2.2.1. The Country-Wide Economic Equilibrium Model
The country-wide layer (CL) is a multi-sectoral, multi-regional, dynamic general equilibrium model (CGE). Compared to a standard economic equilibrium model, the CL introduces formulations to represent the location choices, the inter-regional flows, and the regional economic production and consumption subject to the country-wide (and EU-wide) economic equilibrium.
The CL model is a simultaneous system of mixed-complementarity conditions, derived as Kuhn–Tucker conditions of microeconomic optimization of the agents (i.e., suppliers and consumers) and equilibrium conditions covering all markets for commodities and primary production factors (i.e., labour and capital) simultaneously. The dual variables of the equilibrium conditions determine the prices of commodities and primary production factors. A balance equation acting as a closure of money flows represents the Walras law and, consequently, the model determines all except one of the prices (or a price index), which is the numeraire. The equilibrium runs over time dynamically based on stock-flow relations for capital, labour, and other variables. The optimisation of agents’ behaviours includes foresight, which adjusts over time myopically in the standard model version.
The agents are households, firms, and the government. They are specific to each country. Households derive demand for commodities and supply of labour from utility maximization under a budget constraint, which depends on revenues from wage salaries, dividends, and social benefits. Consumption splits in categories of utility drivers (e.g., food, health, entertainment, housing transport services) and form an aggregated utility function. The model formulates an optimization problem which maximizes utility subject to income to derive consumption by product and supply of labour (in other terms, participation in the labour market). The population and labour forces evolve in the future exogenously, based on assumed demographic factors. Labour mobility across regions is endogenous, depending on wage differentials and the regional attractiveness calculated for each region endogenously. At the country level, labour is mobile only among sectors within national borders.
Firms produce a single representative output by sector of activity. The outputs are differentiated depending on sector and country origin. The outputs are inputs to other sectors used together with primary production factors (labour and capital) to produce the output by sector of activity. Production functions aggregate the inputs to represent the possibility frontiers of production technologies. The production possibilities evolve over time driven by exogenous productivity factors. The model formulates an optimization problem by sector to represent the choice of the production factor mix as a result of minimization of total production cost, assuming price-taking for all inputs, subject to the production technology possibility frontier. Within a year, capital stock acts as a restriction of the volume of production by sector. The model includes investment functions to project the increase or decrease in capital stock over time. The investment functions depend on the rate of return of existing capital stock, derived from the static equilibrium, and a foresight of the likely evolution of demand for the sector’s output in the future. The implementation of investment employs fixed proportions of goods and services, such equipment goods, construction, and several types of services, which further constitute part of the demand for goods and services in the economy.
The commodities, being distinct by sector and country of origin, are non-perfect substitutes for each other in the formulation of commodity trading, and in all choices for consumption or production mix. The model provides the possibility to specify some commodities as non-tradeable. Furthermore, the model assumes that all markets of commodities and primary production factors operate under perfect competition conditions, which imply that prices derive from marginal costs and all agents are price takers.
The government acts as a final consumer of commodities for public consumption and investment. The model includes the non-market services sector to formulate production, choice of inputs, and investment of the public sector. The model included a variety of fiscal policy instruments, which aim to collect revenues to finance public investment (determined exogenously) and subsidies to firms or households, along with policy instruments for social benefits, social security, and transfer payments.
Labour, distinguishing a few typical skills, is fully endogenous regarding both demand and supply. Demand for labour skills derives from cost optimization of production by sector of activity. A Phillips curve, depending on labour force availability and unemployment, determines the relationship between the real wage rate and the labour market unemployment equilibrium by type of skill. The Philips curve represents frictions in the labour market due to the trade unions’ market power and other imperfections, such as mismatches in skill demand and supply. Thus, the curve supports a shift of labour supply function driving equilibrium unemployment, see Reference [
53], which is higher than natural unemployment. The skill types are distinguished in the model only for the purpose of capturing wage differentials and productivity differences.
The CL model covers the global economy. The country resolution distinguishes all European countries individually, as well as major neighbouring countries (
Table 1). The model distinguishes the G20 countries individually and divides the rest of the world into regions by grouping together the respective countries. The CL model segments the economy into 37 distinct sectors of activity, hence, 37 distinct goods and services for each country or group of countries.
