Workers’ Earnings Losses Due to the Low-Carbon Transition. Theory and Application in a CGE Model

Consider a worker who must decide between allocating his or her labour across K different sectors. That worker’s productivity could differ between sectors. More specifically, following Hsieh et al. (2019), we assume that the worker possesses some amount of efficiency units ⁽²⁾ in one sector and a potentially different amount of efficiency units in another sector. In addition, we allow workers to be heterogeneous; that is, within one sector two workers might have different productivities. We assume that wage per efficiency unit in sector k, w_k, is the same for all workers in that sector. Let w = (w₁,...,w_K) denote the vector of wages per amount of efficiency units for sectors 1,...,K. The compensation a worker receives from working in sector k is w_k times the amount of efficiency unit the worker possesses in sector k. The worker simply chooses the sector which offers the highest compensation.

Throughout the paper we assume that changes in wages do not alter the total supply of labour and that the supply of labour is always equal to the demand by firms; that is, there is no unemployment. Furthermore, we do not distinguish between cohorts of workers – in each period all workers can make a decision about where to work and this decision has no consequences for their opportunities or payoffs in the future. These assumptions are made to simplify the presentation of the main argument. We briefly discuss the potential bias in our results caused by these assumptions in Section 4.

Let n_j(w) be the share of workers who wish to work in sector j and let N_j(w) be the sum of their efficiency units. Before specifying the relationship between N and n, consider the following experiment: if we start with w_j = 0 and then gradually increase it, n_k increases. We would expect that the first workers who join sector j are those with the highest amount of efficiency units in that sector. As we continue to increase w_j, the sector is approached by other workers with a lower amount of efficiency units. The last workers to join are those for whom the sector is a particularly unfavourable match. This means that as n_j increases, the average productivity of workers decreases. To reflect this logic, we assume that the average amount of efficiency units in a sector is described by equation (1): (1) Njnj=ajnjγμ {{{N_j}} \over {{n_j}}} = {a_j}n_j^\gamma \mu where a_j and γ < 0 are parameters (which we will soon determine using the parameters of the supply curve) and μ is the parameter we will use to reflect productivity growth due to exogenous technological progress ⁽³⁾. The parameter a_j represents the average productivity in the sector in a hypothetical situation if all workers are employed in sector j (i.e. if n_j =1) and μ=1. The parameter γ can be interpreted as the elasticity of average labour productivity in the sector with respect to changes in the share of workers employed in the sector.

Next, we specify the shape of the supply curve, n_j (w). Since n_j represents the share of workers, we must choose a function that satisfies Σ_jn_j(w)=1. Ideally, we would also choose a function that permits constant elasticity of the labour supply, dlogn_j/dlogw_j. Satisfying both conditions at the same time is not possible. However, we can choose a functional form that uses an elasticity that is nearly constant, providing that the number of sectors is large and no sector has very large n_j (conditions which are usually satisfied in CGE models): (2) nj=(sjwj)αΣk(skwk)α {n_j} = {{{{({s_j}{w_j})}^\alpha }} \over {{\Sigma _k}{{({s_k}{w_k})}^\alpha }}} where s_j and α are the parameters determining the shape of the supply curve. The parameter α governs the response of labour supply to a change in wages. The parameters s_j is a share parameter: (s_j)^α represents the share of workers willing to work in sector j in the base year, when all wages per efficiency unit, w_j, are normalised to unity.

