Brexit: an economy-wide impact assessment on trade, immigration, and foreign direct investment

Abstract

We assess the impacts of Brexit on the UK economy along three dimensions: EU market access based on consideration of tariffs and non-tariff barriers, reduced numbers of EU citizens working in the UK, and reductions in FDI. Using a Computable General Equilibrium model with an integrated Melitz (Econometrica 71(6): 1695–1725, 2003) framework, we consider capital accumulation and population size feedback on tax income and demand for public services. In the worst-case scenario where all dimensions are considered simultaneously, welfare losses of approximately 1100 USD per UK citizen are predicted, exceeding the results of many other studies, which we attribute to our relatively comprehensive scenario design and application of the Melitz framework for manufacturing sectors. We find that a reduced labor force as a consequence of expected reduced immigration to the UK has the greatest welfare impact. Therefore, policy measures that increase labor participation and/or allow for continued immigration of workers could mitigate negative impacts of Brexit.

Appendices

Appendix 1: Technical appendix

1.1 Implementation of Melitz model into the standard GTAP model

We incorporate a module based on Melitz (2003) into the flexible and modular CGE model CGEBox (Britz 2017). The actual implementation of the Melitz model into CGEBox draws largely on the empirical method by Balistreri et al. (2011), Balistreri and Rutherford (2013) and Akgul et al. (2016) to introduce the Melitz (2003) model into an applied equilibrium model.

The Melitz framework focuses on intra-industry differentiation where each firm produces a single unique variety. However, data at the firm level are limited and applied equilibrium models work at aggregate levels. Fortunately, Melitz offers a numerical framework build around (marginal changes in) the average firm operating within a trade linkage. That average firm’s productivity comprises all necessary information on the distribution of productivity levels of firms active in that link. That vastly eases the model’s implementation by effectively eliminating any data needs at individual firm level as detailed below. Against a background of that definition of an average firm within each trade linkage, we now focus on the formulation of an empirically computable version of Melitz model and its linkages with the GTAP model.

Assume that a representative agent ‘a’ (private households, government, investors, intermediate inputs by the different firms) in region ‘r’ obtains utility ${\text{U}}_{\text{air}}$ from consumption of the range of differentiated varieties of product ‘i’ and considering the CES utility function as proposed by Dixit and Stiglitz (1977), the aggregate demand by each agent $a$ for commodity ${\text{i}}$ in region $r$ (${\text{Q}}_{\text{air}}$) which is equivalent to utility (${\text{Q}}_{\text{air}} \equiv {\text{U}}_{\text{iar}}$) can be represented as:

$$Q_{air} = \left( { \mathop \sum \limits_{s} \mathop \int \limits_{{\omega \in\Omega _{isr} }}^{{}} \lambda_{aisr}^{{\frac{1}{{\sigma_{ir} }}}} Q_{aisr} \left( \omega \right)^{{\frac{{\sigma_{ir} - 1}}{{\sigma_{ir} }}}} d\omega } \right)^{{\frac{{\sigma_{ir} }}{{\sigma_{ir} - 1}}}}$$

(1)

where $\Omega _{isr}$ represents the set of products $i$ sourced from region $s$ to $r$ and $\omega \in \Omega _{isr}$ index the varieties in the set $\Omega _{isr}$ In this context, $Q_{aisr} \left( \omega \right)$ represents the demand quantity of commodity $i$ for variety $\omega$ in region $r$ by agent a which is sourced from region $s$, $\sigma_{i}$ represents the constant elasticity of substitution for each commodity, and $\lambda_{aisr}$ are preference weights (share parameters)⁷ that reflect differences between origins not linked to diversity in varieties. Note that substitution elasticities might be differentiated by destination region $r$, but are uniform across agents in each region in our implementation.

