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The Volatility of the Socially Optimal Level of InvestmentRoss S. Guest^{[a]} and Ian M. McDonald^{[b],}^{a}Monash University and the ^{b}University of Melbourne ABSTRACTIn this paper an annual series for the socially optimal level of investment from 1960-61 to 1993-94 for Australia is derived from a vintage production function and compared to the actual series for investment over the same period. The vintage production function can be expected to yield a smoother socially optimal investment series than that derived from a non- vintage, malleable capital production function. Even so, the resulting series for socially optimal investment is much more volatile than the series for actual investment. Several alternative assumptions are tried in an attempt to further smooth the socially optimal investment series. These include smoothing the assumed values of the exogenous variables, modelling adjustment costs and delivery lags, and changing the form of the production function. While these approaches do succeed in smoothing the investment series, in order for the socially optimal investment series to be as smooth as the actual series the assumed values of the parameters must be quite unrealistic. In the conclusion we suggest that in future research on the socially optimal level of investment the role of liquidity constraints and of irreversible investment should be investigated. 1. Introduction.In public debate and discussion of macroeconomic performance, the level of investment is subject to frequent scrutiny and comment. Often concerns are raised that the level of investment is too low. Higher levels of investment are urged in the expectation that they will, by raising labour productivity, raise living standards. In spite of these concerns, economists have devoted little attention to calculating the socially optimal level of investment spending. It is true that there is a considerable literature on modelling the investment behaviour of privately optimising firms in an attempt to explain the pattern of actual investment (see Chirinko, 1993, for a survey). However, the privately optimal level of investment for the individual firm may be different from the socially optimal level. Some reasons for this are discussed in the conclusion to this paper. An empirical paper that has used the analysis of the social optimum to evaluate the actual level of investment is Abel et.al. (1989). They developed a criterion for determining whether an economy is dynamically efficient. Their rule is a generalisation of the rule, in Diamond (1965), for dynamic efficiency in a steady state. However, the criterion in Abel et. al. does not determine the socially optimal level of investment - it simply tests for whether the actual level of investment has exceeded the golden rule level. In this paper we develop and apply to Australia a method for calculating the socially optimal level of investment. This exercise is part of a broader project in which we are attempting to calculate the socially optimal level of aggregate saving and its socially optimal disposition between investment expenditure and the accumulation of foreign assets. Our calculations yield a series for the socially optimal level of investment which has, compared with the series of the actual level of investment, a very high coefficient of variation^{[1]}. This is notwithstanding the use of a vintage putty-clay production function to calculate our series. Such a production function will yield a smoother series than a production function with malleable capital. It seems, at first sight at least, unreasonable to argue for a socially optimal investment series with such a high level of volatility because, while one can think of reasons which suggest that the level of socially optimal investment may differ from the level of actual investment, it is more difficult to think of reasons for a large difference in the variances of these two series. Because of this we investigate various ways in which the series can be smoothed. These include smoothing the exogenous variables, varying the elasticity of substitution between labour and capital and introducing adjustment costs of investment. Even with the choice of reasonable degrees of smoothing using these techniques we find that the series for the socially optimal level of investment spending is still very volatile compared with the series for the actual level of investment. This result is discussed in the conclusion. 