Francesco Busato
email: fbusato@econ.au.dk.

Bruno Chiarini

Elisabetta Marzano

E -Macroeconomics and Monetary Economics, D91 -Intertemporal Consumer Choice|Life Cycle Models and Saving < D9 -Intertemporal Choice and Growth < D -Microeconomics, D92 -Intertemporal Firm Choice and Growth, Investment, or Financing < D9 -Intertemporal Choice

Keywords: Dynamic equilibrium models; Intertemporal Consumer Choice, Intertemporal Firm Choice, underground economy. JEL classification numbers: E32, D91, D92, E260

This paper presents a two sector dynamic general equilibrium model in which income smoothing takes place within the households (intra-temporally), and consumption smoothing takes place among the households (inter-temporally). Idiosyncratic risk sharing within the family is based on an income smoothing contract. There are two sectors in the model, the regular sector and the underground sector, and the smoothing comes from the underground sector, which is countercyclical with respect aggregate GDP. The paper shows that the simulated disaggregated consumption and income series (that are the regular and underground consumption flows) are more sensitive to exogenous changes in sector-specific productivity and tax rates than regular and underground income flows, and that this picture is reversed when the aggregate series are considered.

The growth and the size of the underground economy in the OECD countries have been illustrated in many studies. In this paper we analyze the implications on consumption and income dynamics over a sizeable underground economy. This paper presents a two sector equilibrium growth model, in which one sector represents the underground economy, and the other stands for underground activities [START_REF] Bovi | Un miglioramento del metodo di Tanzi per la stima dell'economia sommersa in Italia[END_REF] .

The model joins together three phenomena to improve the understanding of the dynamics of consumption and production decision making: consumption smoothing, income smoothing, and risk sharing using a contract based upon a countercyclical underground sector. The contract provides the risk sharing opportunity for smoothing income and consumption. Intra-sector production smoothing, as well as the non-market activities, are generated by existence of distortionary taxation.

There are three main results:

First, the model shows that the simulated disaggregated consumption and income series (that are the regular and underground consumption flows in this context) are more sensitive to exogenous changes in sector-specific productivity and tax rates than regular and underground income flows. Selected volatility measures for disaggregated consumption series are greater than or equal to those of the corresponding income series.

Second, the picture is reversed when the aggregate series (regular plus underground) are considered. The impulse response function of aggregate consumption to innovations in productivity and tax rates is below the response of aggregate income. In addition, aggregate consumption is less volatile than aggregate income (over 1000 simulations), which is one of the most robust empirical stylized facts matched by equilibrium growth models.

Third, we show that in our model agents are able to disentangle consumption and income by relying on the countercyclical behavior of the underground sector. In this sense, the underground sector offers risk sharing opportunities to household's members, which is exploited by entering the income-smoothing contract.

The model is calibrated for the Italian economy because it presents a large underground sector, which allows appreciating better the impact of underground activities on the overall [START_REF] Bovi | Un miglioramento del metodo di Tanzi per la stima dell'economia sommersa in Italia[END_REF] Underground activities are especially significant in many European countries where the underground sector represents from 15 to 35 percent of the GDP. In the U.S. the underground sector is about 8 or 9 percent of the GDP. (see, for instance, Schneider and Enste, 2000). There is no universal agreement on what defines the underground economy. Most recent studies use one of more of the following definitions: (a) unrecorded economy (failing to fully or properly record economic activity, such as hiring workers offthe-book); (b) unreported economy (legal activity meant to evade the tax code); (c) illegal economy (trading in illegal goods and services). We are interested about the size of the underground economy as encompassing those activities which are otherwise legal but go unreported or unrecorded, i.e. bits (a) and (b). Needless to say, this analysis is addressed to European countries like Belgium, Denmark, Greece, Portugal and Spain (see Busato and Chiarini, 2004). Moreover, the simulation outcomes are robust for many countries presenting a large size of underground activities. [START_REF] Busato | Market and Underground Activities in a two-sector dynamic equilibrium model[END_REF] .

The paper is organized as follows. In Section 2 we summarize the empirical evidence about underground activities, consumption smoothing and income smoothing. Section 3 describes the structure of the model, and Section 4 discusses our choice of functional forms and parameters values. Section 5 outlines simulation results, while Section 6 presents our conclusions.

Selected stylized facts

In all industrialized countries the underground sector is large both in terms of the labor input and in terms of the output produced. Schneider and Enste (2000) show that the underground economic activity represents on average 16.9% of the GNP for the OECD countries.

Different measurement techniques provide similar approximate magnitudes of the size and development of the underground economy. [START_REF] Campbell | Inspecting the Mechanism[END_REF] In addition, what is most interesting from the short run perspective is the countercyclical behavior of the underground component. Busato and Chiarini (2004) present evidence of this phenomenon for Italy, New Zealand, the United States and the United Kingdom.

Income smoothing is a relevant phenomenon, as well. Mordoch (1995) suggests that a great deal of risk is averted by households in inputs allocation. [START_REF] Chiarini | Business Cycle, Unemployment Benefits and Productivity Shocks[END_REF] This means that the members of a family can pool together their resources, in particular their labor supplies, and allocate them more efficiently over time, and across sectors (see also Gallie and Paugam, 2000). [START_REF] Chiarini | Dimensione e dinamica dell'economia sommersa in Italia[END_REF] Between consumption data and the majority of theoretical models, there exist two critical discrepancies, frequently neglected. First, consumption data usually refers to households, while majority of theoretical models is cast into the representative agent framework. Second, since a 6 0ur model is calibrated to match selected moments for the Italian economy within the sample 1970-1996. We choose Italy because of the availability of a complete data set on the non-market sector. Note however, that Italy presents a share of underground economy equally to approximately 20 or 30 percent of the GDP, which according to Schneider and Enste (2000), corresponds to several European countries like Norway, Belgium, Portugal, Spain, Greece and Denmark. [START_REF] Campbell | Inspecting the Mechanism[END_REF] There exist several methods of estimating the size of the underground economy. A detailed survey of the three most widely used methods to measure the hidden activity (the direct approaches, the indicator approach and the model approaches) are discussed in Schneider and Enste (2000). See also Feige (1989), Thomas (1992; 1999) and Chiarini and Marzano (2004) among others. [START_REF] Chiarini | Business Cycle, Unemployment Benefits and Productivity Shocks[END_REF] Note, however, that altruism within extended family is still controversial, at least for the United States. Altonji, Hayashi and Kotlikoff (1992) for instance reject the hypothesis of altruism within the extended family. But, in the conclusions, the authors underline how they do believe that significant altruistically motivated transfers occur in United States, especially among wealthy, who are underrepresented in the Panel Study on Income Dynamics (PSID). [START_REF] Chiarini | Dimensione e dinamica dell'economia sommersa in Italia[END_REF] For example we may think to the fact that in a large percentage of married and/or cohabiting persons between 25 The model presented in the following Section specifically addresses these issues. We tell a simple story of a two-person household, and we compare the individual consumption and income profiles, with those of the family.

Structure of the Model

The model draws from Busato and Chiarini (2004), while differing in two important aspects.

First, we generalize the household structure by introducing a simple form of heterogeneity; 10 second, we model income smoothing and within-family consumption reallocation by a specific contract.

There are three agents in the model: the firm, the extended family, and the government. In addition there are the regular and the underground sectors.

The members of the extended family choose consumption, investment, and hours worked at each date and in each sector to maximize the expected discounted value of the family's utility subject to an income smoothing contract, a budget constraint, a proportional tax rate on the regular wage, and the law of motion for the capital stock.

In each period, the firm rents capital only in the regular sector, but hires labor in both sectors. [START_REF] Cho | The business cyc1e with nominal contracts[END_REF] It produces output in both the sectors by solving a period-by-period profit maximization problem. The firm solves this series of one-period problems subject to a probability that it may be discovered producing in the underground economy, convicted of tax evasion and subject to a penalty surcharge.

Finally, a government levies proportional taxes on output and labor income, and balances its budget at each point in time. We assume that government spending on goods and services does not contribute to either production or to extended family utility.

The Extended Families.

Consider a production economy populated by many consumers. Each consumer works in only one of the two sectors. They receive incomes that are functions of idiosyncratic shocks. Within the economy there exist extended families, exogenously determined and of fixed size. Within and 54 years old, there is a single job-holder (i.e. Gallie and Paugam, 2000). [START_REF] Chiarini | Market Consumption and Hidden Consumption, A test for Substitutability[END_REF] We will see that, given the structure of our income smoothing contract, we may rewrite the heterogeneous agent model as a representative agent model. This claim is proved in Proposition 1. [START_REF] Cho | The business cyc1e with nominal contracts[END_REF] This assumption reflects a basic stylized fact of many underground economies. to each sector. [START_REF] Cho | Employment and hours over the business cyc1e[END_REF] For simplicity we assume that there exists one family, which is composed by two working individuals, Mr. κ and Miss. l. [START_REF] Cho | Family labor supply and aggregate fluctuations[END_REF] Without loss of generality, we assume that Mr. κ works in the regular sector, while Miss. l works in the underground sector.

