1 Introduction

Iran is a semi-industrialized country that adopted the industrialization policy for development in the 1960s and expanded its manufacturing activities such as steel and car manufacturing, petrochemical and refineries during the oil boom in the 1970s. In the aftermath of Iran–Iraq war, the government of Iran found itself with a substantial disruption of supplied chains caused by the freezing of the country’s foreign assets, the destructive 8-year war with Iraq (1980–1988) and economic sanctions (Pesaran 1995; Amuzegar 2000; Valadkhani 2001). In the face of such intense challenges, the government was forced to spend billions of dollars to compensate for the destruction of production capacity and infrastructure. The result of these policies, as argued by Pesaran (2000), ‘had been an economy in a state of acute disequilibrium with highly distorted price signals’. These economic conditions together with the absence of a strong private sector may help explain the government’s role in the economy.

After the 1979 revolution, the government began the nationalization process.Footnote 1 The result of this policy was the formation of a high degree of government ownership and its associated institutional incentives which prohibited the private sector from participating in the financing, construction, operation and maintenance of public infrastructure. Since then, the government has actively involved in a variety of investment projects aimed at meeting the country’s identified needs such as education, electricity and water utilities, medical care, transportation and waste disposal. As a resource-rich country, Iran’s economic performance has been largely driven by oil and gas revenues. The windfall income from natural resources creates fertile ground for rent-seeking activities (Farhadi et al. 2015). It is worth noting that the share of oil revenue in GDP declined from 9% in 1990 to 6% in 2015, while the share of tax revenue in GDP marginally increased from 4.5% in 1990 to 6% in 2015. Thus, it is reasonable to believe that the decline in oil revenue may tie the government’s hands to some extent. We therefore expect the government to make viable investment decisions and to become more accountable to the public.

It is important to emphasize at this point that, in the case of Iran, the government participation in investment projects is most likely to be prone to X-inefficiency. According to the monitoring report of the Iranian Planning and Budget Organization for 2014, the expected life of a new public project is around 2.8 years, while the weighted average of a completed project is 12.7 years.Footnote 2 This gap is associated with a mass of unfinished projects which has weakened the financial ability of the Iranian government to meet its objectives efficiently. Given the increasing importance of cost–benefit analysis for public investment decisions, the choice of a proper social discount rate is a crucial issue. The social discount rate is an established measure of a society’s preference for present consumption versus future consumption. In other words, it reflects the extent to which a society is willing to give up a unit of consumption now in return for more consumption later. Zhuang et al. (2007) argue that when dealing with very long-lived projects, the choice of the social discount rate should not only consider economic efficiency, but also intergenerational equity. Itoh (2016) points out that implementation of cost–benefit analysis has significant implications for the financial evaluation of economic sectors. This is particularly important for policymakers in the government decision-making process to ensure significant and long-term benefits for the country.

To this end, the rest of the paper proceeds as follows. Section 2 will explain and justify the social time preference rate method. In Sect. 3, we will present the data and methodology. The empirical analysis and the derivation of the social discount rate will be displayed in Sect. 4. Section 5 will highlight our conclusions.

2 Theoretical model

The use of social time preference rate (STPR) approach as an appropriate measure of the social discount rate is well documented in the literature (Marglin 1963; Feldstein 1964; Kula 1984, 2004; Pearce and Ulph 1995; Evans and Sezer 2002, 2005; Percoco 2008; Schad and John 2012; Halicioglu and Karatas 2013; Moore et al. 2013). Kossova and Sheluntcova (2016) convincingly argue that in the absence of competition for resources between public and private projects due to a lower rate of return on investment, the STPR approach is recommended. As a starting point, consider the following model where the representative infinitely lived agent maximizes inter-temporal utility subject to the flow budget constraint:

$${}_{{c_{t} }}^{\text{max}} W = \mathop{\int}\limits_{0}^{\infty } U\left( {c_{t} } \right){\text{e}}^{ - \rho t} {\text{d}}t,$$
(1)
$${\text{s}} . {\text{t}} .\,\,\,\,\,\,\dot{k}_{t} = f\left( {k_{t} } \right) - c_{t} ,$$
(2)

where U(.) represents a time-invariant iso-elastic utility function, \(U\left( {c_{t} } \right) = \frac{{c_{t}^{1 - \eta } }}{1 - \eta }\) with the following properties: \(U^{\prime}\left( . \right) > 0, U^{\prime\prime}\left( . \right) \le 0\); e−ρt is the discount factor for utility of consumption; \(\dot{k}_{t}\) is the rate of change of the capital stock (net investment); and f(.) represents a production function owned by the agent according to an AK technology. The Euler equation corresponding to this maximization problem requires that:

$$U^{\prime}\left( {c_{t} } \right)f^{\prime}\left( {k_{t} } \right) + U^{\prime\prime}\left( {c_{t} } \right)\dot{c}_{t} - \rho U^{\prime}\left( {c_{t} } \right) = 0$$
(3)