The segmentation of industrial sectors (
Table 2) is appropriate for capturing the changes implied by transport decarbonization and electrification. For this purpose, the model distinguishes between electricity generation, electricity networks, and the fuelling sectors, including biofuels, and between car manufacturing and electric goods industries. The regional dataset specifies different regional endowments in the type and intensity of renewable resources, as some regions having high solar potential differ from those having high wind potential and those having high biomass feedstock production possibilities.
The database of the model uses statistics from Eurostat (National Accounts, Public Finance, Input-Output tables, Labour statistics), WIOD (World Input-Output Table) and GTAP (Global Trade Analysis Project). A special routine performs model calibration, that is calculation of scale parameters of production and consumption functions to allow the model reproducing the base year statistics exactly. The model projects the entire structure of production, consumption and market equilibrium for all sectors and countries listed in
Table 1 and
Table 2.
2.2.2. Overview of Functional Forms Used in the Model
The production functions in the model represent nested choices of the production factor mix. The structure of the nesting (
Figure 1), often called a nested scheme, is important for the magnitude of substitution or complementarity among the production factors. At each level of the nest, a single Allen substitution elasticity applies, but the values of elasticities differ across the nesting levels. In all levels, the functions aggregating the respective production factors followed the constant elasticity of substitution (CES) algebraic form, which involves scale parameters and an elasticity of substitution. The scale parameters are determined during calibration using the value shares of production inputs. Using dynamic calibration techniques, we vary the scale parameters over time, in particular for transport and energy choices, to make restructuring options possible, for example, regarding alternative fuels and electric vehicles. The same techniques are used to link the detailed transport or energy models to the economic model, so as to make the latter able to mimic the restructuring projections of the detailed models. In this manner, the production functions can produce input mixes like the fuel and technology mix suggested by transport and energy models in the context of decarbonization scenarios. We use this technique to link the general equilibrium model with projections using the PRIMES-TREMOVE models, which also operate in the E3MLab laboratory. In this manner, we introduce the transport and energy sector restructuring projections, calculated in dedicated models, in the economic and regional models to assess economic impacts adequately.
The production model solves for cost minimization to determine the input mix. Assuming constant returns to scale, the indirect minimum cost function separates the unit cost function from the volume of output. We thus apply the Shephard’s lemma to derive the optimum quantities of production inputs per unit of output volume. The typical formulation for a sector
i producing
at unit cost
c using
as an input priced at
is as follows (
and
are scale and elasticity of substitution parameters, respectively, whereas
and
are productivity factors):
The
j index in the above formulation spans the production inputs. This index spans all goods and services with a distinction by country of origin. The product varieties are not perfect substitutes for each other, following the well-known Armington assumption, as in Reference [
54].
The cost of using labour and capital in production depends, respectively, on wage rates (w) and the unit capital cost (r), both derived as equilibrium prices of the respective markets. Exogenously determined tax rates and social security contribution rates also affect the costs of primary production factors.
The investment functions are specified by sector of activity and determine the evolution of the stock of capital in the future. Within the static equilibrium the stock of capital by sector is given, acting as a constraint on potential output. The investment behaviour depends on anticipation of profitability by sector in the future, depicted by the endogenously determined rate of return on capital, the anticipation of future demand for the output of the sector, the cost of building new capital, and the rental cost of capital. The latter depends on interest rates derived in the model from the equilibrium of capital markets in an aim to represent how financial conditions influence profitability and thus investment. The formulations allow for capital mobility, but the allocation of capital to sectors and countries depends on assumptions that may differentiate interest rates and policy options obstructing or facilitating capital mobility. The model uses an investment matrix with fixed technical coefficients to transform investment by sector into demand for goods and services that implement the investment.