Since Mr. κ, and Miss. l are good friends until many years, we may argue that their preferences are not too far. Hence, assuming that they have the same utility function for consumption should not be seen as a forcing. They have, however, a different preference structure for labor supply, which is consistent with the fact they work in different sectors. We can use a variant of the Cho and Rogerson's (1988) extended family labor supply model. Specifically, extended family composed of Mr. κ and Mss. l is characterized by the following instantaneous utility function:

( ) ( ) ( ) ( ) ( ) , , , l l l l l t t t t t l t t t t U c c l l u c u c l l l κ κ κ κ κ φ φ ν µ = + - - , (1)
where u(c κ t ) and u(c l t ) represent utility from Mr. κ and Miss. l consumption, and v(l κ t )l l t describes the disutility of working in both sectors. We interpret the last term, µ(l l t ), as reflecting the idiosyncratic cost of working in the underground sector. This cost may be associated in particular with the lack of any social and health insurance in the underground sector. [START_REF] Eichembaum | Current real business cyc1e theory and aggregate labor market fluctuations[END_REF] Given their long standing friendship, Mr. κ shares part of Miss. l concerns of working in that sector.

For the same reason, we assume: φ k = φ l = ½. [START_REF]Frontiers of Business Cycle Research[END_REF] An aspect of primary interest in our labor market is workers' labor supply in the two sectors of the economy. Mr. κ , which works in the regular sector, supplies l k t , and receive a wage w k t = w t (1-τ t ), where τ t is the stochastic tax rate on wage income. Miss l, who works in the other sector, offers l l t , and earns a wage w l t = w t . The family budget constraint is:

tot t tot t tot t t l t t k t t t X C K R l w l w + = + + -) 1 ( τ , (2)
where

l t k t tot t c c C + =
and X tot t represents total consumption and total investment by the family, respectively. Eventually they pool their savings together, and rent the grand total, X tot t , to the firms, wh capital stock evolves according to the following state equation:

( ) tot t tot t tot t X K K + - = + δ 1 1 , (3)
where δ denotes the exogenous and constant depreciation rate. We refer to equations (1) . [START_REF] Deaton | Understanding Consumption[END_REF] When agents have the same utility function for consumption, ω*=1/2. [START_REF] Feige | The Underground Economies. Tax Evasion and Information Distortion[END_REF] 3. We assume that agents accept the contract, that it holds for each period in time, and that it is incentive compatible and perfectly enforceable. [START_REF] Fiorito | Stylized facts of business cyc1es in the G7 from a real business cyc1es perspective[END_REF] In this paper we do not consider strategic interaction among agents. The contract has the simple goal to pool together labor supply, and consumption, insuring the family against idiosyncratic shocks. In addition, its structure serves as foundation of Proposition 1.

Proposition 1 (Representative Agent Model and Heterogeneous Agents Model) Under the income smoothing contract, as in Definition 1, the Heterogeneous Agent Model (Equations (1) to (3)) is equivalent to a Representative Agent Model characterized by instantaneous utility function

( ) ( ) ( )( ) ( ) j t j t j t j t j t j t v c u c U θ µ θ θ θ - - - - = 1 1 , , budget constraint ( ) j t j t j t t t j t t X c K R w + = + -τ θ 1
and capital accumulation constrain (3). Equivalence is in the sense of having the same First Order Conditions.

Proof. See Appendix A .

Remark 1 ("A Transparent Representative Agent Model") By Proposition 1, we transform

the HAM into a Representative Household model of a special kind. [START_REF]Welfare regimes and the experience of unemployment in Europe[END_REF] The novelty of our [START_REF] Deaton | Understanding Consumption[END_REF] In this way individual consumption is disentangled from individual income. It may be interesting to note that this is the argument behind the risk sharing and consumption literature (see Deaton, 1992 for a survey). [START_REF] Feige | The Underground Economies. Tax Evasion and Information Distortion[END_REF] This claim can be showed quite easily, but for completeness we precise the argument in Appendix B. [START_REF] Fiorito | Stylized facts of business cyc1es in the G7 from a real business cyc1es perspective[END_REF] By definition, an implicit contract will need to be sustained as an equilibrium in the interaction between the parties (Salanie', 1997). The contract we present in this model has the very simple goal to provide insurance against production idiosyncratic risk. For this reasons we assume that agents accept the contract. [START_REF]Welfare regimes and the experience of unemployment in Europe[END_REF] Notice that this an application of an already known result. Indeed Rubinstein (1974) shows that when agents have homogenous beliefs, and time additive utility function with linear risk tolerance, and same exponent, these Specifically, our model generates, for both income and consumption, three series: two "pre-contract" or disaggregated series (regular and underground), and one "after-contract" or aggregated series. The former series refer to individual consumers, and we interpret them as consumption and income profile which arise without contract. The latter ones belong to the household. Then we can explicitly compare the stochastic properties of the different series. We may think to ours, as to a "Transparent Representative Agent Model". [START_REF] King | a, Production, growth and business cycles: I. The basic neoclassical model[END_REF] Relying on Proposition 1, and assuming that there exists a continuum of households, which are uniformly distributed over a unit interval, we specify a following functional form for the jth household momentary utility function. [START_REF] King | Production, growth and business cycles: II New directions[END_REF] Specifically (1) becomes:22

( ) ( ) ( ) ( ) ( )           + - - - + - - - = + + - η θ θ γ θ θ η γ 1 1 1 1 1 1 ; 1 1 1 j t j t j t q j t j t j t f h q c c u (4)
To have a well behaved utility function, we assume that

1 , ; 0 , - > ≥ η γ f h
, that all the parts of the momentary utility function are well behaved. [START_REF] Hansen | Indivisible labor and supply and the business cycle[END_REF] The first quantity denotes the utility from aggregate consumption stream, while the second term represents the overall disutility of working; [START_REF] Lucas | On the Mechanics of Development Planning[END_REF] the last term reflects the idiosyncratic cost of working in the underground sector. In particular, this cost may be associated with the lack of any social and health insurance in the underground sector. Finally, we formalize productivity and tax rates as a stochastic vector of variables that follow univariate AR( 1) processes in log:

t t t A A ε + Ω = +1
where A t is a vector [M t , Z t , t t , τ t ] containing the productivity shocks, M t , Z t , the stochastic corporate tax rate, t t , and the stochastic personal income tax rate τ t . Ω =diag (ρ i ), where i = m, z, t, τ, is a 4 x 4 matrix describing the autoregressive components of the disturbances relative to each of the four shocks. The innovation,

ε ' t = [ε m , ε z , ε t , ε τ ,] is a vector of i.i.d. random variables.

The Stochastic Dynamic Programming Problem for Households.

Let Vt(K j t , A t ) be the value function for the household problem:

( ) ( ) ( ) [ ] { } 1 1 1 , ; , max ; 1 +
+ + + = + t j t t j t j t K t j t t A K V E c u A K V j t j t β θ θ , (5)
subject to momentary utility function (4), to budget constraint (A.5 in Appendix A), and the low of motion for the household capital stock (3). The optimality conditions for the problem are the Euler equation (6.1) and the intra-temporal consumption-Labor allocation condition (6.2): 25

( ) ( ) ( ) ( ) η γ γ θ θ γ γ θ τ β j t j t j t q j t t t t t q j t j t f h c w R c c E - +         + + + - - =         ℑ         = + - + - + 1 1 2 0 1 1 1 1 / (6.1; 6.2) where ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 + + - + - + + + ≡ + - = - + - t t i mt i t t t R r K M t δ θ α δ α α
from firm profit maximization (see below).

The Firms.

There are I firms. Each firm I i ∈ produces both in the regular and in the underground sector using two different production functions:

( ) ( ) i ut t i ut i mt i t t i mt l Z y l k M y = = -; 1 α α (7)
The regular output, y i mt , is the result of capital, k i t and regular labor, l i mt , applied to a Cobb Douglas production function. The underground output, y i ut , is produced with a production [START_REF] Mace | Full Insurance in the Presence of Aggregate Uncertainty[END_REF] Appendix A characterizes in details the model, and states precisely the solution procedure. function which uses only underground labor, l i ut ,. Finally, M t and Z t are the idiosyncratic stochastic productivity shocks. This formulation is consistent with the behavior underling the existence of an underground sector. Indeed the firms have no incentive to invest capital in the underground sector [START_REF] Mare | L'evasione in Italia e nei paesi OECD: evidenze empiriche, determinanti ed effetti economici[END_REF] .

In equilibrium each firm allocates a share, θ i t of the total labor, L t , to regular production (therefore l i mt = θ i t L t ) and the remainder, l -θ i t to the other sector (therefore l i ut = (1-θ i t )L t ). [START_REF] Mordoch | Income Smoothing and Consumption Smoothing Harvard -Institute for International Development[END_REF] Normalizing L t to unity, we can rewrite (7) as:

( ) ( ) ( ) i t t i ut i t i t t i mt Z y k M y θ θ α α - = = - 1 ; 1 (8)
When the firm produces in the regular sector, its output is taxed with certainty at the stochastic rate t t . When producing in the underground sector, the firm may be discovered, with probability p, and forced to pay the stochastic tax rate, t t , increased by a surcharge factor, s> l, applied to the standard tax rate.