The resulting CES unit expenditure function which is the dual price index on Dixit–Stiglitz composite demand in region r ($P_{air} )$ is given by:

$$P_{air} = \left( { \mathop \sum \limits_{s} \mathop \int \limits_{{\omega \in\Omega _{isr} }}^{{}} \lambda_{aisr} PA_{aisr} \left( \omega \right)^{{1 - \sigma_{ir} }} d\omega } \right)^{{\frac{1}{{1 - \sigma_{ir} }}}}$$

(2)

where $PA_{aisr} \left( \omega \right)$ is agent’s $a$ (purchase) price of product $i$ for variety $\omega$ in region $r$ sourced from region $s$. Using the aggregate price index in Melitz (2003) based on the definition of the average firm and considering that varieties do not differ in their marginal utility for the first unit, one can define the price index as equivalent to the dual price defined in Eq. (2)

$$P_{air} = \left( {\mathop \sum \limits_{s} \lambda_{aisr} N_{isr} \widetilde{PA}_{aisr}^{{1 - \sigma_{ir} }} } \right)^{{\frac{1}{{1 - \sigma_{ir} }}}}$$

(3)

where $\widetilde{PA}_{aisr}$ denotes the agent price inclusive of export, import and consumption taxes for the average firm, and $N_{isr}$ refers to the number of firms operating on the trade link s-r. Equation 3 generates the top level Armington price for each agent, replacing the Armington price aggregator in GTAP from the agents’ domestic and import prices. Consistent with Melitz (2003), there is a one-to-one mapping among firms and varieties such that the number of firms is equal to the number of varieties on each trade linkage. In comparison to Eq. (2), which is based on the individual varieties, Eq. (3) summarizes the compositional change (i.e., change in the number of varieties), which goes along with an update of the average price. Again we assume the same substitution elasticities across agents.

The total ($Q_{aisr} )$ and average per firm ($\tilde{Q}_{aisr} )$ demand for the average variety by an agent to be shipped from s to r ($\tilde{Q}_{aisr} )$ can be obtained by applying Shephard’s Lema on the expenditure function:

$$Q_{aisr} = \tilde{Q}_{aisr} N_{isr} = \lambda_{aisr} N_{isr} Q_{air} \left( {\frac{{P_{air} }}{{ \widetilde{PA}_{aisr} }}} \right)^{{\sigma_{ir} }}$$

(4)

This equation replaces the equations determining the agent specific Armington demands for the domestic and imported good as well as the equations for bi-lateral import demand which are not agent specific in the GTAP standard model.

This reveals the main differences to a standard Armington composite: the share parameters vary with the number of operating firms (i.e., the number of varieties comprised in the bilateral trade bundles). As the agent demand for the average firm’s output in region s in each industry $i$ in region $r$ ($Q_{aisr} )$ depends on the aggregate regional demand for that industry $Q_{air}$, we need to determine this in equilibrium for each agent. In other word, we need to determine the demand for use of $i$ as an intermediate input in each sector separately, and as final demand for household consumption, government consumption, investment, and for international transport margins. In the standard GTAP model, each agent has a specific preference function which determines the demand for each Armington commodity; the government and saving sector based CES preferences, while households use a CDE indirect demand function. The Armington demand for each agent and commodity is then decomposed into a domestic and import component in the first Armington nest. The second nest decomposes import demand by each region by origin, independent of the agent.

The implementation of the Melitz model thus simplifies the demand structure present in the standard GTAP model by aggregating the two Armington nests into a single one, however, note that the GTAP database does not yet differentiate in the SAM bi-lateral flows by agent. Therefore we use the same shares by origin to separate import demand for the different agents.

Assume that a small profit maximizing firm facing the constant elasticity of demand according to Eq. (4) for its variety and based on the assumption of the large group monopolistic competition, a firm will not consider its impact on the average price index and therefore follow the usual markup rule to translate their marginal cost of production ($c_{is} )$ to the optimal price.

Firms in Melitz (2003) face different types of cost: sunk fixed cost of entry $f^{ie}$, fixed cost of operating on a trade linkage $f_{isr}$ and marginal cost $c_{is}$. Let $\varphi_{isr}$ indicate the firm’s specific productivity, which measures the amount of “variable composite unit” needed per unit of output $Q_{isr}$. Accordingly, the marginal cost per unit is the amount of “composite input” required per unit $( {\frac{1}{{\varphi_{isr} }}})$ times the unit cost of the “variable composite input” ($c_{is} )$ in industry $i$ of region $s$. Therefore, a firm wishing to supply $Q_{isr}$ units from region $s$ to $r$ employs ($f_{isr}$ + $\frac{{Q_{isr} }}{{\varphi_{isr} }}$) units of “variable composite input.” The structure of fixed costs and variable composite input demand is discussed in detail below. Let, $\tau_{isr}$ denote the iceberg cost of trade, which represent domestic production costs and not the international trade margins present in GTAP. Focusing on the average firm with a productivity $\tilde{\varphi }_{isr}$ operating within a trade linkage and solving the firm’s profit maximization problem, the price charged by the average firm in region s to supply region r $\widetilde{PF}_{isr}$ (inclusive of domestic transport margin) is:

$$\widetilde{PF}_{isr} = \frac{{\sigma_{ir} }}{{\sigma_{ir} - 1}}\frac{{\tau_{isr} c_{is} }}{{\tilde{\varphi }_{isr} }}$$

