Public Capital Spillovers and Growth Revisited: A long-run and Dynamic Structural Analysis*

(i) Technology and Household Choice

Let Y be aggregate output, K be aggregate private capital stock, L be the total number of workers or population and be a measure of congestion-adjusted public capital stock to be defined later. The Harrod-neutral rate of technological progress is denoted by x and some constant level of total factor productivity is given by A > 1. We would like public capital to enter aggregate production so that potentially there would be a spillover effect.³ Assume that the aggregate production function takes the Cobb-Douglas form

(1)

which yields the production function in per worker terms as

((1a))

where the lower case variables, y and k, denote per worker output and private capital, respectively. Thus, the model nests the possibilities of exogenous and/or endogenous growth. The production technology is subject to i.i.d. shocks, , assumed to be multiplicative in this model. Aggregate public capital, _t, enters as an input into production (implying the spillovers or externality effect) and it is taken by the representative agent as given. Furthermore, aggregate public capital is subject to congestion from its use by private production:

(2)

where G_t is the aggregate stock of public infrastructure investment and φ and (1 – φ) denote the degree of congestion arising from private capital stock and labour force, respectively. This is contrary the usual notion that public goods are nonexclusive and non-rival.

We can further detrend Equation (1a). Let ŷ_t := y_t/(l + x)^t, k̂_t := k_t/(1 + x)^t and ĝ_t := G_t/L_t(1 + x)^t. Thus Equation (1a) can be written in per efficiency unit worker terms as

(3)

Assume that there is 100-per cent depreciation at the end of each period for private capital. Then private per capita investment will give the following period's capital stock per efficiency unit worker:

(4)

where i is investment per efficiency unit worker and k₀ is given. Similarly, aggregate public infrastructure investment is assumed to depreciate fully at the end of the period such that

(5)

where I^G is aggregate public expenditure on infrastructure and G₀ is given.

Let τ be the uniform income tax rate. The household solves

(6)

subject to

for all .

It is shown in Appendix I, by restating the problem in Equation (6) as a dynamic program, that the solution to the household problem taking government policy as given, yields the optimal paths of consumption and private capital as

(7)

(8)

for all states and dates , given k₀. The analytical solutions were obtainable by assuming logarithmic utility, Cobb-Douglas technology, 100 per cent depreciation of private and public capital, a uniform tax structure and a balanced budget. This also simplifies the model's restriction on the co-integrating properties of the variables.

(ii) Public Sector

The government budget is such that public investment demand each period is exactly financed by income tax revenue:

(9)

We assume a government policy to be one that implements a Ramsey optimal fiscal plan. The government maximises the same objective function as households but it also takes into account the optimal behaviour of private agents with respect to the policy plan in a competitive equilibrium. The optimal policy is a sequence of tax functions solving the following problem:

(10)

for all . Notice that we have replaced per period consumption in the objective with the competitive behaviour in Equation (7) and also encoded households’ optimal capital investment decision Equation (8) into constraint (c) above.

It is assumed that the benevolent government maximises household welfare when it maximises household consumption growth. A further assumption is that the sequence is bounded above by for some value η ≥ 1, to ensure that the infinite horizon household objective is bounded above for all feasible consumption paths. In other words, the optimal paths in Equations (7) and (8) will be unique: Glomm and Ravikumar (1994).

(iii) Optimal Public Policy

Solving the government's problem by dynamic programming (Appendix II), it is found that the optimal tax rate is a function of constants. Specifically, the optimal tax rate is defined by the function

for all states and dates . Thus, the optimal tax rate is equal to the one-period discounted share of public capital in output, where the government faces the same subjective discount rate, β, as the household.

Second, given the optimal choice of public policy, the evolution of private capital per efficiency unit worker in Equation (8) can be described by the first-order stochastic difference equation

(12)

The evolution of public capital per efficiency unit worker is

(13)

Consequently, the ratio of the optimal paths for private and public capital stays constant over time. This can be observed by taking the ratio of Equation (13) to Equation (12), which yields

(14)

for all states and dates .