Assuming that the firm produces in both sectors, we can describe its revenues as follow:

( ) i ut t i mt t i t D y st y t y - + - = 1 ) 1 ( , with prob. p i ut i mt t i t ND y y t y + - = ) 1 ( , with prob. (1-p),
where y i D,t is the output when the firm's underground activity is detected and y i ND,t is the output produced when the firm's underground activities go undetected. The expected value of the output is then given by ( )

i y p py y E t ND i t D t i t , , ) 1 ( / - + = ℑ .
The production costs come from the labor hired in both sectors, and from rented capital. The cost of regular labor is represented by wage paid for hours worked; a wage that is augmented by social security taxes at a rate that we will assume is equal to the corporate income tax rate, t t . In accordance with the rationale behind underground activities, we assume that the firm does not [START_REF] Mare | L'evasione in Italia e nei paesi OECD: evidenze empiriche, determinanti ed effetti economici[END_REF] Notice that this structure is equivalent to a more general set up with two production functions which use both the inputs, like for example y i mt = M t (k i t ) α (l i mt ) 1-α and y i ut = Z t (k i ut ) β (l i ut ) 1-β . According to Uzawa (1965) and Lucas (1988) if β<α we can set the smaller elasticity to zero without loss of any generality. It follows that the share of capital in the labor intensive production function is null and therefore an optimizing firm would choose k ut = 0 for each t. Because we assume that the underground sector is labor intensive, we rely on this argument. [START_REF] Mordoch | Income Smoothing and Consumption Smoothing Harvard -Institute for International Development[END_REF] The use of the share is also consistent both with the fact that labor supply per person is approximately stationary in many economies although the real wage grows, and with the utility function, homogenous in consumption, that we adopt to model the household preferences. The aim is, therefore, to analyze the movement of resources between the two sectors, to understand how agents want to move inputs out of the market and into the underground. The reallocation of hours from market to informal sector rather than exclusively from leisure to labor, increases the volatility of the official labor input for a given technology shock. [START_REF] Prescott | Recursive Competitive Equilibrium: the Case of Homogenous Household[END_REF] Formally, firm costs are defined by:29 ( )

i t t i t t t t i t i t K r t w w K CO + + = θ θ , (9)
At each date t, firm i maximizes period expected profits:

( ) ( ) ( ) i t i t t i t k k CO y E i t i t , / max 0 , θ θ - ℑ ≥ ,
to derive the regular share of aggregate labor demand ( ) ( ) ( )( )

( ) α α θ 1 * * 1 1 1       + - - - = t t t t t t i t i t t w Z pst M t k , the underground component ( ) ( ) ( )( ) ( ) 1 * * 1 1 1 1 1 t t i i t t t t t t t M k pst Z w t α α θ   - - - = -   - +    
and the capital demand

( ) ( ) ( ) α α θ -       - = 1 1 * * 1 t t t i t i t R M t k 30

The Government.

Under Proposition 1 the flow government budget constraint is: [START_REF] Schneider | Shadow Economies: Size, Causes, and Consequences[END_REF] ( )

t mt t ut t t t t G y t y pst w = + + θ τ , (10)
where G G t = . [START_REF] Thomas | Informal Economic Activity[END_REF]

Equilibrium

Equilibrium for our model is described as a variant on a Recursive Competitive Equilibrium The system of equations we use to compute the dynamic equilibrium of the model depends on a set of 12 parameters. Six pertain to household preferences, (q, h, f, η, γ, β), four to the structuralinstitutional context (the probability of a firm being detected p, the surcharge factor s, the equilibrium income and corporate tax rates t and τ), and the remaining two parameters to technology (the capital elasticity α, and the capital depreciation rate δ). The fact that the data on the underground economy is difficult to obtain substantially complicates the calibration.

Because we are not aware of other studies which calibrate the parameters of a general equilibrium model augmented with an underground sector, we precisely detail our calibration procedure below.

1. The probability of being detected, p. We calibrate this parameter by estimating the unconditional mean of the ratio of number of inspected firms to their total number. [START_REF] Thomas | Quantifying the black economy: 'measurement without theory' yet again?[END_REF] For Italy, as well as for the majority of countries, only a portion of these data are publicly . For the Italian economy p = 0.03. [START_REF] Uhlig | A toolkit for analysing nonlinear dynamic stochastic models easily[END_REF] Even though this is not an efficient estimate, it represents the best possible calibration for this parameter, given the available data.

2. The surcharge factor s, the income tax rate t, and the corporate tax rate τ. The parameter s represents the surcharge on the standard tax rate that a firm, detected employing workers in underground sector, must pay. According to the Italian Tax Law (Legislative Decree 471/97, Section 13, paragraph l) the surcharge equals 30 percent of the statutory tax rate if the firm pays the fine when detected, or 200 percent when the firm refuses to pay. [START_REF] Uzawa | Optimal technical change in an aggregative model of economic growth[END_REF] We present results for both the values, s = 1.3 and s = 2.00.

In Italy, corporations are subject to a progressive tax rate. A tax rate of 19 percent is [START_REF] Thomas | Quantifying the black economy: 'measurement without theory' yet again?[END_REF] Note that an inspected firm is not necessarily convicted of evasion and therefore fined. Since inspections are based either on private information of Institutions, or randomly, it may happen that behavior of a perfectly honest firm will be inspected. [START_REF] Uhlig | A toolkit for analysing nonlinear dynamic stochastic models easily[END_REF] These data are available on line at the web site of the Italian Ministry of Labor, at the URL http://www.minlavoro.it/Personale/div7-conferenzastampa..Ol032001.htm. [START_REF] Uzawa | Optimal technical change in an aggregative model of economic growth[END_REF] In this case the firm will prosecuted under Criminal Law perspective, and if condemned pay 200 percent. 12 applied to the share of profits that represents 7 percent of the firm's capitalization; the remaining portion is then subjected to an increased tax rate of 36 percent. We calibrate the steady state value of the corporate tax rate as the average of these two numbers, i.e. t = 0.275.

The personal income tax system is more complex, since we have five tax rates, spanning from 18.5 percent to 45.5 percent. The calibration of the income tax rate may be undertaken in two ways. 36 It may be estimated as the average tax rate, weighted by the relative share of population in each income class. It may also be estimated as the tax rate associated with the average income of the working population (Adults 15-64 years old). We rely on the second procedure and since the average income equals 18,246

Euros we estimate the income tax rate at 33.5 percent.

3. The share of underground sector, l -θ. To calibrate this parameter we refer to Schneider and Enste (2000) who estimate the share of the underground sector for a panel of OECD countries. The value for the Italian Economy, l -θ = 0.30, is also consistent with Mare's (1996) estimates.

4. The preference parameters, q and β, the capital share, α, and the capital de- preciation rate δ. These parameters are set to values commonplace in this literature (e.g. Fiorito and Kollintzas, 1994, or Chiarini and Piselli 2005). More precisely, we set q = l, β = 0.98 and δ = 0.025.

5.

Stochastic Shocks autocorrelation coefficients, ρ m , ρ u , ρ t , ρ τ and innovation amplitudes, σ m , σ u , σ t , σ τ . The ρ' s are set to .90 and the σ's to 0.003. As we stress in Busato and Chiarini (2004) these values are much lower than the classical ones (see King and Rebelo, 1999). This means that the model has a particularly efficient amplification mechanism which allows us to employ very small shocks.

6. The utility function parameters h, f, η and γ. The calibration of these parameters is a not easy (see Cho and Cooley, 1994). We select them to match four moments: the ratio between standard deviation of total output σ (Y t tot ), and the standard deviation of total consumption, σ (C t tot ), the correlation between total output and total consumption ρ(C t tot , Y t tot ) , the correlation between underground income and total consumption ρ(C t tot , y l t ) and the correlation between regular income and total consumption ρ(C t tot ,y k t ). The calibrated values are h = 0.55, f = 1.99, η = 1.40, γ = 3.00. 37 36 More precisely, the structure of the tax rates is the following as of 2001. For incomes less than 10,331 Euros tax rate is 18.5 percent, for incomes between 10,331 Euros and 15,496 Euros tax rate is 25.5 percent, for incomes between 15,496 Euros and 30,992 Euros tax rate is 33.5 percent, for incomes between 30,992 Euros and 63,283 Euros tax rate is 39.5 percent and, eventually, for incomes above 63,283 Euros tax rate is 45.5 percent. 37 Cho and Cooley (1994) calibrate these parameters for the United States, and choose h = 6.0, f = 0.87, η = 0.62, γ = 2.00. Note, however, that their formulation of the model addresses issues different from matching market and This section shows that an equilibrium growth model, augmented with an underground sector and an income smoothing contract operating within the households, generates consumption and income series consistent both with smoothness properties of actual data. Specifically, it is here shown that the "individual", "pre-contract", or disaggregated series (c k t , y k t , c l t and y l t ) are sufficiently volatile as in the actual data, and that the "household", "aftercontract" or aggregated consumption and income profiles (C tot t and Y t tot ) satisfy smoothness properties presented by actual data. 38 Note, however, that "pre-contract" series do not arise as actual income and consumption profiles, because household smooth pre-contract income and consumption series on a period by period basis. 39 For this reason, these series have been generated by numerically simulating the model.

Comparing the stochastic properties of these two sets of variables, we can explicitly capture the reallocation mechanism of consumption and income between agents and between sectors, which is usually implicit in data collection, and in previous consumption studies. We argue that this is the driving force that ties our results to the countercyclicality of underground activities.

Numerical Results.