(5)

where $\frac{{\sigma_{ir} }}{{\sigma_{ir} - 1}}$ represents the constant markup ratio in industry $i$, which reflects market power due to product differentiation into varieties. This newly introduced equation translates the variable per unit cost function for each sector and region into offer prices by the firms on each trade link including domestic sales. In the standard GTAP model, offer prices are not differentiated by destination and are equal to per unit cost corrected for output taxes.

The average price in Eq. (5) therefore depends on the price of variable composite input $c_{is}$, which is a function of the price of intermediates and primary factors. Given the assumption of constant return to scale and the way technology is presented in the standard GTAP model, the unit cost function for sector $i$ in region $s$$c_{is}$ in GTAP is given by the Leontief composite of the value added bundle (CES aggregate of factors of production) and the aggregate of intermediate demand (Leontief aggregate of intermediate demands). In the CGEBox, m_px⁸ is a macro defined as producer price, which constitute per unit costs corrected for production taxes. To be consistent with our Melitz formulation, the unit cost inclusive of production tax is directly introduced in the markup Eq. (5). It should be emphasized that the presence of fixed cost in the Melitz model is the source of increasing returns to scale in a monopolistically competitive industry: if firms expand production, the fixed cost can be distributed over a greater quantity of outputs such that per unit cost decreases.

While observed data on quantities traded and related prices allow identifying the necessary attributes of an average firm, additional information is needed to gain information about the marginal firm (i.e., the firm that earns zero profit). Obviously, the distance in productivity between the average and marginal firm reflects properties of the underlying distribution. We rely here on a Pareto Productivity distribution, which has analytical tractability.

Let $M_{s}$ indicate the number of firms choosing to incur the fixed entry cost (i.e., total industry size), each individual firm receives its productivity $\varphi$ draws from a Pareto distribution with Probability Density Function (PDF):

$$g\left( \varphi \right) = \frac{a}{\varphi }\left( {\frac{b}{\varphi }} \right)^{a}$$

(6)

and Cumulative Distribution Function (CDF):

$$G\left( \varphi \right) = 1 - \left( {\frac{b}{\varphi }} \right)^{a}$$

(7)

where $b$ is the minimum productivity and $a$ is a shape parameter. Lower values of the shape parameter imply higher productivity dispersion among firms. As discussed in Melitz (2003), $a > \sigma_{ir} - 1$ should be applied in order to ensure a finite average productivity level in the industry.

On each bilateral trade linkage, the given fixed bilateral trade cost, variable costs and demand define jointly a certain cut off productivity level ($\varphi_{sr}^{*}$) at which firms will receive zero profit. A firm with the productivity equal to that threshold level ($\varphi_{sr} = \varphi_{sr}^{*}$) will therefore face zero profits and act as the marginal firm from region $s$ supplying $r$. Those firm whose productivity is above the threshold level ($\varphi_{sr} > \varphi_{sr}^{*}$) will receive a positive profit and will operate on the $s - r$ link and those firm with productivity that is below the threshold level ($\varphi_{sr} > \varphi_{sr}^{*}$) will not operate on the $s - r$ linkage. Focusing on the fixed operating cost $f_{isr}$ in composite input units, the marginal firm on s-r linkage receives zero profit at:

$$c_{is} f_{isr} = \frac{{r\left( {\varphi_{isr}^{*} } \right)}}{{\sigma_{ir} }}$$

(8)

where $r\left( {\varphi_{isr}^{*} } \right) = p\left( {\varphi_{isr}^{*} } \right)q\left( {\varphi_{isr}^{*} } \right)$ denotes the revenue of marginal firm at the productivity equal to the cut off level ($\varphi_{isr} = \varphi_{isr}^{*} ).$