2. A Neoclassical Vintage Model: The Base Case2.1 Why a vintage model?In neoclassical theory, the (non-vintage) production function is of the general form: (1) where K_{j} and L_{j} are the capital stock and employment levels at time j, respectively. The early neoclassical model of investment was developed by Jorgenson (1963). It assumes perfect competition and determines the optimal capital stock at the level of the firm by assuming either that output is exogenously given and the firm faces constant returns to scale, or that there are diminishing returns to scale and output is endogenous. One of these alternative assumptions is necessary for the capital stock to be determinate at the level of the firm, which establishes microfoundations for aggregate investment. However, if one is prepared to model aggregate investment assuming that aggregate employment is exogenous, then the problem of an indeterminate capital stock no longer exists. Instead, the optimal capital stock is determined by only one first order condition: that the marginal product of capital equals the (real) user cost of capital. The problem is that if optimal investment were derived from this first order condition, it would exhibit massive volatility. A simple example gives the intuition for this result. Suppose that the ratio of investment to output is 0.2 and the ratio of the capital stock to output is 4.0, plausible ratios for Australia. A change in the user cost of capital which causes the desired capital stock to change by say 1% will cause investment to change by 20%, from 20% of output to 24% of output. The early neoclassical models tried to overcome this problem by defining investment as a distributed lag function of the optimal capital stock. The lag function is intended to capture the observation that investment projects take time to complete as a result of, for instance, unforseen delivery lags. The effect is to reduce the volatility of investment which would occur if the entire capital stock were to respond instantly to a change in the user cost of capital. Since the lag function is not derived explicitly from optimising behaviour, it introduces an ad hoc element into the investment model. In later models capital is assumed to be costly to adjust, which is captured by an adjustment cost function. Like delivery lags, the adjustment cost model also has the effect of smoothing the series for aggregate optimal investment. A vintage production function, as used in this paper, will yield a smoother investment series than the non-vintage model, ceteris paribus^{[2]}. This is because the vintage model requires a change in only the marginal vintage of capital, rather than the entire stock of capital, in response to a change in the user cost of capital. Aggregate investment in each year is defined to constitute a vintage of capital. Hence, in the simple numerical example given above, it would not be the aggregate capital stock that changes by 1%, but simply aggregate investment. The resulting series for investment will clearly exhibit less volatility than the non-vintage model. We can then apply modifications, such as adjustment costs, to the base case in order to further reduce investment volatility. 2.2 Determining the socially optimal level of investmentAn equation determining the socially optimal level of investment can be derived from the following optimisation problem. A representative agent maximises a concave utility function: (2) where C_{j} is aggregate consumption in year j, h is the length of the planning horizon in years, W_{h} is the level of real wealth at the end of the planning horizon. Equation (2) is maximised subject to an output constraint given by the vintage production function: (3) where Y_{j} is output in year j, I_{k} is the investment in capital, which has a I year gestation, installed in year k, A_{k} is the efficiency parameter; T is the age of the oldest plant in use at time j and 6 is the depreciation rate; and subject to an international borrowing constraint: (4) where D_{0} is the level of overseas debt inherited in year 1; in is the (constant) proportion of the debt to be repaid in each year; r_{k} is the interest rate in year k; B_{j }is the level of overseas borrowing in year j. Terminal wealth is defined as (5) where K_{0} is the capital stock inherited in year 1. The first order condition for the determination of socially optimal investment is (6) where, as shown in Appendix A of Guest and McDonald (1996a), F_{j} and W_{j} depend on the pattern of interest rates over the horizon in the following way: (7) (8) Because the interest rates are exogenous, the model is separable in that the socially optimal value for investment is determined from (6) which is independent of the utility function. For the CES case the production function (3) can be written: (9) where l_{k,j} is the labour in period j working on capital installed in period k and the elasticity of substitution between capital and labour, s, is equal to 1/(1+q). The putty-clay case assumes that the labour requirement of each vintage of capital is exogenous to the firm between the time it is installed and the time it is scrapped. The vintage production function (9) is not a convenient form for empirical applications for the following reasons. Firstly, it contains the sequence of all past values of investment in vintages of capital currently in use. The data thereby required may severely limit the empirical applications. Secondly, l_{k,j} is not readily observable. We observe aggregate limit the empirical applications. Secondly, l_{k,j} is