We present our results from four different perspectives. First, we show that c k t , and c l t exhibit volatility greater than or equal to y k t and y l t respectively. 40 Moreover, the former variables exhibit larger impact-response after innovations in productivity and tax rates. Second, the previous relationship is reversed when looking at aggregated series: C tot t is smoother and less sensitive to innovations than Y tot t . Third, we compare volatility, sensitivity to innovations, and correlation among the three consumption definitions, and among the three income components generated in our model. 41 Fourth, we analyze correlations between consumption and income series to draw additional evidence on the volatility of disaggregated components, and on the smoothness of their aggregate counterparts. underground moments. Specifically, they study the implications of this kind of utility function for the volatility of hours, employment and productivity in the United States. 38 To precisely describe the series, we denote y k t , as "pre-contract" or "market income" earned from by Mr. k, from his job in the market sector, before pooling it together with Miss l "after-contract" or "underground income", y l t . Then c k t and c l t represent Mr. k and Mrs, l, pre-contract individual consumption profiles, respectively. Finally, Y tot t and C tot t define total income and total, after contract, consumption. 39 Indeed, from consumption and incombe smoothing, Definition 1, point 2, it turns out that c k t = c l t =1/2 C t tot . See also Appendix B. 40 Where volatility is measured as the standard deviation of the Hodrick-Prescott filtered series. Figure 1 shows the first 32 quarter response of y k t , c k t and X tot t to a one standard deviation innovation in regular-sector productivity, underground sector productivity, corporate and income tax rates. The curves are the quarterly percentage deviations from a baseline scenario where all innovations are set to zero. As the four figures show, the response of regular consumption series is larger or equal than that of regular income component. Also Table l

.Aggregate consumption is less volatile than aggregate income

The analysis of aggregate variables presents a completely reversed picture.

(Figure 2 about here)

Figure 2 shows the first 32 quarter response of Y t tot , C tot t and X tot t to a one standard deviation innovation in regular-sector productivity, underground sector productivity, corporate and income tax rates. Notice that impulse response of aggregate consumption is smaller than or equal to that of aggregate income.

A further interesting result concerns volatility measures for both aggregate series, σ(C tot t ) and σ(Y tot t ) (see Table l in Section 6.1.1). Note how aggregate consumption and aggregate income are less volatile than disaggregated counterparts, and, more importantly, that the former is smoother than the latter. Precisely, σ(Y tot t ) = 1.45 and σ(C tot t ) = 1.17. Moreover, it is important to stress that we generate all these results with a low risk aversion coefficient, q = 1, consistent with empirical micro-studies (e.g. Attanasio, 1999.).

These results are consistent with the widespread empirical evidence that aggregate consumption is smoother than aggregate income, which is one of the most robust empirical evidences matched by equilibrium growth models.

Smoothing and correlations.

Table 2 presents correlations for consumption and income series, at a disaggregated and aggregated level.

It is worth to notice that the correlation between aggregate consumption and output, ρ(C tot t , Y tot t ) = 0.69, decomposes into correlations between total output and the two disaggregated consumption components, ρ(c k t , Y tot t ) = 0.95 and ρ(c l t , Y tot t ) = -0.96, respectively. Total and regular consumption are both pro-cyclical, but the former presents a weaker (positive) correlation with aggregate income. 43 In the logic of our model, this comes from the fact that consumers allocate a share of income to underground consumption, which is countercyclical.

Since we calibrate the weight of regular sector (0.725) to be larger than that of underground sector (0.275), total consumption ends up being pro-cyclical.

Second, Proposition 2 shows that a sufficient condition for aggregate consumption smoothing, is that correlation between regular and underground consumption, ρ(c k t , c l t ) should be smaller, in absolute value, than correlation between regular and underground output, ρ(y k t ,y l t ). 44

43 Pro-cyclicality and Counter-cyclicality are defined in this contest with respect to total income, Y tot t . 44 The proof is trivial, but for completeness we present it in Appendix A. Mace (1991) or Abel and Kotlikoff (1989). The difference is that in this case the insurance comes from income smoothing contract (i.e. a "real side" of the market) while in the works quoted above insurance originates from investing in financial securities. Specifically, we compare impulse response functions and volatility measures for all income definitions, Y tot t , y k , y l , and for all consumption components C t , c k , c l .

(Figure 3 about here)

In Figure 3 the impact on the endogenous variables of a one-standard-deviation shock to regular sector productivity, directly increases capital investment, regular output and regular consumption. Note that total consumption rises only gradually. In particular, the household composed of Mr. and Mrs. k does not choose to adjust consumption completely after an innovation. This is due to inter-temporal substitution and wealth effects, as in a traditional

Robinson Crusoe economy, but in addition we see the redistribution effect within the family, previously defined. Indeed, regular and underground consumption components move always in opposite directions, and the former is much more responsive than aggregate variables. 45 Productivity shocks in the underground sector, and increases in tax rates, reverse the picture, yielding opposite effects. Now aggregate consumption and production are reduced, together with investment. In spite of the remarkable jumps in underground output and consumption, a rise in income and corporate taxes reduces regular consumption, total consumption and investment, thereby impoverishing the economy and causing a recession.

Notice that these patters are consistent with a traditional equilibrium growth model (e.g. the contributions in Cooley, 1995 or King and Rebelo, 1999). The new insight of our approach consists in the opportunity to understand the composition of aggregate in terms of disaggregated variables. Concluding, the most interesting results we observe from the four panels of Figure 3 are that c k t responds to innovations more than C tot t does, and, in absolute terms, also more than c l t . In addition, total consumption presents a highly persistent response after the shocks.

(Figure 4 about here)

Impulse response functions of production series (Figure 4) display an analogous picture, 45 We are suggesting that market and underground consumption profiles, which are defined both as a precontract series, are highly negatively correlated. Table 2 (see section 6. 1.3) shows that correlation between market and underground consumption equals -0.91. To see more clearly forces' interaction, consider the following example. Suppose we have a positive productivity innovation in market sector: market income and labor demand increase, and, since we know the two sectors have negative correlation, underground income and labor demand fall. Then the family reallocates its labor supply to the more productive sector, subtracting it from the less productive. Since labor supply cannot be traded (e.g. we cannot short sell H hours worked in one the sector), consumption would follow approximately income dynamic. Chiarini and Marzano (2006) examine the relationship between market and underground consumption profiles in Italy using econometric techniques in a partial equilibrium framework. Their empirical estimates suggest that the two forms of consumption are complements, so implying that when underground consumption rises, marginal utility of the market consumption rises too. This finding do not contrast with the evidence shown in this paper. In fact, given the negative correlation reported in These results are confirmed by the graphical inspection of Hodrick-Prescott filtered series for consumption and income (Figure 5). The model generates pro-cyclical regular consumption movements, which are positively correlated with regular output. Underground consumption, instead, is countercyclical with respect to total consumption, and to regular consumption. Note that total consumption is always in between its regular and underground component. Analogous comments hold for production series. The purpose of this paper is to provide an original interpretation for consumption and income smoothing. The underground economy constitute a remarkable size of the GDP in many countries and, often, the related behaviors produce a second cycle which gives to households the opportunity to ensure themselves against bad times, by entering an income smoothing contract.

Specifically, workers belonging to same extended family can insure themselves against fluctuations in regular and underground income, by entering the income smoothing contract. Specifically, each consumer-worker-investor can smooth aggregate income, even though disaggregated income and consumption components are more volatile by relying on this risk sharing mechanism. The introduction of the underground sector makes expansions less bright, and recessions less dark.

For a given institutional and productive structure, firms smooth production across sectors, and households smooth consumption inter-temporally and across sector. Agents diversify both economic activities and labor input across sectors, and, by this end (partially) protect the consumption flow from income volatility.

Our income smoothing contract represents, however, one out of many different ways to model income smoothing. We think that the economic literature would benefit from a more thorough investigation of these composition issues, because they are often ignored in the conventional consumption studies as well as in the dynamic equilibrium model literature. ( (10) in the text.

( ) ( ) ( ) ( )           + - - - + - - - = + + - η θ θ γ θ θ η γ 1 1 1 1 1 1 ; 1 1 1 j t j t j t q j t j t j t f h q c c u (A. )
Finally, the decentralized model we study in this paper is represented by equations (A.5), (A.6), (A.7) for j-th household, (A.lO), (A.lI) for i-th firm, and (A.12) for government.

The solution method used to solve this artificial economy is that suggested by King, Plosser and Rebelo (1988a,b). To this end we transform the equilibrium characterization of the economy into an approximating first order autoregressive linear system, applying linear approximations (e.g. Campbell 1994; Uhlig 1999).

F

  may offer risk sharing opportunities to workers, the conclusions of models that do not explicitly incorporate this sector, may be seriously compromised for economies in which underground economy represents a significant share of actual GDP.

  the members have perfect information concerning each other's idiosyncratic shocks

7 approach

 7 consists in inspecting the composition mechanism of income and consumption flows occurring within the family, and across sectors.

1

 1 Productivity shocks and tax rates.

  for labor input employed in the underground sector.

(

  RCE) of Prescott and Mehra (1980) notion. Specifically, aggregate and individual quantities coincide, and equilibrium can be characterized as the F.O.C. of the Representative Household

  available. For the Italian economy, the Ministry of Labor reports that the number of inspected firms has been 118,119 in 2000, 106,307 in 1999 and 95,676 in 1998. The overall number of firms in the Italian Economy has been 4,639,393 in 2000, 4,472,375 in 1999 and 4,311,369 in 1998. As suggested above, we first compute the probability of being detected in each year, p * t , and then we estimate the aggregate probability as

1

 1 Regular consumption is more volatile than regular income, and...

Figure 1

 1 Figure 1 presents impulse response functions, and Table 1 presents selected time-series properties

4

 4 Consumption and Income Smoothing: Inspecting the CompositionHere we complete the characterization of consumption and income smoothing, inspecting the composition mechanism operating before aggregation of consumption and income series.

(Figure 5

 5 Panel A and Panel B about here) 6 Conclusions.

  .4) we derive equation 4 in the text. Notice (A.3) ≡ (A.6).