The model treats public investment in infrastructure, also in the transport sector, as exogenous. The implementation of investment in infrastructure transforms into demand for goods and services via an investment matrix, as for all other investment. Also, the accumulation of transport infrastructure investment, thus the increase in physical capital, induces positive productivity growth effects. The capital stock driving productivity changes is a mechanism that has rarely been seen in the literature of economic modelling. The aim is to capture the productivity effects of transport infrastructure, for example, the reduction in travelling times and the ensuing increase in productivity of labour and capital. The magnitude of the effects on productivity is specified exogenously, based on information collected from a vast literature of econometric estimation of productivity trends by sector in developed countries. Public investment also requires financing, which adds to the demand for funding in the economy and may increase the scarcity of capital, thus inducing an increase in interest rates. The model applies options to delimit the broadness of capital markets, as scarcity can be different when capital markets clear in the entire EU or on a country-by-country basis. The current version of the model does not allow for financial disequilibrium within the static equilibrium; thus, savings and investment must be balanced every year. This is a restrictive assumption as it cannot capture the mechanism of raising public investment in infrastructure expecting long-term repayment from accumulation of savings enabled by derived productivity growth. The static balancing implies that the rise of interest rates may obstruct public investment in case the funding affects capital scarcity. We plan to revise this assumption in future versions of the model.
The model formulates a representative household by country or region for the determination of consumption in durable and non-durable goods, together with savings and labour supply, as a result of intertemporal utility maximization subject to revenue constraint, which also depends on labour supply indirectly. As the simulation over time is sequential, the model transforms the intertemporal utility into a steady-state utility formulation and then applies optimization to derive consumption. A utility function aggregates the volumes of consumption of goods and services (
Figure 2) using parameters that represent preferences. The formulation makes sure that the mix of goods and services meet requirements for minimum subsistence consumption levels and derives utility from the additional amounts of consumed goods. The revenues depend on wages, dividends, taxes, social benefits, and social security contributions, as well as transfers to or from abroad. The utility function is a nested linear expenditure system (LES) model. The first level of the nest combines savings and aggregated consumption, whereas at the second level, aggregated consumption splits into nested choices of product or services types. The formulation of private consumption distinguishes between durable and non-durable goods and considers durable goods by type as well as their consumption of non-durable goods.
The general form of the LES formulation for a household in country
involves minimum subsistence amounts
and marginal utility parameters
for every consumption by product category (
) and savings. Savings can be considered as utility-enabling with zero subsistence amounts. The constraint represents income, which partly depends on labour supply. The utility function has the following general form (often called Stone-Geary utility function):
Consumption is valued at market prices, assuming that households are price-takers, and savings are valued using a subjective discount rate influenced by market interest rates.
2.2.3. The Regional Economic Equilibrium Model
The purpose of the sub-country layer (SCL) model is to downscale the projections of the country-wide (CL) model to the regions. The SCL model includes the following main formulations for downscaling: (i) the SCL model determines the regional location of population/labour and production capacities of sectors depending on the relative attractiveness of the regions; (ii) given the regional endowment of primary production factors, the SCL model calculates production, consumption, income, and bilateral trade at a regional level; (iii) given regional production, the SCL model determines the accumulation of factors (state variables) that affect the valuation of amenities and dis-amenities of regions, which further influence the attractiveness of the regions in a subsequent time period; (iv) given sectoral activity and labour of regions, the model calculates inter-regional flows of goods and services and commuting.
Households derive utility from the consumption of goods and services, as well as from local amenities. The indirect utility (i.e., the maximum utility under income constraint) depends on prices of goods (
), wages
, price of capital (
), and the amenities and dis-amenities measured by a vector
, which is an aggregation of the individual amenities and dis-amenities forming an attractiveness index, as follows (where
is a functional form formulated as a LES utility function). Increasing amenities implies higher utility, while the opposite holds for dis-amenities.
Cost of production by sector of activity depends not only on the prices of primary factors of production and the prices of other intermediate inputs, but also on local amenities and dis-amenities, which influence the productivity of factors. Also, the amenities influence the attractiveness of a region for the location of a new productive investment. Thus, the amenities influence the location of production activities dynamically. Similar mechanisms can be found in References [
33,
40,
41,
43]. The indirect cost function (i.e., the minimum cost for given volume of output) depends on factor prices, such as wages (
), cost of capital (
), and the prices of intermediate goods and services (
), as well as on amenities, denoted by a vector
, that is an aggregation of the individual amenities and dis-amenities forming an attractiveness index.