If supply is given by equation 2, elasticity of supply is given by (3) ∂nj∂wjwjnj=α(sj)α(wj)α−1Σk(skwk)αwjnj−α(sjwj)α(sj)α(wj)α−1(Σk(skwk)α)2wjnj=α(sjwj)αΣk(skwk)α1nj−α(sjwj)αΣk(skwk)α21nj=α1−nj, \matrix{ {{{\partial {n_j}} \over {\partial {w_j}}}{{{w_j}} \over {{n_j}}}} \hfill & { = \alpha {{{{({s_j})}^\alpha }{{({w_j})}^{\alpha - 1}}} \over {{\Sigma _k}{{({s_k}{w_k})}^\alpha }}}{{{w_j}} \over {{n_j}}} - \alpha {{{{({s_j}{w_j})}^\alpha }{{({s_j})}^\alpha }{{({w_j})}^{\alpha - 1}}} \over {{{({\Sigma _k}{{({s_k}{w_k})}^\alpha })}^2}}}{{{w_j}} \over {{n_j}}}} \hfill \cr {} \hfill & { = \alpha {{{{({s_j}{w_j})}^\alpha }} \over {{\Sigma _k}{{({s_k}{w_k})}^\alpha }}}{1 \over {{n_j}}} - \alpha {{\left( {{{{{({s_j}{w_j})}^\alpha }} \over {{\Sigma _k}{{({s_k}{w_k})}^\alpha }}}} \right)}^2}{1 \over {{n_j}}} = \alpha \left( {1 - {n_j}} \right),} \hfill \cr } which approaches α as n_j approaches zero. In fact identical functional form was derived from a micro-founded model with heterogeneous workers by Witajewski-Baltvilks and Boratyński (2021) for the case of two-sector economy.

Now consider two sectors, j and k. After a decline in w_j, n_j drops and n_k increases. This indicates that before the change in sector j, there was a marginal worker who was indifferent about staying in sector j or moving to sector k. For that worker, the payoff in the two sectors must be equal. To compute that payoff, we must first compute the amount of efficiency units of the marginal worker in both sectors. According to (1), that worker leaving sector j causes the total amount of efficiency units in that sector to drop by (4a) dNjdnj=dajnjγ+1μdnj=ajγ+1njγμ {{d{N_j}} \over {d{n_j}}} = {{d\left( {{a_j}n_j^{\gamma + 1}\mu } \right)} \over {d{n_j}}} = {a_j}\left( {\gamma + 1} \right)n_j^\gamma \mu When that worker enters sector k, the total amount of efficiency units in k increases by (4b) dNkdnk=akγ+1nkγμ {{d{N_k}} \over {d{n_k}}} = {a_k}\left( {\gamma + 1} \right)n_k^\gamma \mu If workers maximise their compensation, for workers at the margin compensation must be equal in both sectors: (4c) dNjdnjwj=dNkdnkwk {{d{N_j}} \over {d{n_j}}}{w_j} = {{d{N_k}} \over {d{n_k}}}{w_k} which implies that: (4d) ajaknjnkγ=wjwk−1 {{{a_j}} \over {{a_k}}}{\left( {{{{n_j}} \over {{n_k}}}} \right)^\gamma } = {\left( {{{{w_j}} \over {{w_k}}}} \right)^{ - 1}} However, this last result depends on an additional assumption – that workers’ preferences are homogeneous and that workers are indifferent between jobs offering the same remuneration. To understand the importance of this assumption, consider the alternative setup where all workers have exactly the same productivity, but their preferences are heterogeneous: some workers are highly satisfied with working in a particular sector, while others are not. Suppose also that wages are initially equal in all sectors. A fall in the wage in one sector will give some workers an incentive to leave. All those who stay will be those who find their current job more satisfying than other jobs. This will also be true for the marginal worker: she is indifferent between her current job and the outside option because, on the one hand, her current job gives her more satisfaction, and on the other hand, it gives her lower compensation. However, if the marginal worker’s compensation differs between her current and potential sectors, the condition necessary to derive equation (4c) is not satisfied.

Assuming homogeneous preferences, we can combine (2) and (4d) to understand how the elasticity of labour supply (α) could be translated into the elasticity of average productivity with respect to a change in labour supply (parameter γ). In order to do this, we can derive the labour supply in sector j relative to the labour supply in sector k by using the equation (2): (5) njnk=(sjwj)α(skwk)α {{{n_j}} \over {{n_k}}} = {{{{({s_j}{w_j})}^\alpha }} \over {{{({s_k}{w_k})}^\alpha }}} Since we assume that a_j and a_k are constant (independent of w), then γ = −1/α and a_j/a_k = s_j/s_k. Since s_j can be normalised, then a_j=s_j.