7 )

 7 Concluding, after introduction of consumption and income smoothing contract, the consumers' side of the model is represented by equations (A.5), (A.6), (A.7). Lemma 2 (Firms) Firms are characterized by a production function, and a cost function.

σ

  represents the total production standard deviation, tot C σ is total consumption standard de- viation, ρ(Y tot , C u ) is the correlation coefficient between total production and underground consumption, and ρ (Y tot , C tot ) is the correlation coefficient between total production and total consumption.

Definition 1 (Consumption and Income Smoothing Contract) The

	as to the Heterogeneous Agent Model (HAM).
	But one day, unexpectedly, a spark ignites between the two. Mr. κ and Miss. l fall in love, and
	are offered an "consumption and income smoothing contract (CISC)". Readers unfamiliar with
	Contract theory would call it a "marriage" contract. The contract, defined below, says the
	consumers should pool together income (and thus labor supply) and consumption.
				contract has three
	features:
	1. l k	t = θ t L t and l l	t = (1-θ t )L t . This means, that Mr. κ and Miss. l pool together their labor
	supplies, L t , then they allocate a share θ t to regular sector, and the remaining (1-θ t ) to
	underground sector.
	2. The family will choose total consumption C tot F o be tot t k t C c * ω = and ( ) tot t l t C c * 1 ω -=	t . Then Mr. κ and Miss. l consumption will
			r
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				to (3)
			6

Table 1 (

 1 see section 6.1.1) and Table 2 (see section 6.1.3), and Figures 1 to 5 present the main results.

  . It should be also noted that income components are quite volatile, consistently with the data.42 Finally, Table3(see Appendix C) shows that this result is robust to sensitivity analysis.

							suggests that "individual" series (c k	t , y k	t , c l t and y l	t ) respect the empirical
	evidence. The table shows that	k t c σ >	σ	y	k t	and that	l t c σ =	σ	y	l t	. Precisely,	σ	k t c	=	96 . 2	,
	σ	y	k t	=	07 . 2	, and	σ	l t c	= l t y σ	=	2	22 .

TABLE 1 : STANDARD DEVIATIONS

 1 Since regular and underground consumption data are not available, no statistics are available. The statistics are the means of 1000 simulations, of length 150 time periods. Each simulated series is detrended using Hodrick-Prescott filter before the statistics are calculated. The numbers in brackets are As stressed in many contributions (e.g. Deaton 1992, Attanasio 1999, Attanasio and Rios-Rull 2000) both income and consumption are quite volatile, even though consumption smoothing is strong evidence across countries and datasets.

				σ (y k	t )	σ (y l	t )	σ (Y tot	t )	σ (c k	t )	σ (c l	t )	σ (C tot	t )
	Actual Data	2.27			1.11	1.44			-	-	1.25
	Simul. Data (s=1.3) 2.07			2.22	1.45			2.96	2.22	1.17
				(0.22)			(0.23)	(0.15)			(0.28)	(0.23)	(0.14)
	Sim. Data (s=2.0)	1.99			1.94	1.40			2.71	1.94	1.14
				(0.24)			(0.25)	(0.14)			(0.29)	(0.25)	(0.17)
	the small sample standard deviations. Sources: C tot	t and Y tot t are withdrawn from Istat database, y k	t and y l	t
	are from Bovi (1999), while c k	t and c l	t are generated with our model.
	y k	t , y l t and Y tot	t . In the first point, instead, we compare c k t with y k	t and c l	t with y l t .	k t , c l t , C tot t , and among

Notes: The model is calibrated for Italian economy within the sample 1970-1996. C tot t represent the consumption of non durable goods and services, c k t and c l t represent the regular and underground component of consumption, respectively. Y tot t is the aggregate GDP, y l t is its underground component.

41

What we do here differs from the first point. Here we compare time series properties among c 42

TABLE 2 : CORRELATIONS

 2 Notes:The first block of the table contains the correlations estimated for the actual data; the second blocks presents the correlations estimated on the simulated data. The first one refers to the case s = 1.3, the second to the case s = 2.0. The moments matched in the calibration of utility function parameters are presented in boldface. See Table1notes.

	y k t y l t Y tot t	y k t 1.00	y l t -0.81 1.00	Y tot t 0.89 -0.45 1.00	c k t ---	c l t ---	C tot t 0.77 -0.54 0.80
	y k t	1.00	-0.98 (0.01)	0.95 (0.01)	0.95 (0.02)	-0.97 (0.01)	0.70 (0.11)
	y l t		1.00	-0.96	-0.94	1.00	-0.51
				(0.01)	(0.02)	-	(0.12)
	Y tot t			1.00	0.95	-0.96	0.69
					(0.02)	(0.01)	(0.11)
	c k t				1.00	-0.91	0.75
						(0.02)	(0.10)
	c l t					1.00	0.42
							(0.13)
	C tot t
	y k t	1.00	-0.97	0.99	0.96	-0.97	0.68
			(0.01)	(0.01)	(0.02)	(0.01)	(0.11)
	y l t		1.00	-0.97	-0.95	1.00	-0.61
				(0.01)	(0.02)	-	(0.12)
	Y tot t			1.00	0.96	-0.97	0.69
					(0.02)	(0.01)	(0.11)
	c k t				1.00	-0.91	0.72
						(0.02)	(0.10)
	c l t					1.00	-0.39
							(0.13)
	C tot t						1 . 0 0
	Proposition 2 (Smoothing and Correlation) Aggregate consumption smoothing requires the
	correlation between regular and underground consumption, ( ) l t k t c c , ρ	to be smaller, in
	absolute value, than correlation between regular and underground production, ( ) l t k t y y , ρ	.
	Proof. See Appendix A .

Table 2

 2 shows how the model matches this restriction, since ( ) In words, this means that regular consumption reacts less to innovations, than regular income does. Note that this argument parallels the key concept of the risk sharing arguments discussed by

	ρ	k t c c ,	l t	= 0.91 and

Table 2

 2 , a rise in underground consumption reduces market consumption, rising its marginal utility. These results are robust, consistent with time series behavior of income and consumption, and support volatility and correlation measures already presented.

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22 Proof of Proposition 2 (Smoothing and Correlations).

  Assume σ 2 (c l t ) = σ 2 (y l t ) (Assumption 1), and σ 2 (c k t ) > σ 2 (y k t ) (Assumption 2).46 Let C tot where σ 2 (x) represents the variance of x, and σ(x, y) represents the covariance between x and y.

	Appendix C: Sensitivity Analysis
	Then σ 2 (C tot h = 0.35	t )= σ 2 (c k tot Y σ 1.05 t )+σ 2 (c l	t )+2 σ(c k tot t , c l C σ 0.96	t ) and σ 2 (Y tot σ C σ t )= σ 2 (y k tot tot Y 0.91	t =c k t )+σ 2 (y l t + c l t )+2 σ(y k t , and let Y tot t ,y l ρ(Y tot , C u ) ρ(Y tot , C tot ) t =y k t +y l t . t ), Aggregate -0.97 0.45
	consumption smoothing implies σ 2 (C tot h = 0.38 1.02 0.91	t )< σ 2 (Y tot	t ), or, equivalently σ 2 (c k 0.90 -0.93	t )+σ 2 (c l	t )+2 σ(c k 0.32	t , c l	t )<
	σ 2 (y k h = 0.44 t )+σ 2 (y l	t )+2 σ(y k 1.27 t ,y l	t ). Rewrite σ(c k 0.97	t , c l	t ) as ρ(c k 0.76 t , c l t )σ(c k	t ) σ(c l	t ) and σ(y k -0.94	t , y l	t ) as ρ(y k 0.68 t , y l	t )σ(y k t )
	σ(y l h = 0.46 t ) where ρ(x, y) stands for the correlation between x and y, and σ(x) is the standard deviation 1.32 1.20 0.91 -0.98 0.77
	of x. Therefore: σ 2 (c k t )+σ 2 (c l h = 0.48 2.03	t )+2 ρ(c k 1.86 t , c l	t )σ(c k	t ) σ(c l 0.92 t )< σ 2 (y k	t )+σ 2 (y l -0.98 t )+2 ρ(y k	t , y l	t )σ(y k 0.71 t ) σ(y l	t ). By
	construction we have σ 2 (c l	t ) = σ 2 (y l	t ) and obviously σ(c l	t ) = σ(y l	t ). Since σ 2 (c k	t ) > σ 2 (y k	t ) and
	o r t ) > σ(y k t ), consumption smoothing now implies that ρ(c k t ), or equivalently obviously σ(c k y l t )σ(y k ( ) ( ) ( ) ( ) 1 ; ; < < k t k t l t k t l t k t c y y y c c σ σ ρ ρ . Therefore η = 1.25 1.67 1.30 0.78 -0.98 t , c l t )σ(c k ( ) ( ) 1 ; ; < l t k l t k t y y c c η = 1.27 1.34 1.21 0.90 -0.95 ρ or η = 1.29 1.1 0.90 0.82 -0.94 ρ η = 1.31 1.13 1.11 0.98 -0.93 t ( ) ( ) l t k t l t k t y y c c ; ; ρ ρ < . Remark 2 Notice that since y l t =c l t , we may rewrite Proposition 2 statement as follows t ) < ρ(y k t , 0.71 0.74 0.54 0.58 ( ) ( ) l t k t l t k t y y y c ; ; ρ ρ < or analogously as ( ) ( ) l t k t l t k t c y c c ; ; ρ ρ < . η = 1.33 1.35 1.25 0.92 -0.95 0.57 η = 1.40 1.26 1.10 0.87 -0.95 0.17 F f = 1.90 1.36 1.18 0.86 -0.94 0.45 o f = 1.92 1.41 1.19 0.84 -0.94 0.52 r f = 1.94 1.25 1.13 0.90 -0.96 0.55
	f = 1.96 f = 1.98 f = 2.00		1.25 1.49 1.10				P e e r 1.04 0.83 1.36 0.90 P r e e 1.05 0.95	-0.95 -0.96 -0.98	0.71 0.73 0.70
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											23

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

This hypothesis will be important since it simplifies agents' interaction after the contract introduction.