not readily observable. We observe aggregate employment in a given period but not the labour employed on capital of a given vintage. This problem is solved rather easily in the putty-putty case, since, in that case, labour is allocated competitively such that the marginal product of labour is equated across all vintages. This means that l_{k,j} can be expressed in terms of the (common) marginal product. This in turn implies that the aggregate production can be expressed in terms of the observable variable, L_{j}, aggregate employment. In the putty-clay model, however, labour cannot be allocated such that the marginal product of labour is equated across all vintages. However, it can be shown, see Appendix A of Guest and McDonald (1996a), that if the age of the oldest capital good in use is large, past values of investment and the labour working on new capital can be eliminated from (9). This yields the following equation for the production function: (10) We initially adopt the special case of a Cobb-Douglas functional form, where s=1: (11) By following a procedure similar to that for the CES case, I_{k}, k=j-1, j-2,...j-T, and l_{k,j} can be eliminated to yield: (12) For the Cobb-Douglas case, the socially optimal level of investment, using the first order condition (6) and the production function (12), is determined by (13) To calculate the level of socially optimal investment, measures of A_{k}, r_{k}, a, d, m and the planning horizon, h, are needed. The calculation of the parameters a, d, m and h are explained in Appendix C of Guest and McDonald (1996b). From these calculations the values are 0.37 for a , 0.051 for d, 0.15 for m and 130 years for h. These calculations are based partly on econometric estimation reported in that paper and partly on estimates in the literature. The value for h is based on the principle that the value chosen be such that a longer planning horizon does not after optimal outcomes in the initial time period of the plan by more than a specified degree of tolerance (see Guest and McDonald (1996a) for further discussion of this method for choosing the length of the planning horizon). The values for the efficiency parameter, Ak, are determined for the Cobb-Douglas case by: (14) Where Y_{k}, I_{k} and L_{k} are the observed values described above. The values for A_{k} are shown in Chart 1. The calculation of the interest rate, r_{k}, is described in McDonald, Tacconi and Kaur (1991). It is a real world rate and should be interpreted as the socially optimal rate at which Australia can trade current consumption for future consumption. Essentially, the calculation of r_{k} involves taking the 10 year government bond rates for the USA, UK and West Germany, then adjusting the rates for errors in forecasting inflation, and deflating the resulting series by the Australian consumer price index expressed in the currency of the respective country. A weighted average of the rates calculated by this procedure gives an interest rate for each year from 1960-61 to 1993-94, illustrated in Chart 2. It can be interpreted as the social opportunity cost of consumption. Over the period 1960-61 to 1993-94 the average value of the interest rate is 4.57%. This value is adopted for all future years beyond 1993-94. 3. Interpreting volatility: in the socially optimal investment series.The series for socially optimal investment from 1961 to 1994 is calculated using equation (13) with the values for the parameters and the exogenous variables described above. In Chart 3 this series is compared to the series for actual investment, where both series are expressed as a proportion of GDP. The socially optimal series is obviously much more volatile than the series for actual investment. We measure volatility as the degree of relative dispersion around the mean, given by the coefficient of variation, V^{[3]}. The value of V for the actual and socially optimal investment series is 0.085 and 0.452, respectively. Also, we measure the degree of persistence in the series by the first order serial correlation coefficient, p. The value of p for the actual series is 0.793 (with a t statistic of 6.18), while that for the socially optimal series is 0.367 (with a t statistic of 0.167). These facts indicate the much lower volatility and higher degree of persistence in the actual investment series compared to the socially optimal series. Also, there is a considerable difference between the levels of the actual and socially optimal investment series. The average actual I/Y ratio for the period is 0.249, while the average socially optimal IN ratio is 0.430. This issue is addressed in section 3.1.2 below. The purpose of this paper is to investigate ways of smoothing the socially optimal investment series illustrated in Chart 3. The effectiveness in smoothing the series is judged by how much the volatility is reduced. It is worth emphasising that our socially optimal series in Chart 3 is already a less volatile series than would have been generated using a non-vintage production function, as discussed in section 2. 