The aggregation of amenities (
) forming the attractiveness index employs a CES (constant elasticity of substitution) non-linear function
, having slopes that are increasing for amenities and decreasing for dis-amenities. The function is linearized in the calibration of the model, as a linear combination of amenity values weighted by marginal utilities attributed to the amenities.
Similarly, the amenities (
) that influence production costs form the attractiveness index for industries
using a constant elasticity of substitution (CES) aggregation function
, having slopes that are increasing for amenities and decreasing for dis-amenities. The function is linearized in the model calibration as a linear combination of amenity values weighted by the marginal productivities attributed to the amenities.
In the above formulas, the use of CES aggregation functions implies a non-perfect substitution among amenities.
The model valuates the regional amenities using stock-flow equations that involve state-variables and accumulation of resources (
). The changes in stock variables,
, affect the value of amenities non-linearly to capture economies or dis-economies of scale that further influence location choices. The stock variables depend on these location choices dynamically over time. The relationship between stock variables and amenity values follows a Fréchet function to capture saturation and acceleration effects, depending on parameter values, as in Reference [
36].
Among, the stock and resource variables, it is worth mentioning that transport possibilities among regions is among the drivers of location choices, depending on transport infrastructure, congestion, and transport costs, as in References [
55,
56]. The model quantifies a regional accessibility index, formulated as a Cobb-Douglas aggregation function, to measure accessibility
of a region
, depending on explanatory factors denoted by the vector
.
The location choices are discrete when decided by individual households and firms. At the aggregate level, the model calculates the frequencies of discrete choices by households and thus captures heterogeneity of preferences and technologies, including for the location decisions of households and firms. The model formulates the frequency of location choices and not the discrete choices to represent idiosyncratic preferences, as in References [
36,
37]. The frequencies of location choices follow a Gumbel probability distribution function, which depends on the valuation of utility-enabling attributes, such as amenities and dis-amenities that form regional attractiveness. The combination of idiosyncratic preferences and returns to scale linking amenities to state variables can lead to regional specialization and non-convergence of the regions, as in References [
57,
58,
59]. The combination of the discrete choice model for locations and the dynamic stock-flow relationships constitutes the agglomeration and dispersion mechanisms formulated in the regional economic model.
The abovementioned formulation of amenities and their valuation influenced by the dynamics of state variables concerns the choice of regional location of investment and labour, but not the choice of investment in infrastructure. The latter has social and environmental effects that also affect the state variables. However, the model captures only the positive effects of investment in infrastructure, which derive from improved accessibility of a region. Other works in the literature, such as Reference [
60], evaluate social, environmental, and other factors in the choice of investment infrastructure.
The model employs a generalized extreme value (GEV) distribution function to derive the frequencies of location choices. In other words, the indirect utility and the cost functions follow a GEV distribution, as in Reference [
37]. Thus, the frequency of choosing region
as a location by households and firms is as follows, where
represent scale parameters and
are elasticities:
The model uses a dynamic partial adjustment mechanism, which applies the desired location, seen in Equation (10), gradually over time. In this manner, the model avoids abrupt changes of location.
Among the attributes influencing the choice of location, we mention the following:
The human capital availability which is approximated by skilled labour;
Physical capital availability;
Capital profitability which is approximated by the ratio of the investment cost to the rental price of capital;
Natural resources (the model maps the resource-based activities to the location of the natural resources);
Vertical integration which simulates the incentive of certain industries to form clusters;
The market size, which is often mentioned as the home market effect in the New Trade theory.
The choice of location of households is also assumed to be influenced by the environmental quality, approximated by the CO2 emissions per NUTS-3 zone, disposable income, and population density. We use the CO2 emissions, calculated by the model, as a proxy of air pollution in a region, as CO2 is due to the combustion of fossil fuels, which is the source of air pollutants, such as sulphur, nitrogen oxides, and particulates. The transport-related indicators, which influence the location choice of both businesses and individuals, are used to evaluate the accessibility index by region.