We choose to restrict the analysis to one family to keep notation simple. The size and the number of the extended family can easily be enlarged.

Note that since working in the underground sector is costly, as we can see from the last term of the instantaneous utility function, we rule out equilibria were all labor supply goes to the underground sector. In other words, if l l t →∞, then µ(l l t )→∞, making this decision too costly for the family.

The choice of ½ is without loss of generality, it just simplifies the algebra.

We choose to end up with a representative agent model since the collected data on income and consumption refers, more or less implicitly, to a representative household. In other words, we harmonize the theoretical scheme with the data. If, for instance, we had chosen to calibrate direct1y the heterogeneous agent model, we could not be sure anymore of equivalence between theory and data.

The generalization to continuum of households is not necessary, but is consistent with the traditional set up of equilibrium growth models (see[START_REF] Prescott | Recursive Competitive Equilibrium: the Case of Homogenous Household[END_REF].

This specification is adapted from[START_REF] Cho | Family labor supply and aggregate fluctuations[END_REF] and[START_REF] Cho | Employment and hours over the business cyc1e[END_REF]. Unlike the extreme cases of indivisible labor[START_REF] Hansen | Indivisible labor and supply and the business cycle[END_REF], where all the fluctuations occur on the extensive margin, and the divisible labor in which the fluctuations take place on the intensive margin, in this formulation of preferences households may allocate their time along both margins (intensive-hours and extensive-employment margin). Cho and Rogerson achieve this feature by introducing heterogeneity into the opportunity sets of household decision makers, and Cho and Cooley introduce some fixed costs of going to work that are not explicitly modelled. This allows us to capture changes in labor in both the market and the underground sector simultaneously, and it is consistent with the data, where we observe substantial variations in both the markets.

Restriction on the utility function to make the inter-temporal optimization problem well defined are derived in[START_REF] Busato | Market and Underground Activities in a two-sector dynamic equilibrium model[END_REF].

Notice that there exist perfect substitutability across sector, in the sense that there are no adjustment costs while transferring labor supply (demand) from a sector to an other. Each sector, however, has its own peculiar characteristics that the instantaneous utility function tries to capture with regard to consumer's behavior.

Note that the tax structure is critical for the existence of underground activities, and therefore for the source of risk sharing.

Here we have already implemented the features of consumption and income smoothing contract into i-th firm objective. For details, see Appendix A, Proposition 1, Lemma 2.

Notice that in this context the regular share of aggregate labor demand equals the regular labor demand itself, because we normalize total labor to unity. The same holds for the underground labor market segment.

See Appendix A, Proposition 1, Lemma 3.

Notice that the Government balances its budget only in expectation, since with probability 1 -p some firms and workers are evading. Hence equation[START_REF] Chiarini | Market Consumption and Hidden Consumption, A test for Substitutability[END_REF] will not be satisfied on a state by state basis.

Appendix A: Proofs of Propositions in the Text

Proof of Proposition 1 (Representative Agent and Extended Family).

Lemma 1 (Households and Extended Families) The consumer side of the heterogeneous agent model is represented as the following three equations:

Then consumption and income smoothing contract (Definition 1) dictates following two conditions:

1. Income Pooling:

2. Consumption Pooling:

Now, normalizing L t to unity and implementing these features into (A.1)-(A.

3), we rewrite them as:

Appendix B: Perfect Risk Sharing Scheme

After entering the contract, consumers agree on a perfect risk sharing scheme, in the sense that they set ratio between marginal utilities equal to a constant, i.e.

( ) ( )

, where u'() represents the first derivative of the utility function with respect individual consumption stream. Next, notice that ( ) ( ) ( )

Assuming that both consumers have the same weight, we can set φ k = φ l and therefore c k t = c l t . The two consumers will have an equal consumption profile. In terms of total consumption, we have c k t = c l t =1/2 c j t , where c j t represents consumption chosen by j-th household at time t. 46 Both assumptions are derived from empirical evidences, robust across countries and data sets, and matched by our model, see Deaton (1992), Attanasio (1999), Schneider and Enste (2000).


Jan P A M Jacobs
email: j.p.a.m.jacobs@rug.nl

Gerard H Kuper

Daan P Van Soest

Michael Funke

Shelby Gerking

Willem Van Groenendaal

Simon Kuipers

On the Effect of High Energy Prices on Investment

Keywords: thresholds, investment behaviour, panel data, energy policy Thresholds, investment behaviour, panel data, energy policy. JEL-classification: C23, E22, Q43

Empirical analyses of firm behaviour typically assume that there is a stable relationship between investment on the one hand, and changes in the relative prices of inputs, output demand, and other determinants on the other hand. However, because of the lumpy nature and irreversibility of investments and the presence of uncertainty about future economic developments, a specific percentage change in relative prices and output demand may not always lead to the same percentage change in capital stocks. That means that different regimes may exist in investment behaviour. We test whether such regimes exist using high-quality data on eight manufacturing industries in the Netherlands. Three different regimes can be identified that are characterised by differences in the relative input price levels, and we find that if relative prices take on extreme values, the propensity to adjust the scale of production to changes in demand is very low.

Introduction

This paper focuses on discontinuities in the investment relationship. The relevance of this analysisfor researchers and policy makers alike -is immediately clear when considering the past pattern of changes in the prices of two main variable inputs in production, labour and energy. From the early 1970s until the mid-1980s, the relative price of energy to the wage rate increased substantially, whereas after the mid-1980s this relative price dropped sharply. Typically, researchers have coped with this discontinuity by making either the regression equation's intercept or its elasticities timedependent. Although such an approach may give a reasonably good fit for past investment behaviour, it is difficult to determine which coefficient values should be used for forecasting purposes. For instance, do firms expect the current energy price increases to be temporary or more permanent? By identifying thresholds that result in different investment regimes, we may be able to better analyse the future effects of energy saving policies. This is especially relevant for policy makers today because of the surge in energy prices as observed in the first half of the current decade.

Changes in output demand and changes in relative input prices are the main determinants of investment rates (e.g. Blanchard and Fischer 1989, p. 301;[START_REF] Abel | Consumption and Investment[END_REF]). Typically, empirical analyses are based on the implicit assumption that the relationships under investigation are stable over the estimation period (Berndt 1991, p. 277). In the case of energy policy modelling, however, [START_REF] Kuper | Path-Dependence and Input Substitution: Implications for Energy Policy Modelling[END_REF] argue that the parameters obtained from time-series estimation are not robust to rapid structural changes. Also the degree of energy price uncertainty may affect investment rates. [START_REF] Kuper | Does Oil Price Uncertainty Affect Economic Activity?[END_REF] show that higher levels of energy price uncertainty render changes in energy intensity more sluggish, because high uncertainty implies that there is a probability that energy price change reversals take place. The possibility of adverse price changes makes firms less willing to invest in new technologies (including, for example, energy saving equipment) because these investments may turn out to be unprofitable ex-post. They conclude that managing uncertainty should be high up on the policy agenda. 1 1 [START_REF] Pawlina | Investment under Uncertainty and Policy Change[END_REF] discuss the effects on investment of policies aimed at managing uncertainty. We search for discontinuities in investment behaviour using high-quality data on a panel of eight Dutch manufacturing industries over the period 1979-1994. We use industry data (as opposed to firm level data) because government policy tends to be industry generic rather than firm specific. The fact that we use fairly aggregated data imposes strong demands on the quality of especially the capital stock data series. Our decision to use the Netherlands' data set is motivated by exactly that consideration as capital is measured by means of detailed surveys rather than on the basis of Perpetual Inventory Methods. We apply [START_REF] Hansen | Threshold Effects in Non-Dynamic Panels: Estimation, Testing, and Inference[END_REF] panel data threshold estimation procedure that allows us to estimate the threshold parameters demarcating the different regimes simultaneously with the other parameters. This approach has become quite popular,2 however to the best of our knowledge the application is new to the field of energy economics. Our attempt to measure the threshold at which investment is triggered is advocated by [START_REF] Carruth | What do we know about Investment under Uncertainty?[END_REF] as a way to successfully test the relation between uncertainty and investment. We do find evidence for the existence of thresholds in terms of input prices. This implies that industry responsiveness to changes in its economic environment does depend on the value of input prices. Our findings suggest that uncertainty makes firms less responsive to increases in demand. Hence, firms invest less if uncertainty is high (see also [START_REF] Guiso | Investment and Demand Uncertainty[END_REF].

The set-up of the paper is as follows. In Section 2 we explain why investment behaviour can exhibit discontinuities. The focus is on the fixed-costs nature of investment (the existence of which implies that investments are, at least to some extent, irreversible) and on uncertainty. Section 3 explains the threshold estimation method used. In Section 4 threshold effects are estimated for eight sectors of the Dutch industry. Section 5 argues that the observed effects are not simply industry-and/or timespecific, but are indeed related to the value of energy prices. Section 6 concludes.