(iv) Long-Run Growth

Substitution of Equation (14) into (12) gives the essential difference equation for the evolution of private capital:

(15)

Under an assumption of constant returns to scale to reproducible factors, where α + (1 – φ)θ = 1, the steady-state ( = 1; ∀t) growth rate of private capital will be given by [(1 – θβ)(α – θφ)]^1–θθ^θβA, which is perpetual and nonexplosive. Also, output and public capital will grow at the same rate as private capital, with constant returns to scale Cobb-Douglas technology.

(v) Testable Time Series Properties of the Model

Equations (12) and (13) can be written in natural logarithm and substitution of these into the stochastic investment equations in (4) and (5) yields

(16)

and

(17)

Multiplying Equation (16) by (1 – θL) on both side, where L is the lag operator, and substituting for (1 – θL) ln ĝ_t from Equation (17) yields an equilibrium dynamic equation of the log of per capita private capital expressed in terms of its own lags and the external shocks:

(18)

Multiplying Equation (17) by [1 – (α – θφ)L] and substituting for [1 – (α – θφ)L]ln k_t from Equation (16) yields the equilibrium path for aggregate public capital:

(19)

Also, taking logs of the equation for the private production function in Equation (3), multiplying this by {1 – [α + (1 – φ)θ]L} and expressing this in per worker terms, yields

(20)

This equation describes the equilibrium path of the log of output per worker, ln y_t.

Derivation of Co-integrating Relationships

If there are three I(1) variables in the system, there can be a maximum of two linearly independent co-integrating vectors. For nonexplosive, perpetual endogenous growth, it was concluded that α+ (1 – φ)θ = 1. Using this fact in Equations (19) and (20), and then subtracting the former from the latter, and performing the same again on Equations (18) and (20) gives the co-integrating space as

(21)

(22)

where κ₁ and κ₂ are constants.

If the co-integrating space in Equation (21) and (22) is rejected, then there may be at most one co-integrating vector. This co-integrating equation is a linear combination of all the variables. This can be shown by multiplying Equation (21) on both sides by (1 – θ), and and Equation (22) on both sides by θ, and then summing the two equations, to obtain

(23)

where κ₃ is some constant.

The co-integrating equation in (23) also represents the production function at steady state with nonexplosive, perpetual growth. In general, without assuming α+ (1 – φ)θ = 1, the single unrestricted co-integrating equation can be derived from Equation (3) yielding

(24)

Note that (23) is a nested case of (24) where (23) was derived under the hypothesis of α+ (1 – φ)θ = 1. These possible co-integrating relationships will be tested in Section IV of this paper.

(i) Weak Stationarity of the Series

From Table 1, the augmented Dickey and Fuller (1979, 1981) unit root test reveals that all the variables appear to be non-stationary in levels (contain a unit root), but will be stationary after taking first differences.

(i) Testing for Two Co-integrating Vectors

We consider first the possibility of two co-integrating relationships. Table 2 shows that we can reject the pairwise co-integrating relationships for Equations (21) and (22) as the residual from each estimated linear combination of the variables fails to be stationary. Therefore, we cannot have a maximum of two linearly independent cointegrating vectors in our sample. Alternatively, we also use the Johansen (1991) cointegration rank test which is fully reported in the next section. Again the conclusion is identical.

(ii) Testing for a Single Cointegrating Vector

Next, we consider whether there exists one cointegrating vector among the three variables in Australia. To motivate this possibility, consider a scatter plot of {ln y, ln k, ln g} in Figure 1. This raw and informal plot suggests that all three series occur along some common vector at least in the long-run. In other words, there is some informal evidence of a single cointegrating vector for all three variables.

Based on the combination of the theoretical relationship in Equations (23) and (24), or more generally on (24), this informal evidence can be tested by the following reduced-form regression:

(25)

The reduced-form coefficients β₁, β₂ and β₃ in Equation (25), respectively, denote (α – θφ), θ and {1 – [α + (1 – φ)θ]x} originating from Equation (24). Specifically, if the estimated coefficients follow the conditions that β₁ + β₂ = 1 and β₃ = 0, the engines of economic growth are endogenously determined by both private and public capital. That is, Equation (23) holds such that the production function in Equation (3) exhibits constant returns to scale. Conversely, if β₁ + β₂ is estimated to be < 1, together with a positive β₃, the long-run relationship between the variables {ln y, ln k, ln g} is in favour of Equation (24) and the economy will grow exogenously along the time trend. For completeness, we provide two tests and three alternative estimates of the possible cointegrating relationship.