The zero cut off productivity level in each bilateral market $\varphi_{isr}^{*}$ can be obtained by solving Eq. (8). However, it is numerically easier to define this condition in terms of the average rather than the marginal firm. To do this, we define the productivity and revenue of the average firm relative to that of the marginal firm. Following Melitz (2003), average productivity is defined as the CES aggregation of productivities of all firms operating on a given trade link:

$$\widetilde{\varphi }_{isr} = \left[ {\frac{1}{{1 - G\left( {\varphi_{isr}^{*} } \right)}}\mathop \int \limits_{{\varphi_{isr}^{*} }}^{\infty } \varphi_{isr}^{{\sigma_{ir} - 1}} g\left( \varphi \right) d\varphi } \right]^{{\frac{1}{{1 - \sigma_{ir} }}}}$$

(9)

If these productivities are Pareto distributed, we can write⁹:

$$\widetilde{\varphi }_{isr} = \left[ {\frac{a}{{\left( {a + 1 - \sigma_{ir} } \right)}}} \right]^{{\frac{1}{{1 - \sigma_{ir} }}}} *\varphi_{isr}^{*}$$

(10)

Equation (10) provides the relationship between the productivities of the average and marginal firm

Using optimal firm pricing according to Eq. (5) and given the input technology, the ratio of revenues of the firms with marginal productivity $r_{isr} \left( {\varphi^{*} } \right)$ in relation to the revenue of the firm with the average productivity $r_{isr} \left( {\tilde{\varphi }} \right)$ is defined as:

$$\frac{{r_{isr} \left( {\varphi^{*} } \right) }}{{r_{isr} \left( {\tilde{\varphi }} \right) }} = \left( {\frac{{\varphi^{*} }}{{\tilde{\varphi }}}} \right)^{{\sigma_{ir} }}$$

(11)

Solving Eq. (10) for $\frac{{\varphi^{*} }}{{\tilde{\varphi }}}$, substituting it into (11), and then solving the resulting equation for $r_{isr} \left( {\varphi^{*} } \right)$ and replacing its value in Eq. (8), defines a relation between the bilateral fixed cost at current composite input price [the LHS of (12) below], the average firms revenue ($\widetilde{PF}_{isr} \tilde{Q}_{isr}$), the shape parameter of the Pareto distribution of the productivities and the substitution elasticity of demand:

$$c_{is} f_{isr} = \frac{{\left( {a + 1 - \sigma_{ir} } \right)}}{{a\sigma_{ir} }} \widetilde{PF}_{isr} \tilde{Q}_{isr}$$

(12)

Note that average firm’s sale in region s in each industry $i$ to region r ($\tilde{Q}_{isr} )$ at the equilibrium is composed of the demand for use of $i$ by different agents.¹⁰

The optimal pricing in the markup Eq. (5) requires information on the average productivity on each bilateral trade link. In Melitz (2003), the probability that a firm will operate is $1 - G\left( {\varphi_{isr}^{*} } \right)$, which is equal to the fraction of operating firms over total number of firms choosing to draw their productivity $\left( {\frac{{N_{is} }}{{M_{is} }}} \right)$. Using the Pareto cumulative distribution function from Eq. (7) and inverting it we have:

$$\varphi_{isr}^{*} = \frac{b}{{\left( {\frac{{N_{isr} }}{{M_{is} }}} \right)^{{\frac{1}{a}}} }}$$

(13)

Substituting Eqs. (13) into (10) results in Eq. (14) introduced into the model:

$$\tilde{\varphi }_{isr} = b\left[ {\frac{a}{{\left( {a + 1 - \sigma_{ir} } \right)}}} \right]^{{\frac{1}{{1 - \sigma_{ir} }}}} *\left( {\frac{{M_{is} }}{{N_{isr} }}} \right)^{{ - \frac{1}{a}}}$$

(14)

Next, the number of firms selecting to enter the market $M_{is}$ is determined. Based on the free entry condition, the last firm that enters has expected profits over its life time, which just offset the sunk cost of entry. Industry entry of a firm requires a one-time payment of $f^{ie}$. A firm that enters a market faces a probability of $\delta$ to suffer a shock, which forces its exit in each future period. Therefore $\delta M_{is }$ firms are lost in each period. Based on Melitz (2003), in a stationary equilibrium, the number of aggregate variables must remain constant over time, including industry size. This requires that the number of new entrants in every period is equal to the number of firms lost $\delta M_{is }$. Therefore, total entry cost is equal to $c_{is} \delta M_{is } f^{ie}$. Each firm faces the same expected share on that cost (i.e., $c_{ir} \delta f^{ie}$ if risk neutral behavior and no time discounting is assumed). The firm’s expected share of entry costs must be equal to the flow of expected profit on the condition that firm will operate:

$$\tilde{\pi }_{isr} = \frac{{\widetilde{PF}_{isr} \tilde{Q}_{isr} }}{{\sigma_{ir} }} - c_{is}\,f_{isr}$$