Discontinuities in investment behaviour

The main hypothesis we wish to test in this paper is whether the same percentage change in each of the various variables influencing investment behaviour always result in the same percentage change in investment, irrespective of the absolute level of (relative) input prices, state of the economy, etc. We expect this hypothesis not to be supported by the data, because of the existence of fixed adjustment costs, irreversibility of investments and other indivisibilities in the investment process, financial constraints or a combination (see [START_REF] Holt | Investment and Dividends under Irreversibility and Financial Constraints[END_REF].

There are two reasons why we expect the elasticities of investment with respect to either relative input prices or industry demand not be constant. First, we expect that changes in relative input prices or aggregate demand do not always induce firms to purchase new capital goods to adjust its input mix to the new relative price structure. Due to the fixed-costs nature of such adjustments and the irreversibility of investments, there is a range of relative prices and percentage increases in output demand for which the net present value of such an investment is negative. Hence, theory predicts that within that range, relative price changes do not result in changes in the composition of the capital stock.

Second, the responsiveness of firms with respect to changes in relative prices or aggregate demand may also be affected by uncertainty about future economic developments. Whereas at intermediate ('normal') levels of relative prices the technology choice is fairly easy, additional investments are more risky when relative prices take on more extreme values. Depending on whether the shocks in relative prices are expected to be only temporary or more permanent, the firm will select a technology that is either better equipped for the more 'normal' relative price levels or for the more extreme price levels currently observed. However, uncertainty with respect to the nature of the new price structure may be such that it is optimal to postpone increasing the scale of production, even though demand has increased (see also [START_REF] Bernanke | Irreversibility, Uncertainty, and Cyclical Investment[END_REF]. More information may be obtained (for example by just wait and see) before purchasing additional capital goods. If mean reversion is expected to be strong, uncertainty 6 with respect to future prices is negligible and firms are equally responsive to relative price changes independent of the current relative price level. If divergence from the mean is more persistent, postponing the investment may be optimal. For an overview of the literature on investment under uncertainty and on the role of mean reversion, see [START_REF] Dixit | Investment under Uncertainty[END_REF] or [START_REF] Pindyck | Irreversibility, Uncertainty, and Investment[END_REF].

The threshold estimation model

Before discussing the data and the estimation results, we briefly explain the estimation procedure. We start from the assumption that changes in the variable input mix or in the scale of production require investments in new capital goods. We test for the existence of relative price ranges for which the responsiveness of industries to changes in market circumstances differs. If no thresholds are found to exist, we can reject the hypothesis that indivisibilities matter at the industry level and hence that the response of firms to changes in relative prices is the same over the entire estimation period. To avoid an arbitrary selection of relative price ranges we estimate an investment function using the threshold regression method with industry-specific fixed effects as proposed by [START_REF] Hansen | Threshold Effects in Non-Dynamic Panels: Estimation, Testing, and Inference[END_REF]. The standard investment relation is adjusted to allow for, for example, two thresholds in the following way:3

( ) ( ) ( ) it it it it it it it i it e q I x q I x q I x y + > ′ + ≤ < ′ + ≤ ′ + = 2 3 2 1 2 1 1 γ β γ γ β γ β µ , (1)
where subscripts i and t denote industries and time periods respectively, y it is the growth rate of the stock of capital, i.e. net investment divided by the stock of capital, µ i captures industry fixed effects, x it is a vector of k explanatory variables, q it is the threshold variable, γ 1 and γ 2 are the thresholds, and I is an indicator function that has a value one if its argument is true and zero otherwise. Therefore, the elasticities with respect to the explanatory variables are regime-dependent as the coefficient vectors β j (j=1,…,3) are allowed to differ. The variables x it and q it as used in our application are defined below.

The error term is i.i.d. with mean zero and finite variance, which rules out lagged dependent variables. The threshold is estimated using conditional least squares. The observations are first sorted on the threshold variable. The sum of squared residuals is computed for all values of the threshold variable.

The optimal value of the threshold is the value that minimises the sum of squared residuals.

Bootstrapping simulates the asymptotic distribution of the likelihood ratio test that is used to determine whether the threshold effect is statistically significant under the null of no threshold, i.e. a linear investment model. We use 300 bootstrap replications to construct asymptotically valid p-values. If the null is rejected, similar tests are used to determine whether there are one, two or even more thresholds.

Estimation results

We use a panel of eight Dutch manufacturing industries and our estimation period covers 1979-1994, the longest time period (sixteen years) for which a consistent and balanced data set is available for the Netherlands (see Appendix A). Unfortunately, this high quality set is discontinued. This period covers a period of energy price increases as well as a period in which energy prices dropped again.

The inputs of production are labour, energy and capital. The 3-dimensional vector of explanatory variables (x it ) consists of the rates of growth of two relevant relative prices (the ratio of the user cost of capital and the wage rate, R/W, and the ratio of the price of energy and the wage rate, P E /W)4 , and the growth rate of gross value added (X). We carefully constructed the rental price of capital as capital income (in current prices) resulting from production (which is given by the gross operating surplus corrected for wage income of self-employed) divided by the capital stock.

Thus, the reduced-form equation (1) captures a substitution effect as well as an accelerator effect, and accounts for the (relative) costliness of investments through the inclusion of the cost of capital. Before energy divided by the wage rate 5 and the growth rate of gross value added. Other combinations of interaction terms were not significant. These linear models assume no thresholds and thus serve as benchmark models. The industry-specific fixed effects pick up the rate of depreciation, so that the dependent variable can be interpreted as gross investment over the stock of capital. The results are in Table 1.

<Insert Table 1 about here>

The rate of growth of the rental price of capital relative to the wage rate (R/W) is not found to be significant at 5%. However, the rate of growth of the relative price of energy as well as the percentage increase in output demand are found to positively and significantly affect the rate of growth of capital.

Model (b) indicates that the effect of the rate of growth of value added on investment is not constant but depends of the relative price of energy.

The regression results presented in Table 1 indicate that not all parameters are constant, that is independent for instance of whether the price of energy relative to the wage rate is high or low.

However, the second model does not identify different regimes. Let us now allow the elasticities to differ between regimes, where the thresholds demarcating these regimes are estimated jointly with all other coefficients of the model.

Our threshold variable is the relative price of the variable inputs (calculated as the energy price divided by the wage rate; that is, q it =P E /W). The hypothesis is that firms respond differently when relative prices take on extreme values. In principle the coefficients for all elements of vector x it may be regimedependent, i.e. dependent on the threshold variable. The threshold variable is lagged one period for statistical reasons. An economic argument might be that it takes time to order and install new capital goods. In our search for thresholds, we tested for the existence of up to three thresholds for all three 5 This price ratio is lagged one period based on statistical grounds. price of capital R/W). We did not find evidence for thresholds for the two relative price variables, which implies that the impact of changes in these prices on investment behaviour is not regimedependent. However, there is strong evidence for the existence of thresholds when considering the accelerator effect. For this variable, the null of no thresholds is rejected at the 5% significance level.

The actual number of thresholds is found to be two: the F-statistic testing the possibility of one versus two thresholds equals 14.1. Using bootstrapped critical values we can reject the null of one threshold at a 1% level of significance in favour of the two threshold hypothesis. Finally, we do not reject the null of two thresholds versus three thresholds (the associated p-value is about 0.12).

The existence of the two thresholds can also be illustrated graphically. Figures 1 and2 show the values of the Likelihood Ratio tests for the first threshold (γ 1 ) and second threshold (γ 2 ), respectively. The optimal threshold values are those for which the Likelihood Ratio test is equal to zero, and the dotted horizontal lines identify the 95% confidence intervals around the thresholds' point estimates. The two point estimates are 1.41 and 2.60, and the associated 95% confidence intervals are [1.32, 2.29] and [2.49, 2.76]. The confidence intervals do not overlap, so we do find three different regimes.

<Insert Figures 1 and2

about here>

Given the existence of three regimes for the accelerator and the absence of thresholds for the other two explanatory variables, we can now present the coefficient values of the investment equation ( 1). The relative price of energy is found to be positive and significant, as can be seen from the second row in Table 2. The fact that there is no evidence for the existence of thresholds for this explanatory variable implies that its coefficient does not differ significantly across the regimes. The rental price of capital relative to the wage rate (R/W) is not found to be significant at 5%. The elasticities for the accelerator in the different regimes are presented in the bottom three rows. If the price of energy relative to the wage rate is between 1.41 and 2.60, the accelerator is highly significant.

<Insert Table 2 about here>

Discussion

Our findings indicate that the rental price of capital is not an important determinant for investment behaviour. Moreover, the relative price of energy with respect to labour does affect the propensity to invest throughout the sample. Furthermore, increases in the demand for output (X) only result in net increases in periods where the relative variable input prices are at an intermediate level. Thus, in periods where the relative price takes on a more extreme value, only relative price changes trigger additional investments to adjust the input mix, whereas changes in industry demand do not result in significant changes in the capital stock. Although we cannot test what causes this latter result, it is helpful to realise that changes in the relative price of energy with respect to labour are driven by the energy price. The wage rate increases gradually, whereas the development of the energy price is much more volatile. 6 So, one explanation for the different regimes might be that in periods of extreme energy prices there is too much uncertainty to warrant additional investments in response to demand fluctuations (see also [START_REF] Kuper | Does Oil Price Uncertainty Affect Economic Activity?[END_REF]. Additional investments, if any, do not arise from increased demand but from firms adjusting the energy intensity of production.