As one robustness check for the existence of possible cointegrating relationships between {ln y, ln k, ln g} we apply the Johansen (1991) test. Our test is based on a vector autoregressive (VAR) model with lag length 2, which is optimally chosen using the Schwarz information criterion. Johansen's cointegration rank (trace) test indicates that there exists only one cointegrating equation (with an intercept and a time trend) at the usual 5 per cent level of significance. The Trace test statistic for a null hypothesis of no cointegration (against an alternative hypothesis of one cointegrating relationship) is 44.67, and the 5 per cent-significance level critical value is 42.92. There is no further evidence for at most two cointegrating vectors. The estimated single cointegrating relationship is:

((25a))

The cointegration estimators of ln k_t, ln g_t and time trend are significant even at 1 per cent level of significance.

Alternatively, we can perform the cointegration test amongst {ln y, ln k, ln g} with a time trend based on the Engle and Granger (1987) two-step procedure. In the first step, we estimate the long-run relationship as

((25b))

In the second step of the Engle–Granger procedure, the existence of cointegrating relationship amongst the three variables with a time trend requires the OLS residuals from Equation (25b) to be stationary. Otherwise, we would have a spurious regression. The augmented Dickey–Fuller (ADF) test statistic of the regression residual from Equation (25b) is –5.25 and is beyond the 1 per cent critical value equaling –4.73.⁶ Therefore, we can conclude that the regression residual from Equation (25b) is a stationary process such that this cointegrating equation holds.

However, the inference on the OLS estimators provided in Equation (25b) is not reliable since the estimators are super-consistent and the distribution of the estimators’t-statistics are non-standard. However, proper inference can be made if we refine the estimation using the dynamic OLS (DOLS) estimation method proposed by Stock and Watson (1993). Specifically, the DOLS method takes into account leads and lags of the first-differenced regressors in Equation (25b). We can further ensure that the estimation errors do not exhibit serial correlation and heteroskedasticity. The estimates of Equation (25b) is refined using DOLS as:

((25c))

where the Newey-West heteroskedasticity and autocorrelation consistent (HAC) standard errors are given in parentheses, and denote the estimated coefficients of the leads and lags of first-differenced regressors.⁷ The optimal lengths of leads and lags are chosen based on the minimal Schwarz information criterion.

The estimated shares of public and private capital from Equation (25c) suggest that the former is more than double that of the latter. This implies that public capital has dominated the process of economic development in Australia in our sample period. This result is opposite to the findings from Lau and Sin (1997), who used US data for a similar regression and estimated the coefficients of ln g_t and ln k_t to be 0.11 and 0.43, respectively. They therefore emphasised the dominance of private capital in the process of economic growth in the USA. Another study for Australia was carried out by Otto and Voss (1994a) using a shorter sample period of 1966/1967–1989/1990. Although their findings revealed a higher share of public capital in the production process than Lau and Sin (1997), and which is much closer to our findings here, their share of private capital was estimated to be negative. This negative share does not quite fit well with most growth models.

(iii) Joint Hypothesis Testing of Growth Conjectures

Next, we conduct some hypothesis tests on the long-run relationship based on the DOLS regression result (25c). The tests are reported in Table 3. The first test considers the null hypothesis of β₁ + β₂ = 1, which is the case of perpetual growth with public capital spillover but not ruling out the existence of exogenous trend growth. The second test is for the hypothesis of no exogenous growth, or β₃ = 0. The third joint test is for the hypothesis of strict endogenous growth, or β₁ + β₂ = 1 and β₃ = 0. We can reject all three null hypotheses at the usual 5 per cent level of significance. In particular, the hypothesis of strict endogenous growth due to public capital spillover into aggregate production does not hold in Australia during our sample period. In other words, while there is public capital spillover into aggregate production, there is no endogenous growth effect from it. Instead, growth is driven by an exogenous component. It should also be noted that the trend coefficient estimate of about 0.015, which is the implied annual rate of technical progress, is very close to the Australian Bureau of Statistic's calculations.⁸

In summary, growth is driven by exogenous factors, along with public capital spillovers that have no endogenous growth effects, when we confront the model with Australian data. Output per capita grows exogenously at a rate about 0.015, and public capital spillovers contribute higher share to the output than private capital in our sample period.