(15)

The probability that a firm will operate on the s-r trade linkage is given by the ratio of $\frac{{M_{isr} }}{{N_{is} }}$.¹¹ Thus, the free entry condition ensures that the expected industry profits (i.e., the profits summed up over all potential bilateral trade links) is equal to the annualized flow of the fixed costs of entry

$$c_{is} \delta f^{ie} M_{s} = \mathop \sum \limits_{r} N_{isr} \widetilde{PF}_{isr} \tilde{Q}_{isr} \frac{{\sigma_{ir} - 1}}{{a\sigma_{ir} }}$$

(16)

where zero profit condition in Eq. (12) is used to replace the fixed operating cost $c_{is} f_{isr} .$ With the number of entered firm established in Eq. (16), we now turn to total composite input demand of the industry Y,¹² which consists of three components: sunk entry costs of all entrants ($\delta M_{is } f^{ie}$), operating fixed cost ($\mathop \sum \limits_{s} N_{isr } f_{isr }$) on each trade linkage, and variable costs ($\mathop \sum \limits_{r} N_{isr } \frac{{\tau_{isr} \tilde{Q}_{isr} }}{{\tilde{\varphi }_{isr} }}$). Therefore, composite input demand is defined as:

$$Y_{is } = \delta M_{is } f^{ie} + \mathop \sum \limits_{r} N_{isr } \left( {f_{isr } + \frac{{\tau_{isr} \tilde{Q}_{isr} }}{{\tilde{\varphi }_{isr} }}} \right)$$

(17)

This equation replaces the output balance equation in the standard GTAP model which ensures that exports and domestic sales for each sector are equal to input composite demand. Table 7 summarizes the full set of Melitz equations, and shows the variables through which Melitz model is linked into the structure of the GTAP standard model.

Table 7 Equilibrium conditions

1.2 Production structure for sectors in the Melitz module

This section briefly introduces the production technology for the Melitz commodities which is based on Akgul et al. (2016). The main differences to the GTAP standard model are production function nestings specific to variable and fixed costs. The production nesting for variable and fixed cost is shown in Fig. 1 where the total cost is the sum of variable and fixed costs, the latter is added up from fixed cost per firm which entered the industry and fixed cost on each trade link per firm operating on that link.

Fig. 1

Production structure in Melitz sectors

The variable cost nest uses both a value-added and an intermediate composite based on constant returns to scale (CRS) technology, and in the default case, the fixed cost nest only uses value-added. However, if the overall total cost share of value-added in a sector is small, also the fixed nests comprise a share of intermediate composite. This alternative is identified with the intermediate composite in brackets. The value added and intermediate bundles are CES composite of primary factors of production and intermediate inputs, respectively. The total value added (not shown here) is the sum of value added used in both variable and fixed cost nestings. Similarly, the total use of intermediate commodities and primary factors (not shown here) are the sum of their use in fixed and variable cost nestings

In the variable cost nests for the Melitz sectors, primary factors or intermediate inputs used are proportional to output, i.e. average output per firm on each trade link times firm operating on the link, summed up over all trade links, corrected for average productivity. On the other hand, the fixed cost nests reflect solely the number of firms which entered the industry and the trade links, and not average output per firm.

Each intermediate commodity used by a Melitz sector could be either a Melitz or Armington commodity, and the intermediate input bundle is a CES composite of these commodities, in the default configuration based on fixed input coefficients. For the Armington goods, we retain the standard GTAP model assumption of domestic and import distinction where imports are sourced at the border (not shown). There is no separate domestic and imports distinction for Melitz commodities, but only one aggregator with love-variety effects which comprises domestic sales and all bi-lateral imports.

Appendix 2: Supplemental tables

Tables 8 and 9.

Table 8 Sectoral correspondence of GTAP sectors to new sectors

Table 9 Welfare and GDP impact of alternative scenarios