After some simplification, the resulting equation is known as the ‘Ramsey equation’ and is shown in Eq. (4):

$$r = \eta g + \rho ,$$
(4)

where r is the marginal return on investment, g is the growth rate of per capita real consumption (with mean μ and variance \(\sigma^{2}\)), η is the elasticity of marginal utility of consumption and ρ is the pure rate of time preference describing impatience. The parameter, ρ, could be determined as the instantaneous probability of death (Boardman et al. 2010). Eventually, we extend the Ramsey formula to encompass uncertainty in the rate of growth in consumption, adapting a formulation suggested by Mankiw (1981) as follows:

$${\text{STPR}} = \eta \bar{g} + \rho - 0.5\eta \left( {\eta + 1} \right)\sigma^{2} ,$$
(5)

where \(\bar{g}\) is expected growth rate of consumption (Gollier 2012).Footnote 3 The last term in Eq. (5) is uncertainty about the rate of growth in consumption known as the ‘precautionary saving effect’ and is the product of three factors: relative risk aversion η, relative prudence η + 1 and the variance of the growth rate of consumption σ2 (Gollier 2012). This extension of the Ramsey rule is consistent with the uncertainty attached to the Iranian economy as a whole.

There are basically four alternative methodologies to estimate the elasticity of marginal utility of consumption (Groom and Maddison 2018): the equal-sacrifice income tax approach, relative risk aversion in insurance markets, the Euler-equation approach and the Frisch additive-preferences approach. Each of the four methods has its advocates and critics (for a survey, see Lopez 2008; Percoco 2008; Evans and Kula 2011; Moore et al. 2013; Groom and Maddison 2018; Freeman et al. 2018). On the basis of previous reviews, Groom and Maddison (2018), among others, argue that in the case of some developing countries, the Frisch additive-preferences approach might be the only feasible means of estimating the elasticity of the marginal utility of consumption, η. The calculation of parameter η is based on an adopted version of the Fellner demand model (see Fellner 1967) extended by Kula (1984). To obtain the parameter η from the Fellner demand model, an important assumption has to be satisfied: the ‘Frisch elasticities’ formula has to be assumed to be based on the principle of the presumed existence of additive preferences since food is deemed to be a ‘want-independent’ or ‘preference-independent’ good (Schad and John 2012; Groom and Maddison 2018).

According to this, the elasticity of marginal utility of consumption can be approximated by the ratio of the income elasticity to the compensated price elasticity of demand for food. So:

$$\eta = \frac{{\eta_{1} }}{{\hat{\eta }_{2} }},$$
(6)

where η1 is income elasticity of the food demand function and \(\hat{\eta }_{2}\) is the compensated price elasticity of food after elimination of the income effect. Following the empirical literature, we specify our model of food demand in log-linear form. Therefore, our long-run food demand takes the following form:

$$\ln F_{t} = \alpha_{0} + \eta_{1} \ln Y_{t} + \eta_{2} \ln \left( {{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {P^{*} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P^{*} }$}}} \right)_{t} + \varepsilon_{t} ,$$
(7)

where Ft is the (per capita) real consumption of food, Yt is the (per capita) real income, P and P* are price indices for food and non-food, respectively. \(\varepsilon_{t }\) is error term which has normal distribution with zero mean and constant variance. A number of authors (Stone 1954; Fellner 1967; Kula 1984, 2004; Evans and Sezer 2002) have estimated \(\hat{\eta }_{2}\) by using \(\left| {\hat{\eta }_{2} } \right| = \left| {\eta_{2} } \right| - \omega \eta_{1}\), where ω is the share of food in consumers’ budget.