1. 3.1.1 The effect of smoothing the exogenous variables.In this section we consider the smoothing effect on socially optimal investment of smoothing the series for the exogenous variables: r_{j}, A_{j} and employment growth. We then consider the smoothing effect on socially optimal investment of lowering the value of the elasticity of substitution, followed by an adjustment cost model and a model of delivery tags. Consider three scenarios illustrated in charts 4, 5 and 6. In each scenario one of the three exogenous variables contributing to investment volatility is held constant from 1960-61 to 1993-94, while the other variables are allowed to vary as in the original socially optimal investment series in chart 3. In chart 4, employment growth is held constant at 1.141%^{[4]}. In chart 5, the rate of interest is held constant at 4.57% - the average of the interest rates for the period 1960-61 to 1993-94. In chart 6, technical progress, the rate of growth of the efficiency parameter, A_{j}, is held constant at 0.59%^{[5]}. In this case the initial value, in 1961, for A_{j} is derived from econometric estimation of the production function, with parameters a, d and the growth rate of A_{j} constrained to equal their chosen values (see Appendix C of Guest and McDonald (1996b)). Table I gives the values for V and r of the socially optimal investment series for each of the three scenarios. Holding employment growth constant (Chart 4) has almost no effect on volatility. In this case, V is reduced from 0.452 to 0.446. The degree of persistence in the series is lower and, hence, further away from the degree of persistence shown in the actual series. The series with a constant interest rate (Chart 5) has a greater dampening effect on volatility, although the value of V is still four times that of the actual investment series. The series based on constant technical progress is in fact more volatile than the unsmoothed series (the base case). This is partly because the calculated value of A, is a residual which varies inversely with the employment term in (13). Hence, from (13) the effect of A_{j} multiplied by the employment term is to reduce volatility in the investment series. Paradoxically, this smoothing effect of A_{j} on investment is lost when A_{j} itself is smoothed. However, the series based on constant technical progress has a first order serial correlation coefficient that is much higher than the other smoothed series considered so far and is closest to that for the actual series. Table 1 Despite the greater degree of volatility in the series based on a constant rate of technical progress, there are good reasons for using this basis. The measured rate of technical progress varies due to cyclical factors, in particular labour hoarding in economic downturns, as well as variations in "true" technical progress. Therefore, the unobservable "true" rate of technical progress may reasonably be assumed to vary less than the measured rate. Chart I shows that the calculated values of A_{j}, from which the measured rate of technical progress is derived, exhibit an implausible degree of volatility. On the other hand, arbitrarily smoothing employment and the interest rate is less justified. On these grounds, the simulations in the rest of this paper assume constant technical progress, but variable employment and interest rate as described above. 3.1.2 A mean-adjusted series with constant technical progressNow consider the mean level of socially optimal investment. The mean socially optimal level of investment for the base case is 18 percentage points of GDP higher than the average actual level of investment (Chart 3). Similarly, in the case where technical progress is assumed constant from 1960-61 to 1993-94 (Chart 6), the mean level of socially optimal investment is 19 percentage points above the mean actual level. Because we are concerned with the volatility and persistence of the investment series, rather than the mean, for all further simulations we assume that firms do not persistently "under-invest" (or "over-invest") by imposing the condition that the mean actual investment level is equal to the mean socially optimal level^{[6]}. This is achieved by determining the value for the efficiency parameter, A_{j}, in the first year of the horizon (1960-61) such that this condition holds. In subsequent years, constant technical progress at the rate a = 0.59% is assumed. The resulting mean-adjusted series for socially optimal investment is compared to the series for actual investment in Chart 7. The assumptions for mean-adjustment and constant technical progress are maintained for all further simulations. It is clear from Chart 7 and Table I that the socially optimal investment series with constant technical progress and mean-adjustment still exhibits much greater volatility than the actual series. (Note that mean-adjustment does not affect V or r). 3.3 Sensitivity to a, the output elasticity of investmentFrom equation (13), socially optimal investment is a function of a, the output elasticity of investment. In this section we conduct simulations to determine the sensitivity of the volatility in the socially optimal investment series to variations in the value of a from its chosen value of 0.37. The results are illustrated in Chart 8. Any value of a in the range from 0 to 0.75 yields a value of V very close to the value for the base case assumption of a = 0.37. For instance, the implausibly low value of a of 0.05 yields V = 0.509, a reduction of only 0.03 from the case where a = 0.37. We conclude that reductions in a have a negligible effect in bringing the degree of volatility of socially optimal investment closer to the degree of volatility of actual investment. 3.4 The CES production functionIt might be thought that a lower value of the elasticity of substitution, s, would generate a less volatile series for socially optimal investment. This is because if capital is less substitutable for labour then the choice of the capital-labour ratio will be less responsive to changes in the rate of interest. In pursuing this possibility simulations with a CES production function, equation (10), were used. In the case where s < 1, the first order condition (6) yields the socially optimal level of investment as (15) For purposes of comparison, in these simulations the values of all exogenous variables and parameter values, except for s, are equal to the values in the Cobb-Douglas case with constant technical progress and adjusted mean. This isolates the impact of lowering the value of s. Calculations reveal that lower values of s have a limited impact on volatility and persistence in the series. For instance, Chart 9 compares the case where s = 0.5 with the Cobb-Douglas case where s = 1.0. The value of V falls from 0.539 for s = 1.0 to 0.480 for s = 0.5. The value of r falls from 0.656 to 0.608. Chart 10 illustrates the sensitivity of V to the value of s. Starting from the Cobb-Douglas case (s = 1.0), reductions in the value of a have a small impact on the measure of volatility, V. The value of V falls from 0.539 for s = l to a value of 0.458 for s = 0. 1, which is in any case an implausibly low value of s. This is not a significant reduction in volatility and is still considerably greater than the value for the actual investment series, which is a value of V of 0.085. To seek an explanation for why a lower value of s has little impact on V, consider the elasticity of investment with respect to the interest rate, e_{r}, calculated from (15): (16) This elasticity is an important determinant of investment volatility^{[7]}. From inspection of (16), the responsiveness of e_{r} to a change in q, and therefore s, is not unit free. Rather, it is dependent on the units in which A_{j} is measured. It appears that for the units in which the Australian data used in this paper are measured, a change in the value of q has little impact on e_{r}, and therefore on V. The conclusion from these simulations is that with a CES production function, reducing the elasticity of substitution does not significantly reduce the volatility in the socially optimal investment series. 3.5 An adjustment cost modelAn approach which has been used extensively in the literature is to model adjustment costs in the accumulation of capital. The model presented here is similar to that used in McKibbin and Siegloff (1988), who broadly follow the approach established by Lucas (1967) and Hayashi (1982). Assume one dollar of investment expenditure effectively yields less than one dollar of new capital, because real resources are used up by the disruptions to the existing production process caused by the installation of new capital goods and the retraining of workers. These costs are assumed to increase at an increasing rate, so that marginal adjustment costs are increasing, which ensures a solution for socially optimal investment. Also, adjustment costs are negatively affected by the size of the firm as measured by its capital stock in year k, K_{k}. Let J_{k} equal the effective accumulation of capital in year k after adjustment costs have been deducted. Then: (17) where 0.5 m (I^{2}/K)_{k} is the cost of installing J_{k} units of capital. The production function with adjustment costs is: (18) and the first order condition for determining the socially optimal level of investment is: (19) For the Cobb-Douglas production function (12): (20) Equating (19) and (20) yields the following non-linear equation in I_{j} to be solved for I_{j}: (21) The value for m is chosen to equal 3.89. This implies that one dollar of investment expenditure yields an average of eighty cents of capital net of adjustment costs. The size of the adjustment costs chosen here is approximately in the mid-range of the magnitudes used by McKibbin and Siegloff (1988). The resulting series for socially optimal investment is shown in chart 11. Table I shows that the introduction of adjustment costs reduces the volatility of the socially optimal investment series by a large amount and reduces the degree of persistence in the series by a small amount. However, the volatility of the socially optimal series is still approximately 3 times higher than the volatility of the actual investment series. The degree of persistence in the series is below the degree of persistence for the actual series. A value of m could be chosen such that the volatility of the socially optimal investment series is equal to the volatility of the actual series. However, simulations showed that this value is approximately 20, which implies nearly 100% adjustment costs - clearly a nonsensical proposition. Thus, the plausible value of 4, as chosen above, generates a socially optimal investment series with considerably more volatility than the actual series. 