One might argue that the observed effects can simply be captured if we allow the accelerator's coefficient to be industry-and/or time-specific. For instance, the business cycle coincides with specific relative price levels, which explains the observation that industry demand triggers investments in some periods but not in others. Similarly, not all industries pay the same prices for their inputs (especially 6 The coefficient of variation of the rate of growth of energy prices equals10.7 while the coefficient of variation of the rate of growth of the wage rate is 0.7. prices (as compared to the wage rate). Inspection of the number of observations that fall in the three regimes for each year and industry gives insight into the validity of these arguments. Table 3 shows that to a certain extent, the thresholds capture the past pattern of relative price changes. More specifically, the first regime is found to coincide with post-1986 observations whereas the third is found to coincide with the pre-1986 period. Therefore, this analysis indicates that there is a structural break in investment behaviour between the period of high and increasing relative energy prices (the first period) and that of lower, relatively stable relative energy prices (the second period). However, simply imposing a dummy to capture that break will not do, as the second regime (in which the accelerator is found to be particularly important) exists in both subperiods, albeit slightly more dominant in the post-1986 period than in the preceding one.

<Insert Table 3

about here>

The thresholds might also reflect differences in industry behaviour rather than point at more structural differences. However, Table 4 rules out this possibility. All observations of the energy-intensive sectors fall in the lower two regimes, whereas the bulk of the other six sectors occur in the higher two regimes. So, although we find that relatively energy-intensive industries (chemicals and the basic metal industry) generally face lower relative energy prices than do the other industries, the number of observations for each of the two groups falls for about 50% in the middle regime. Consequently, it is not simply the case that -in response to changes in industry demand -relatively energy-extensive industries would be more apt to invest than the relatively intensive ones. This would have been the case if, for example, all observations of the latter were concentrated in the lower relative price regime.

<Insert

Conclusion

The main driving forces of investments in new technologies are changes in the demand for the firm's or industry's output and changes in relative variable input prices. In this paper we argue that the elasticities of these explanatory variables may not be constant over time due to, for example, uncertainty about future market circumstances and the fixed-cost nature of investments. We use [START_REF] Hansen | Threshold Effects in Non-Dynamic Panels: Estimation, Testing, and Inference[END_REF] threshold estimation procedure to determine whether discontinuities in investment behaviour exist using a data set of eight Dutch manufacturing industries for the period 1979-1994. The data set is rather short because the high quality data set we use in this paper is discontinued and can not be extended to cover more recent years.

We find that the propensity to invest in response to changes in relative input prices is about constant over the entire estimation period. However, we identify three regimes with different responses on investment to changes in output demand. We find that if relative prices take on extreme values, the propensity to adjust the scale of production to changes in demand is very low. These relative price changes are caused by changes in energy prices because the wage rate develops rather gradually.

The implications for the current situation of high and volatile energy prices are clear. Firms do not expand their production capacity if demand increases. Our conjecture is that in periods of extreme relative prices, for instance relative high energy prices, there is too much uncertainty about the future energy prices to warrant additional investments in response to demand fluctuations. If energy price uncertainty could be reduced, firms would be more inclined to expand capacity and invest in new, energy-saving, equipment even in periods when energy prices are high.

  testing for the existence of thresholds, we first present the results for two linear models. Model (a) , whereas Model (b) includes an interaction term between the price of

Figure 1 .

 1 Figure 1. Confidence interval construction in a double threshold model.

Page 8 of 21 Editorial

  the accelerator X, the relative variable input price P E /W and the relative rental

	Submitted Manuscript
	explanatory variables (
	F o
	r
	P
	e e r
	R e v i e w
	9

Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  where the relative energy price is very high (P E /W > 2.60) or very low (P E /W ≤ 1.41).

	than for regions
	F o
	r
	P
	e e r
	R e v i e w
	10

The responsiveness of investments to changes in demand is much higher in the intermediate range of the thresholds (1.41 < P E /W ≤ 2.60)

Page 10 of 21 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  Therefore, industries that face intermediate levels of relative prices, simply exhibit a different type of investment behaviour than industries that either pay very high or very low energy

	Submitted Manuscript
	labour and energy).
	F o
	r
	P
	e e r
	R e v i e w
	11

Table 4

 4 Rejecting these alternative explanations leads us to the conclusion that only in periods of relatively stable energy prices demand fluctuations lead to additional investments. Demand fluctuations do not induce higher investment if energy price changes are more extreme. However, in either case energy price changes do effect investments directly, but the nature of these investments may differ across regimes.

	about here>

Page 12 of 21 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	F o F o F o F o
	r r r r
	P P P P
	e e r e e r e e r e e r
	R e v i e w R e v i e w R e v i e w R e v i e w
	16

Table 1 .

 1 The linear model estimates for the investment function (dependent variable is ∆log(K);

	Hubert-White robust standard errors are presented in parenthesis; industry-specific fixed effects are
	not shown).
		Model (a)	Model (b)
	Regressor	Coefficient	Coefficient
	∆log(R/W)	0.037	0.033
		(0.021)	(0.019)
	∆log(P E /W)	0.035 *	0.040 **
		(0.014)	(0.015)
	∆log(X)	0.181 **	0.584 **
		(0.064)	(0.161)
	(P E /W ) -1 × ∆log(X)	-	-0.166 **
			(0.058)
	Number of Observations	128	128
	Sum of Squared Errors	0.143	0.125

Table 2 .

 2 Double threshold model estimates for the investment function (dependent variable is ∆log(K);

		Submitted Manuscript
	Hubert-White robust standard errors are presented in parenthesis; industry-specific fixed effects are
	not shown).
	Regressor	Coefficient
	∆log(R/W)	0.023
		(0.018)
	∆log(P E /W)	0.026 *
		(0.012)
	∆log(X) × I[(P E /W) -1 ≤1.41]	0.017
		(0.044)
	∆log(X) × I[1.41<[(P E /W) -1 ≤2.60]	0.498 **
		(0.102)
	∆log(X) × I[[(P E /W) -1 >2.60]	0.028
		(0.082)
	Number of Observations	128
	Sum of Squared Errors	0.119

* Significant at the 5% level ** Significant at the 1% level. Page 17 of 21 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 3 .

 3 Number of observations per year in each regime.

			Submitted Manuscript	Page 18 of 21
			Regime
	Year	I(P E /W≤1.41)	I(1.41<P E /W≤2.60)	I(P E /W>2.60)	Total
	1979		7	1	8
	1980		3	5	8
	1981		2	6	8
	1982		2	6	8
	1983		2	6	8
	1984		2	6	8
	1985		2	6	8
	1986		7	1	8
	1987	2	6		8
	1988	2	6		8
	1989	2	6		8
	1990	2	6		8
	1991	2	6		8
	1992	4	4		8
	1993	3	5		8
	1994	5	3		8
	Total	22	69	37	128

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 4 .

 4 Number of observations per sector in each regime.

	Page 19 of 21		Submitted Manuscript
			Regime
		I(P E /W≤1.41)	I(1.41<P E /W≤2.60)	I(P E /W>2.60)	Total
	Agriculture		8	8	16
	Textiles and clothing		10	6	16
	Building materials		10	6	16
	Construction	1	10	5	16
	Food and beverages	2	8	6	16
	Paper industry	3	7	6	16
	Chemical industry	8	8		16
	Basic metal industry	8	8		16
	Total	22	69	37	128

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

See for instance[START_REF] Shen | The Impact of Cross-Ownership on the Reaction of Corporate Investment and Financing Constraints: A Panel Threshold Model[END_REF] and[START_REF] Shen | Cost Efficiency and Banking Performances in a Partial Universal Banking System: Application of the Panel Smooth Threshold Model[END_REF]. The latter allows for a smooth transition between regimes.[START_REF] Bo | A Threshold Uncertainty Investment Model for the Netherlands[END_REF] apply the Hansen panel data estimation procedure in a threshold uncertainty model of investment.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Note that the specification of equation (1) allows for two thresholds and hence is able to discern three regimes, but obviously the specification can easily be changed in order to allow for any number of thresholds.

As capital and energy are often found to be complements, we use the wage rate as a deflator of the rental price of capital (see[START_REF] Kemfert | Estimated Substitution Elasticities of a Nested CES Production Function Approach for Germany[END_REF].

Hein Mannaerts and Henry van der Wiel (both CPB, Netherlands

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Appendix A Data

Our data panel covers eight sectors of industry for the Dutch economy: agriculture, food and beverages, textiles and clothing, paper industry, basic metal industry, building materials, chemical industry and construction. These sectors are chosen on the basis of data availability for a longer time period. Data on energy use and energy prices are not (yet) available for the period before 1973 and after 1994 consistent with the 1973-1994 period. Furthermore, not all variables are available for the chemical industry for the period 1973-1976, so we restricted the time period to the period 1977-1994.

There are three main sources of the data. Volumes and prices of value added and labour are taken from the P-series of the National Accounts 1997 of CBS Statistics Netherlands (1998). The CPB Netherlands Bureau for Economic Policy Analysis provided data on the stock of capital in 1990 prices.

Data on the use of energy and the price of energy are based on the publication De Nederlandse Energiehuishouding of the CBS. Based on these data, we have constructed the rental price of capital as capital income (in current prices) resulting from production (which is given by the gross operating surplus corrected for wage income of self-employed) divided by the capital stock.