Finally, we can express the sum of efficiency units of all workers in the sector: (6) Nj=ajnjγ+1μ=sj(sjwj)αΣk(skwk)αα−1αμ {N_j} = {a_j}n_j^{\gamma + 1}\mu = {s_j}{\left( {{{{{({s_j}{w_j})}^\alpha }} \over {{\Sigma _k}{{({s_k}{w_k})}^\alpha }}}} \right)^{{{\alpha - 1} \over \alpha }}}\mu

Next we demonstrate how the information contained in the labour supply curves could be used to recover the costs of phasing out a given sector.

A sectoral phase-out implies that all workers are forced to leave the sector. In our setup, forcing each worker to leave a sector is equivalent to lowering that worker’s wage per efficiency units below the point at which she or he is indifferent to staying in or leaving the sector. We will consider an exercise in which we lower wage in sector j forcing consecutive workers to leave, starting with the workers for whom the costs of leaving the sector are the smallest. For now, we will assume that all other wages are held constant – an assumption we will relax in the general equilibrium setup of Section 4. Each time, the reduction in employment requires lowering wages by dw_j/dn_j. At every point during this process, the reduction in wages does not actually affect workers who are leaving the sector. These workers reached the minimal compensation that they were willing to accept, and when their wages drop, they immediately jump to a new sector, receiving the same compensation that they had previously. However, the drop in wages affects all workers who remain in the sector. Every time the wage drops by one unit, these workers lose the compensation equal to N_j efficiency units. Thus, the total loss of earnings for workers due to a complete sector j phase-out is equal to (7) ∫n¯j0Njdwjdnjdnj \mathop \smallint \nolimits_{{{\bar n}_j}}^0 {N_j}{{d{w_j}} \over {d{n_j}}}d{n_j} We will use n¯j {\bar n_j} and w¯j {\bar w_j} to denote initial labour supply and initial wage, respectively.

Using the expression for N_j derived earlier, this integral can be evaluated as: (8) ∫n¯j0Njdwjdnjdnj=Σkskwkα1αμ−Σk≠jskwkα1αμ \mathop \smallint \nolimits_{{{\bar n}_j}}^0 {N_j}{{d{w_j}} \over {d{n_j}}}d{n_j} = {\left( {{\Sigma _k}{{\left( {{s_k}{w_k}} \right)}^\alpha }} \right)^{{1 \over \alpha }}}\mu - {\left( {{\Sigma _{k \ne j}}{{\left( {{s_k}{w_k}} \right)}^\alpha }} \right)^{{1 \over \alpha }}}\mu

The equations describing the system of supply curves in Section 3.2 can be incorporated directly into the CGE framework. An alternative approach is to specify an optimisation problem with representative workers that gives exactly the same predictions as those described in Sections 3.2 and 3.3. This second approach is more useful in cases where models consist of a set of objective functions and constraints for various actors. In this subsection, we show that the supply system presented in Section 3.2 can easily be incorporated into the CGE framework by assuming the existence of a representative worker with Constant Elasticity of Transformation (CET) constraints on the distribution of labour across sectors.

Consider a representative worker who receives an endowment of labour L from households. The worker can transform this labour into productive units of labour dedicated to each sector i, according to the following transformation possibility frontier: (9) ∑jNjsjρ=μρ \sum\nolimits_j {{{\left( {{{{N_j}} \over {{s_j}}}} \right)}^\rho }} = {\mu ^\rho } where s_j is the share parameter, and the parameter ρ determines the ease with which units of labour across sectors are transformed (the higher the value of ρ, the more difficult the transformation).

Note that in this setup labour is sector-specific in the sense that one unit of labour cannot be freely moved between sectors without a change in productivity. When the representative worker transforms a unit of efficient labour from one sector to fit another sector, the amount of efficiency units that is now available for the new sector is given by: (10) dNkdNj=−NjNkρ−1sjsk−ρ {{d{N_k}} \over {d{N_j}}} = - {\left( {{{{N_j}} \over {{N_k}}}} \right)^{\rho - 1}}{\left( {{{{s_j}} \over {{s_k}}}} \right)^{ - \rho }} The objective of the representative worker is to choose the allocation of efficient labour, N_j, that maximises the total return to labour. Formally, the optimisation problem reads (11) maxNjj=1K∑jwjNj \mathop {\max }\limits_{\left\{ {{N_j}} \right\}_{j = 1}^K} \sum\nolimits_j {{w_j}{N_j}} subject to constraint (9)

The optimal allocation satisfies the first order conditions, given by: (12a) wj−λρsj−ρNjρ−1=0 {w_j} - \lambda \rho s_j^{ - \rho }N_j^{\rho - 1} = 0 where λ is the shadow value of a marginal unit of endowment. Using the constraint (9) we find that (12b) λρ=∑jsjwjρρ−1ρρ−1μ−1ρ−1 \lambda \rho = {\left( {\sum\nolimits_j {{{\left( {{s_j}{w_j}} \right)}^{{\rho \over {\rho - 1}}}}} } \right)^{{\rho \over {\rho - 1}}}}{\mu ^{{{ - 1} \over {\rho - 1}}}} and (12c) sjρρ−1wj1ρ−1μΣkskρρ−1wkρρ−11ρ=Nj {{s_j^{{{\rho \over {\rho - 1}}}}w_j^{{{1 \over {\rho - 1}}}}\mu } \over {{{\left( {{\Sigma _k}s_k^{{{\rho \over {\rho - 1}}}}w_k^{{{\rho \over {\rho - 1}}}}} \right)}^{{1 \over \rho }}}}} = {N_j} The total return to labour for all workers in the economy is given by (13) ∑jNjwj=∑jsjρρ−1wjρρ−1ρ−1ρμ \sum\nolimits_j {{N_j}{w_j}} = {\left( {\sum\nolimits_j {s_j^{{\rho \over {\rho - 1}}}w_j^{{\rho \over {\rho - 1}}}} } \right)^{{{\rho - 1} \over \rho }}}\mu and the total loss due to phase-out of sector j is given by (14) ∑kskρρ−1wkρρ−1ρ−1ρμ−∑k≠jskρρ−1wkρρ−1ρ−1ρμ {\left( {\sum\nolimits_k {s_k^{{\rho \over {\rho - 1}}}w_k^{{\rho \over {\rho - 1}}}} } \right)^{{{\rho - 1} \over \rho }}}\mu - {\left( {\sum\nolimits_{k \ne j} {s_k^{{\rho \over {\rho - 1}}}w_k^{{\rho \over {\rho - 1}}}} } \right)^{{{\rho - 1} \over \rho }}}\mu Note that these predictions are exactly the same as in Sections 3.1 and 3.2, as long as α=ρ/(ρ-1).

We can also compute the loss of workers for any change in the vector of wages. If w¯ \bar w is the vector of initial wages and w is the vector of new wages, the total loss for workers in the economy is given by: (15) ∑kNkw¯w¯k−∑kNkwwk=∑kskρρ−1w¯kρρ−1ρ−1ρμ−∑kskρρ−1wkρρ−1ρ−1ρμ \matrix{ {\sum\nolimits_k {{N_k}\left( {{\boldsymbol{\bar w}}} \right){{\bar w}_k}} - \sum\nolimits_k {{N_k}\left( {\boldsymbol{w}} \right){w_k}} = } \cr {{{\left( {\sum\nolimits_k {s_k^{{\rho \over {\rho - 1}}}\bar w_k^{{\rho \over {\rho - 1}}}} } \right)}^{{{\rho - 1} \over \rho }}}\mu - {{\left( {\sum\nolimits_k {s_k^{{\rho \over {\rho - 1}}}w_k^{{\rho \over {\rho - 1}}}} } \right)}^{{{\rho - 1} \over \rho }}}\mu } \cr }

In Section 3.3 we demonstrated how one could use information on the system of supply curves to compute the total loss for all workers in the economy. Policy makers and public opinion might also be interested in tracing the wages (and losses) of particular groups of workers, e.g., workers who were originally employed in mining. In this section, we demonstrate a strategy for estimating such loss.

We start by noting that estimating the loss of workers employed in sector j after phase-out of this sector is trivial if there is no change in the wages of other sectors. In such a case, workers employed in other sectors suffer no loss; none of them suffer any productivity loss since none of them moves. Hence the total loss could be attributed only to workers from the phased-out sector.