(iv) Economic Relevance of Long-run Estimates

Since one of the parameters {α,φ} cannot be identified from the estimates of β₁ and β₂ = θ, we perform the following informal exercise to check whether our estimates provide some sensible economic parameterisation. We use the following guideline. Since we know very little about the congestion parameter φ in the model, except that it must be theoretically constrained to be within the open set (0, 1), we set α to two scenarios: α = 1/3 and α = 1/4. The latter parameterisation is motivated by the argument that the usual share of private capital stock in levels is lower than the stylized fact of 1/3 in the presence of endogenous growth effects. Table 4 demonstrates a comparison of estimated results between the two previous studies and our findings based on Equation (25c). It should be noted that when considering the congestion effect of aggregate public infrastructure in Equation (2), the estimated coefficients of ln g_t and ln k_t from Lau and Sin (1997) result in a theoretically infeasible congestion parameter φ when we set the private capital share α as a reasonable scale (either 1/3 or 1/4). However, in either parameterisation, our regression output indicates that φ can be calculated with reasonable values. The case where α = 1/3 with φ = 0.5267 suggests that private capital, rather than private labour input, has a slightly higher share in creating congestion effects on public capital in Equation (2). In contrast, the congestion effect of private capital is about three times weaker than private labour input when we set α = 1/4.

Dynamic Programming for the Household Problem

The method of solving the household's intertemporal utility maximisation problem subject to given constraints and public policy in Equation (6) is as follows. The Bellman (1957) principle of optimality dictates that if the sequence of functions is maximising, then it must also be the case that it is maximal for . Hence, the sequence problem in Equation (6) can be written recursively as

((A.1))

subject to constraints (6) (a)–(c). A guess of the solution to (A.1) is the value function of the form

((A.2))

Substituting the form of equation (A.2) into (A.1) gives

((A.3))

subject to constraints (6) (a)–(c). The optimality conditions for a maximum on the RHS of (A.3) are

((A.4))

((A.5))

for all states and dates . Substitute equation (A.4) into (A.5) to obtain

((A.6))

Use the natural constraint (A.5) and (A.6) to derive the stochastic difference equation for private capital per efficiency unit worker

((A.7))

Substitute Equations (A.6) and (A.7) into the RHS of the Bellman Equation (A.3) so that (A.3)

The guess (A.2) is the fixed-point solution of the (A.1), if and only if

((A.8))

and two similar conditions for B₀ and B₂ which are not vital for characterising the optimal decision rules. Substitute (A.8) into Equations (A.6) and (A.7) then we can obtain the characterisations of the optimal household consumption (7) and investment (8) plans with given public policy .

The Government's Problem and Optimal Outcomes

The government's Ramsey optimal fiscal policy problem is recursive by assumption:

((B.1))

subject to constraints (10)(a)–(d). Guess that the solution is of the form below:

((B.2))

Utilising the guess in (B.2), re-write equation (B.l) as

((B.3))

subject to (10) (a)–(d). It is also assumed here that ln() is i.i.d and therefore,  ln() = 0.

The optimality conditions for the RHS problem are

((B.4))

((B.5))

((B.6))

for all states and dates .

Substitute Equations (B.4) and (B.5) into equation (B.6) gives

((B.9))

Further substitution of Equation (B.9) back into constraints (B.4) and (B.5) results in

((B.10))

and

((B.11))

Next, substitute (B.10) and (B.11) into (B.3) and compare with the form of (B.2).

The guess (B.2) is a solution to the government's Bellman equation if and only if B₁ = (α – θφ) (1 + βB₁ + βB₂) and B₂ = θ(1 + βB₁ + βB₂) and a similar condition for B₀ holds. Solving for B₁ and B₂ yields

((B.12))

((B.13))

Substitution of equation (B.12) and (B.13) into (B.9), (B.10) and (B.11) gives the optimal tax rate (11), and the evolutions of private capital (12) and public capital (13).

Data Appendix

The Exact Log-linear System for Short-run Analysis

Equations (16) and (17) characterise the solutions to the decentralised equilibrium under optimal fiscal policy. Along with (3), (7) and (11), we can re-write the exact log-linear equilibrium conditions in state-space form as: