in the upper panel have a correlation of 0.68, and indicate the model can reproduce the spike in ventureinvestmentthatoccurredin2000,albeitoverlyso. Atthesametime,the model does not deliver sufficient venture investment in years further from the peak. This occurs because 0,v must be larger to get more investment outside the peak, and a smaller v is then required to moderate the higher peak that would otherwise result. Both of these adjustments, however, would raise the simulated mean return to venture funds well above the data. The lower panel of Figure 3 shows that the 5 This is also how Prescott and McGrattan (200) treat unmeasured investment it is lost output as in the Ben-Porath model where is built by witholding time from production and not using goods that are produced. 6 Therefore our targets are VC =.79-0.005 =.68 and BO =.9-0.005 =.08. 4
n V (right scale) model 8 2 data 6 6 n B (left scale) 0 4 2 model data 0 990 995 2000 2005 200 205 Figure 3: Returns to buyout, venture, and the S&P 500 in the model and the data, 988-206 model fits the cyclical properties of the intake for buyout funds quite well, with a correlation of 0.76 between model and data. Figure 4 shows the fit between the model and data with respect to the returns series. The model once again fits buyout well with a correlation of 0.28 and, with the exception of the spike in 999, fits the returns to venture funds even more closely with the correlation of model and data of 0.26 including the spike. The model lags the data for the S&P 500 return, but the correlation of the model s prediction with the lagged data is 0.90. 3 Conclusion We document that the returns to venture funds are higher than those of buyout funds, and that venture funds intake responds more strongly to the business cycle 5
.4 R B (left scale) data.0 0.6 model R V (right scale) data model 4 2.4 R S&P (left scale) model 0.0 0.6 data 990 995 2000 2005 200 205 Figure 4: Returns to buyout, venture, and the S&P 500 in the model and the data, 988-206 than buyout funds intake. The model assumes that venture brings in new capital whereas buyout largely reorganizes existing capital, leading venture intake to co-move more strongly with aggregate Tobin s Q. A thicker right tail for venture ideas also relates to venture s higher returns. More venture activity is drawn into the economy when rises and in the model agents consume more when is high. Venture returns are thus more strongly correlated with aggregate consumption than buyout and must pay a higher return. An extension we do not pursue is that of returns to self employment. Perhaps the modeling distinction we make between how VC and buyout funds add value to capital also applies in the domain of self employment choices. One can open a store in an entirely new location, or one can buy someone else out or simply take over a location that someone else has vacated. 6
Appendix A. Data and Methods In this appendix we document the data sources and methods used to construct the seriesdepictedinourfigures and included in the empirical analysis. Figure. The intakes are the sum of investments made annually in U.S. venture capital and buyout funds, divided by annual estimates of gross private domestic investment from the BEA (207, Table 5.2.5, line 4). Venture and buyout investments are from the April 207 version of Thomson One s VentureXpert database, and are the sum of all investments made in a given calendar year at any stage or round across fundsofeachtype. The intakeratio, v b, is the ratio of the respective investment sums in each year For aggregate, we use fourth quarter observations underlying Hall (200) for 987-999, and then join them with estimates underlying Abel and Eberly (20) for 999 to 2005. Abel and Eberly derive aggregate Tobin s from the Federal Reserve Board s Flow of Funds Accounts as the ratio of total market value of equity and bonds to private fixed assets in the non-financial corporate sector. We bring these estimates forward through 206 using the same sources. Hall s measure of in 999 is higher than that of Abel and Eberly (3.376 vs..89), so we use a ratio splicing factor of.856 to adjust the series from 2000 forward. Figure 2. Aggregate returns to U.S. venture capital funds for 987-206 are from Cambridge Associates (206a), and are annualized returns constructed by compounding one quarter horizon pooled returns. Aggregate returns to U.S. buyout funds for 987-206 are from Cambridge Associates (206b), and are also annualized returns constructed from single quarter horizon pooled returns. Both series are net of fees, expenses, and carried interest. We convert each series into ex-post real returns using the annual growth of the consumer price index from the National Income and Product accounts. Annual returns to the S&P 500 are from Damodaran (208, http://www.stern.nyu.edu/~adamodar/pc/datasets/histretsp.xls), and deflated by the consumer price index. We then subtract.05 percent from both venture and buyout returns, which is liquidity premium reported in Sorensen, Wang and Wang (204). Figures 3 and 4. For,weuseprivateoutput,defined as GDP less government expenditures on consumption and investment from the BEA (207) for 987-206. We then divide the result by after adjusting it for inflation during year by averaging the annual inflation factors across the two years that overlap and then using its square root as a deflator. The are end-year stocks of private fixed assets from the BEA (207, table 6., line ) for 987 through 206. The aggregate investment rate is constructed as annual gross private domestic investment from the Bureau of Economic Analysis (BEA 207, table 5.2.5, line 4) for 987-206 divided by. 7
Appendix B. Proofs ProofofProposition2:Let satisfy the equation = µ Z µ + + P +( ) [ +( ) ] () (40) First, suppose that a solution for exists (its existence will be shown at the end of this proof). Since 0 = 0 0, (28) implies µ Z = ( 0 ) [ 0 +( ) 0 ] ( 0 ) (4) 0 Therefore 0 = µ Z ( 0 ) [ 0 +( ) 0 ] ( 0 ) where 0 is defined in (26). ³ Now write = +. From income identity (27) = + P + ( ) thus we have 0 = = + [ + P + ( ) ] ³ + ( + P + ( )) where the second line uses the identity (3) which implies = Therefore µ Z ³ + ( + P = ( 0 ) [ 0 +( ) 0 ] + ( )) i.e., µ + ³ + X + ( ) = µ Z + ( 0 ) [ 0 +( ) 0 ] 8
i.e., = = where is defined in (40),and + ( + P + ( )) ³ + R (0 ) [ 0 +( ) 0 ] ( 0 ) + P + ( )+( ) + R ( 0 ) [ 0 +( ) 0 ] ( 0 ) = + P ()+ ( )+( ) + = + P + ( ) = [ + P + ( )] ( ) + Existence of L:. Next, we show that exists when (29) holds. Divide both sides of (40) by to get = = µ Z µ Z µ µ + + P [ +( ) ] +( ) + + P [ +( ) ] +( ) Since ranges from zero to infinity as ranges over the positive line, and since 0, a necessary and sufficient condition for a solution for to exist is that µ Z µ + P +( ) [ +( ) ] This is equivalent to Z +( ) ( + P ()+( ) ) (42) Since and are positive, for (42) to hold it suffices that Z Z µ +( ) ( +( ) ) = + i.e., (29). 9
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