From a theoretical point of view, earlier works on inter-temporal choice behaviour have proposed the use of survival (or mortality) probabilities as rational bases for discounting in inter-temporal consumption decisions. In an attempt to explore how survival uncertainty may affect individual behaviour, Bommier (2006) has shown that time discounting is directly related to preferences over length of life and inter-temporal choice. This makes sense given the fact that individuals would discount their future utility by the probability of being alive at that date. For this purpose, following Kula (1984) the pure rate of time preference is assumed to be equal the average death rate in the population as a whole.

3 Methodology and data collection

Given the relatively small sample size in the present study, we use the ARDL bounds testing approach developed by Pesaran et al. (2001) to test the presence of cointegration within the variables and also to estimate the long-run and short-run coefficients of the variables. The ARDL cointegration approach has numerous advantages in comparison with the conventional cointegration methods such as Engle and Granger (1987), Johansen (1988) and Johansen and Juselius (1990) procedures. One of the main advantages of this technique is that the underlying regressors are not restrictive irrespective of whether the variables are I(0), I(1) or fractionally cointegrated (Pesaran and Pesaran 2009). Also, as noted by Harris and Sollis (2003), the ARDL technique provides unbiased estimates of the long-run model and valid t-statistics even when some of the regressors are endogenous. The unrestricted error correction model (UECM) version of the ARDL model of Eq. (3) can be written as follows:

$$\begin{aligned} \Delta \ln F_{t} & = \beta_{0} + \beta_{1} {\text{Dum}}_{t} + \mathop \sum \limits_{i = 1}^{a1} \delta_{1i} \Delta \ln F_{t - i} \\ & \quad + \mathop \sum \limits_{j = 0}^{b1} \theta_{1j} \Delta \ln Y_{t - j} + \mathop \sum \limits_{p = 0}^{c1} \phi_{1p} \Delta \ln \left( {{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {P^{*} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P^{*} }$}}} \right)_{t - p} \\ & \quad + \lambda_{1} \ln F_{t - 1} + \lambda_{2} \ln Y_{t - 1} + \lambda_{3} \ln \left( {{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {P^{*} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P^{*} }$}}} \right)_{t - 1} + \varepsilon_{t} \\ \end{aligned}$$
(8)

where \(\Delta\) represents the first difference operator; ɛt is the random error term; δ1i, θ1j and ϕ1p are short-run coefficients; and \(\lambda_{1}\), λ2 and λ3 denote the long-run coefficients. Here, β0 and β1 are deterministic variables in an augmented ARDL model.Footnote 4 The ARDL bounds testing procedure involves two steps. In the first step, the presence of cointegration between the variables is investigated by an F-test for the joint significance of the coefficients of the lagged levels of the variables. In this set-up, the null hypothesis of no cointegration \(H_{0} : \lambda_{1} = \lambda_{2} = \lambda_{3} = 0\) was tested against the alternative hypothesis of \(H_{1} : \lambda_{1} \ne 0,\lambda_{2} \ne 0,\lambda_{3} \ne 0\). Here, the F-test has a non-standard distribution. To overcome this problem, Pesaran et al. (2001) and Pesaran and Pesaran (2009) compute two sets of critical values for a given significance level. One set assumes that all variables are I(0), and the other set assumes they are all I(1). If the calculated F-statistic lies above the upper level of the bound, the null hypothesis of no cointegration is rejected. If the calculated F-statistic is below the lower critical value of the bound, the null hypothesis cannot be rejected. Finally, if it lies between the bounds, the result is inconclusive.

In the second step, the lag orders of the variables are chosen using Schwarz Bayesian Criterion (SBC), as many studies show that SBC yields more parsimonious lag order. Thus, the selected ARDL model determined by SBC is used to estimate the short- and long-run models. Once a long-run relationship has been established, we use estimates of λ1 − λ3 to form the error correction term, \({\text{ECT}}_{t - 1}\). The error correction term shows how quickly variables converge to equilibrium and it should have statistically significant coefficient with a negative sign. As stated by Fuinhas and Marques (2012), the absence of shift dummies to control for outliers and structural breaks could lead to the misinterpretation of elasticities and cointegration between variables. This approach is supported by Pesaran et al. (2001), who argue that the asymptotic theory developed for the bounds test procedure is not affected by the inclusion of ‘one-off’ dummy variables. Therefore, a dummy variable, \({\text{Dum}}_{t}\), that takes the value one in 2012 and zero in all other years has been included in the estimation in order to capture the second phase of subsidy reform in Iran.