3.6 Modelling delivery lagsThe final approach is to include unanticipated delivery lags. These lags refer to the lag between the time the decision to invest is made and the time the expenditure on the investment is actually made. The existence of these lags implies that in any particular period some investment expenditure is determined by decisions made in earlier periods, based on information about interest rates and profitability in the earlier period. Jorgenson (1963) models delivery lags in the following way. Jorgenson assumes that for an investment decision taken in year k, a proportion, u_{k}, of the project is completed in year k, u_{k+1} is completed in year k+l, u_{k+2} in year k+2, and so on. Since these delivery lags are unanticipated they do not enter the optimising decision of the firm in the Jorgenson model. This is a weakness of the model, since firms would presumably begin to anticipate delivery lags, which would then become a constraint in the optimising process. Jorgenson then assumes that S^{k+infinity}_{t=k}v_{t}=1 and he estimates the u's in a distributed lag function for investment. To introduce delivery lags into our calculations of the socially optimal series of investment, assume that the u's decline geometrically so that socially optimal investment in year j, I _{j}*, is an exponentially weighted moving average of socially optimal investment decisions taken in years 1, 2,...j. McKibbin and Siegloff (1988) use this lag distribution function. A Koyck transformation yields the following equation for socially optimal investment, I _{j}*: (22) We choose a value of the smoothing parameter, f, equal to 0.5. This implies that following the decision to invest, half the investment expenditure is made in the first year, a quarter in the second year, and so on. This is approximately equivalent to the time lags estimated by McKibbin and Siegloff (1988). The resulting series is shown in Chart 12. The volatility, V, of this series is 0.456, still approximately 5 times higher than that for the actual investment series, and higher than the volatility of the series with adjustment costs. The degree of persistence in the series is increased considerably and is actually higher than for the actual series (see Table 1). Analagously to the argument in the case of the adjustment cost model, a value could be chosen for f which would generate a series with the same degree of volatility as the actual series. However, experimentation shows that this value is close to zero, implying close to infinite delivery lags, which is again implausible. We then combine adjustment costs of 20% and delivery lags (f = 0.5) as in the series constructed by McKibbin and Siegloff (1988). Their series tracks actual investment quite well. However, an important departure from our approach is their assumption that some firms do not optimise freely but, rather, face liquidity constraints in their investment decisions. In fact, liquidity constraints dominate in their explanation of actual investment, in that they conclude that 90% of investment is explained by the liquidity constraint variable. Nevertheless, we adopt their twin assumptions regarding adjustment costs and delivery lags. The resulting series for socially optimal investment is shown in Chart 13. The value for V is 0. 194. This is still twice the degree of volatility in the actual investment series. The combination of delivery lags, adjustment costs and constant technical progress can yield a series with a degree of volatility equal to the volatility of the actual series. One such scenario is illustrated in Chart 14. This scenario involves 30% adjustment costs (an increase from 20% in Chart 13), constant technical progress and delivery lags where (f = 0.5 (the same value as in Chart 13). ConclusionThis paper compares, for Australia, the volatility of the socially optimal series of investment with the volatility of the actual series. Various series of the socially optimal level of investment are constructed. These series are considerably more volatile than the actual investment series. In constructing the socially optimal series from a vintage production function several factors were included which smooth the series. These are: (i) a constant rate of employment growth; (ii) a constant interest rate; (iii) lowering the values of a, the elasticity of output with respect to capital, and s, the elasticity of substitution; (iv) adjustment costs; and (v) delivery lags. Of these smoothing techniques, only adjustment costs is both defensible on theoretical grounds and yields a significantly lower level of volatility. For instance, smoothing employment is ineffective in smoothing the series and is ad hoc; smoothing the interest rate yields a smoother series but is ad hoc; lowering a and s do not significantly smooth the series; and delivery lags are ad hoc because they are not part of the optimising problem but act simply as a statistical filter. Furthermore, adopting a value for the adjustment cost parameter based on the literature yields a series which is still three times as volatile as the actual investment series. By increasing the adjustment cost parameter to the unreasonably high level of 30% and introducing ad hoc delivery lags of one year the socially optimal investment series can be smoothed to give a degree of volatility equal to that for the actual investment series. This case is illustrated in Chart 14. The main point to be drawn from the simulations is that the volatility of the socially optimal investment series is excessive, relative to the volatility of the actual series. Of course, the actual series will differ from the socially optimal series whenever there is a wedge between the profit maximising level of investment chosen by firms and the socially optimal level. However, some reasons for this wedge would not explain the difference in volatility. These reasons include: (i) distortions due to tax rates which alter the relative prices of capital and labour. It is unlikely that tax rates themselves are sufficiently volatile to explain investment volatility. (ii) Externalities drive a wedge between the social rate of discount and the private rate of discount. This will occur if individuals are myopic in that, for instance, they use a discount rate which is higher than the social discount rate. Another form of market failure results where some risk is diversifiable to the economy but has not been diversified by private agents - this will mean the private rate of discount is above the social rate, which would lower the actual level of investment relative to the socially optimal level. Again it is difficult to imagine why the size of the wedge between the private and social rate of discount should be volatile. (iii) Imperfect competition - both monopoly and monopsony power - results in a level of investment below the socially optimal level. But it seems unlikely that the degree of imperfect competition is volatile. On the other hand, there are two other factors which could possibly reduce the divergence between the actual and socially optimal investment series but which we have not modelled. These factors are: (i) Liquidity constraints on investment, or credit rationing, occur as a result of information asymmetries between borrowers and lenders. De Meza and Webb (1987) show that there may be too little investment under credit rationing. However, Williamson (1986) and Keeton (1979) show that under other circumstances credit rationing is socially optimal. So whether a properly specified process determining the socially optimal level of investment would include liquidity constraints is a moot point. Nevertheless, there is evidence that the actual investment series is influenced by liquidity constraints. Recent empirical work at the Reserve Bank of Australia by Mills et.al. (1994) strongly suggests that financial variables, such as a firm's cash flow and liquid assets, are important determinants of the actual level of investment. McKibbin and Siegloff (1988) also suggest liquidity constraints as one possible interpretation for their model of investment. Binding liquidity constraints will dampen the response of investment to a fall in interest rates, thereby reducing the volatility of the actual investment series. It is interesting to note that the coefficient of variation of the series for the share of gross operating surplus in GDP for Australia for the period 1959-60 to 1993-94 is 0.09, which is very close to 0.085, the coefficient of variation of the actual investment series. Thus, in as far as current profitability, as a measure of the liquidity constraint, determines investment it would have a smoothing effect relative to the more traditional neoclassical factors. (ii) Our model of investment determination has not taken into account uncertainty and irreversibility of investment decisions. Dixit and Pindyck (1994) argue that since investment decisions are often irreversible and undertaken in the face of uncertainty, the opportunity to invest is akin to holding a call option. Hence, it can be optimal to wait longer before investing than would be the case with perfect foresight. Bertola and Caballero (1994) derive an aggregate series of privately optimal investment from a model of irreversible investment and idiosyncratic uncertainty at the level of the firm. Applying this model to U.S. data yields an aggregate investment series which has a degree of volatility close to the degree of volatility of the aggregate series of actual investment. To achieve this correspondence of volatility Bertola and Caballero have to assume a degree of idiosyncratic uncertainty which they consider high. However, as they point out, they do not include other smoothing processes such as the costs of adjustment of investment. One conclusion for future work suggested by the calculations in this paper is that the technique developed here to calculate the socially optimal series of investment should be extended to include liquidity constraints and idiosyncratic uncertainty. 