The formulation of regional attractiveness captures the following important aspects of regionalization: (i) positive externalities stemming from agglomeration due to the coexistence of certain activities (e.g., activity specialization and industrial integration, cities as enablers of social networks, regions able to attract highly skilled labour, etc.); (ii) negative externalities in relation to resources and cross-effects among state variables (e.g., conflicts between tourism and heavy industry); and (iii) limitations deriving from geography, transport or infrastructure (e.g., regions adjacent to more than one country having higher opportunities to attract activity compared to peripheral regions).
Once located, the households supply labour both to local and adjacent regional markets, depending on relative wages and the commuting time. The commuting part of the labour force among any pair of adjacent regions
is determined according to:
Equation (11) also draws on discrete choice theory. The attributes influencing the choices include the regional wages and the commuting costs that further depend on transportation and time costs . The formula uses as weights reflecting the habits of commuters and the elasticity representing the easiness of commuting and the influence of other factors. The model adds transport costs and cost of time for commuting to workplaces as factors that influence residence location in relation to the workplace. The spatial resolution of this representation is, however, too aggregated to represent commuting adequately. However, data limitations and computational complexity do not allow going deeper than the NUTS-3 regional segmentation.
The model considers that the regional origin defines distinct varieties of goods and services, and thus applies imperfect substitution among goods of different regional origin. Both the households and the production sectors determine demand for the regional varieties as part of the nested choices. The choice of varieties depends on relative costs that reflect regional economic features and transport costs, depending also on accessibility. When transport infrastructure develops and improves the accessibility of a region, transport costs reduce, which implies that the menu of varieties available for selection is enlarged, inducing efficiency gains in the aggregation of varieties. The mechanism is similar to the love-of-variety formulation used in economic trade models, which were firstly specified by Dixit and Stiglitz, see Reference [
61], as illustrated by the equation below.
The frequency of an investment by sector
to be implemented in a region
follows a GEV distribution function depending on the attractiveness index
, as seen by the sector
that aggregates the values of the various features of the region (costs, proximity to resources, access to cheap labour or adequate skills and transport costs) entering the cost function
.
The locational choices for investment determine the capital stock dynamically, which acts as a constraint on regional production in the next period. The funding of investment expenditures by region has to match investment expenditures within the same sector, as calculated at a country level. Public investment and consumption are exogenously allocated to regions. Likewise, to the national part of the model, investment by sector implies demand for equipment goods and construction, based on fixed technical coefficients.
To calculate interregional trade flows, the regional economic model employs a formulation based on distances and transportation costs, as well as on production costs and behavioural parameters, the latter representing preferences. The formulation aims to capture the influence of several factors, such as infrastructure development, transport technologies and fuel costs, and the possible improvement in the accessibility of regions. The calculation of trade flows is performed step-wise:
At the first stage, imports are differentiated by country of origin (i.e., intra-national imports versus international imports);
At the second stage, the consumption of imported goods is further disaggregated into consumption by region of origin.
The equation below illustrates the calculation of trade flows,
, of product
between regions
and
.
The equation relates the share of trade flow from to over product demand in the market to the regional share of product in production in the country of origin and the cost factors that include transport costs, and relative regional prices. The elasticity represents trade impediments and imperfect substitution of product varieties depending on the origin. The transport costs derive as the weighted average cost of transport modes including cost of time.
The regional economic equilibrium model is a very large mathematical problem, as it covers NUTS-3 (approximately 3000 regions) and 37 goods and services per country. For a non-linear mixed complementarity problem of this size, computational limitations are considerable (mainly memory limits). To overcome the computational difficulties, we apply an iterative algorithm consisting of running the regional model per region in a parallel computing framework, assuming interregional flows as given in intermediate steps and applying a collection of the results for the regions to run the gravity model for all regions simultaneously and, thus, derive the interregional flows. The adjusted flows update the fixed interregional exchanges in the isolated regional models which run again in parallel. The iterations continue over time using the stock-flow relations, which concern capital stock, labour, and the state and stock variables that affect the valuation of amenities, hence, the attractiveness of regions that are adjusting dynamically.