Annual time-series data, which cover the period 1990–2015, are utilized in this study. Data are collected from the Household Income and Expenditure (HIE) Surveys conducted by Statistical Centre of Iran (www.amar.org.ir). Relevant series for food model are shown in Table 1 in the case of Iran.

Table 1 Time series data for food demand function in Iran
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4 Empirical results and analysis

Prior to implementing the ARDL model, Dickey–Fuller generalized least squares (DF-GLS) test is used to determine the integrating properties of the variables in our study. The DF-GLS test (Elliot et al. 1996) is a modification of the conventional ADF t test as it applies to GLS de-trending prior to testing for stationary. Compared with the ADF tests, the DF-GLS test has superior performance in terms of small sample size and power. The DF-GLS unit root tests results in Table 2 show that the variables have mixed orders of integration such as lnF and \(\ln \left( {{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {P^{*} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${P^{*} }$}}} \right)\) being I(1) and lnY being I(0).

Table 2 DF-GLS unit root tests on variables
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According to Bahmani-Oskooee and Brooks (1999), the result of the F-test could be sensitive to the number of lags imposed on each first differenced variable. Given the small sample size of our data series, the maximum lag length is set at 2 to ensure sufficient degrees of freedom for econometric analysis. The results show that the computed F-statistic is 8.32, which is above the upper bound critical values at 5% significance level (Table 3). This shows that the null hypothesis of no cointegration between the variables is rejected. At this stage, the underlying ARDL model can be established to determine the short- and long-run relationships.

Table 3 Results of bounds test: ARDL model (2, 1, 2)
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As can be seen from Table 4, the food demand model yields satisfactory results in terms of both the short- and long-run estimated coefficients and the fitness of the model. Now, we can calculate \(\hat{\eta }_{2}\) from Eq. (6) using the long-run estimated coefficients from Panel A of Table 4 along with the budget share of food with respect to total consumption in Iran 27%, ω, during the estimation period:

$$\left| {\hat{\eta }_{2} } \right| = \left| { - 0.265} \right| - \left( {0.27*0.524} \right) = 0.124$$

Then, substituting the values of η1 and \(\hat{\eta }_{2}\) in Eq. (6) gives:

$$\eta = \frac{0.524}{0.124} = 4.226$$

The expected annualized consumption growth rate and the average death rate for the entire sample period are 1.3% and 0.53%, respectively. Finally, putting these values into the STPR formula in Eq. (5), we obtain:

$$\begin{aligned} {\text{STPR}} & = \left( {4.226*0.01338} \right) + 0.0053 \\ & \quad - 0.5*\left( {4.226*\left( {4.226 + 1} \right)} \right)*0.00043 = 0.0579 \\ \end{aligned}$$
(9)

Hence, a 5.8% social discount rate is appropriate for application in public sector project appraisal in Iran. This estimate is consistent with previous estimates of STPR for selected Asian and Middle Eastern countries falling in a range of 4.5–7.8% (Zhuang et al. 2007; Halicioglu and Karatas 2013).

Table 4 Full information estimate of ARDL model (2, 1, 2)
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5 Conclusion

In the presence of market distortions, the market interest rate is unlikely to reflect the marginal social opportunity costs of public funds and hence, it is not the appropriate social discount rate for project appraisal. Social discounting plays a critical role in cost–benefit analysis and has important implications for resource allocations in the public sector. A relatively high social discount rate has a tendency to preclude socially valuable public investment projects from being undertaken. This could be due to the reason that a relatively high social discount rate normally favours projects with benefits occurring at near future. The vice versa is also feasible and possible.

This study estimates the social discount rate for Iran using the social rate of time preference method. The elasticity of marginal utility of consumption, an important component of the social discount rate, is estimated using the ARDL bounds testing approach from a demand for food model over the data period 1990–2015. The estimated model is found to be adequate in terms of the statistically valid empirical results. The overall result indicates that a 5.8% social discount rate is appropriate for discounting the benefits and costs of public projects. Despite of the importance of the social discount rate for many policy choices, to our knowledge, there is no official social discount rate proposed by the Iranian authorities. For this reason, our ability to compare our recommended value with that adopted by the government agencies is limited. In our view, further research should be devoted to estimating the social discount rate at the provincial level for the country.