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(1970), "Estimation of the Vintage Cobb-Douglas Production,Function for the United States 1900-1960", The Review of Economics and Statistics, 52, 2, 187-193. Williamson, S.D. (1986), "Costly Monitoring, Financial Intermediation, and Equilibrium Credit Rationing", Journal of Monetary Economics, 18, 2, 159-180. You, J.K. (1976), "Embodied and Disembodied Technical Progress in the United States, 1929-1968", The Review of Economics and Statistics, 58, 123-127. AppendixAppndix: Values of Exogenous Variables ChartsChart 1: The efficiency parameter, A_{j},
for Australia for the period 1960-61 to 1993-94. Chart 2: The interest rate, r_{j}, for
Australia for the period 1960-61 to 1993-94. Chart 3: Optimal investment - the base case - for
Australia for the period 1960-61 to 1993-94. Chart 4: The effect of constant employment growth
on optimal investment for Australia for the period 1960-61 to 1993-94. Chart 5: The effect of a constant interest rate
on optimal investment for Australia for the period 1960-61 to 1993-94. Chart 6: The effect of constant technical
progress on optimal investment for Australia for the period 1960-61 to
1993-94. Chart 7: Comparision of optimal investment
(mean-adjusted, constant technical progress) with actual investment for
Australia for the period 1960-61 to 1993-94. Chart 8: Sensitivity of volatility to values of
the parameter a. Chart 9: Comparison of optimal investment series
using Cobb-Douglas and CES production functions, for Australia for the
period 1960-61 to 1993-94. Chart 10: Sensitivity of volatility to the
elasticity of subst. a. Chart 11: Effect of 20% adjustment costs on
optimal investment for Australia for the period 1960-61 to 1993-94 [Both
series are mean-adjusted and technical progress is constant]. Chart 12: The effect of delivery lags on optimal
investment for Australia for the period 1960-61 to 1993-94 [Both series
are mean-adjusted and technical progress is constant]. Chart 13: Comparison of optimal investment (20%
adjustment costs, delivery lags (f=0.5)
mean-adjustment; const.tech. prog.) with the actual investment series. Chart 14: Comparison of optimal investment (30%
adjustment costs, delivery lags (f=0.5)
mean-adjustment; const.tech. prog.) with the actual investment series. Footnotes^{[1]}The series for the socially optimal level of investment is calculated as a proportion of GDP. This avoids the problems of a non-stationary series. Bertola and Caballero (1994) adopt the same procedure to assess the volatility of investment. ^{[2]}The vintage model allows for a relaxation of the assumptions of homogeneous capital and disembodied technical progress, implicit in the non-vintage model. Instead, technical progress is embodied in successive vintages of capital, which are increasingly efficient. ^{[3]}The coefficient of variation, V, is equal to the standard deviation of the series divided by its mean. Hence, this gives a measure of volatility which is independent of the mean. ^{[4]}The rate of employment growth is assumed to equal the rate of population growth. This implies that the labour force participation rate is constant throughout the time horizon. This prevents, even for very long time horizons, the labour force participation rate either approaching zero or exceeding one. Population is assumed to grow at 1.141%. This is the average annual growth rate from 1991 to 2011 for three series (A/B, C and D) of projected population growth rates in ABS Catalogue No.3204.0: Projections of the Population of Australia. ^{[5]}The rate of 0.59% is derived from Dixon and McDonald (1992), as discussed in Appendix C of Guest and McDonald (1996b)). ^{[6]}The possibility of persistent "under-investment" is discussed in the conclusion. ^{[7]}The other determinants of investment volatility in this model are: (i) changes in employment, the impact of which on volatility is invariant with respect to 0 and therefore a; (ii) changes in A_{j}, the impact of which on volatility varies with respect to a in a similar way to the interest rate (but note that in this section of the paper A_{j} is smoothed to have a constant rate of growth); (iii) changes in the erogenous level of output in year I of each plan, Y_{1}, since it is the investment share of output, (I/Y)_{1}, that is being measured. |