If the transition involves changes in the general equilibrium level of wages in several sectors, movement of labour is more complex. There could be several sectors that experience outflow of labour. Below, we provide a strategy for estimation of the loss of a group of workers who are originally employed in a sector that is being phased out. This group could be defined, e.g., as a pool of workers who would be employed in mining, during a given year, in a reference scenario with no climate policy. Alternatively, it could be a pool of workers employed in mining during the reference year, e.g., at the beginning of the analysed period. The choice of reference does not affect the main steps of the derivations. We will focus on the comparison between compensation of workers employed in mining in the reference scenario and compensation of the same group of workers in a policy scenario.

In the reference scenario at time t, the total compensation of workers employed in sector j can be expressed using (6) by (16) Njw¯w¯j=sjw¯jsjw¯jΣkskw¯kα1αα−1μ {N_j}\left( {{\boldsymbol{\bar w}}} \right){\bar w_j} = {s_j}{\bar w_j}{\left( {{{{s_j}{{\bar w}_j}} \over {{{\left( {{\Sigma _k}{{\left( {{s_k}{{\bar w}_k}} \right)}^\alpha }} \right)}^{{1 \over \alpha }}}}}} \right)^{\alpha - 1}}\mu In the policy scenario, this group of workers is split into two groups. We will trace the compensation and productivity of each group.

The first group consists of workers who are still in the sector within the policy scenario. Their total productivity (sum of efficiency units of all workers) is given by N_j(w), and their compensation is (17) Njwwj=sjwjsjwjΣkskwkα1αα−1μ {N_j}\left( {\boldsymbol{w}} \right){w_j} = {s_j}{w_j}{\left( {{{{s_j}{w_j}} \over {{{\left( {{\Sigma _k}{{\left( {{s_k}{w_k}} \right)}^\alpha }} \right)}^{{1 \over \alpha }}}}}} \right)^{\alpha - 1}}\mu The second group consists of workers who left for another sector due to the change in wages that is induced by the policy. The amount of efficiency units that belongs to workers from that group and that are now employed in sector l can be approximated as the difference between total amount of efficiency units in sector l within the policy scenario and that number in the counterfactual. Here, the counterfactual assumes that the wage in sector j was the same as in the reference scenario: (18) slμslwlΣkskwkα1αα−1−slμslwlΣk≠jskwkα+sjw¯jα1αα−1 {s_l}\mu {\left( {{{{s_l}{w_l}} \over {{{\left( {{\Sigma _k}{{\left( {{s_k}{w_k}} \right)}^\alpha }} \right)}^{{1 \over \alpha }}}}}} \right)^{\alpha - 1}} - {s_l}\mu {\left( {{{{s_l}{w_l}} \over {{{\left( {{\Sigma _{k \ne j}}{{\left( {{s_k}{w_k}} \right)}^\alpha } + {{\left( {{s_j}{{\bar w}_j}} \right)}^\alpha }} \right)}^{{1 \over \alpha }}}}}} \right)^{\alpha - 1}} Detailed derivations and the discussion of the assumptions behind this approximation is presented in the Appendix A1.

The total compensation for workers who were originally employed in sector j is the sum of the compensation of the two groups. Using the equation (6), it can be expressed as: (19) Njwj+1−Σk≠jNkwkΣmNmwm+N¯jw¯jΣmN¯mw¯mΣmN¯mw¯mΣmNmwmα1−ααΣl≠jNlwl {N_j}{w_j} + \left[ {1 - {{\left( {{\Sigma _{k \ne j}}{{{N_k}{w_k}} \over {{\Sigma _m}{N_m}{w_m}}} + {{{{\bar N}_j}{{\bar w}_j}} \over {{\Sigma _m}{{\bar N}_m}{{\bar w}_m}}}{{\left( {{{{\Sigma _m}{{\bar N}_m}{{\bar w}_m}} \over {{\Sigma _m}{N_m}{w_m}}}} \right)}^\alpha }} \right)}^{{{1 - \alpha } \over \alpha }}}} \right]{\Sigma _{l \ne j}}{N_l}{w_l} The total cost for workers who were originally employed in sector j is the difference between their current compensation, stated above, and the compensation in the reference point, N¯j w¯j {\bar N_j}\;{\bar w_j} .

The supply system has K+1 parameters: K share parameters (s_j’s) and the parameter α. We calibrate share parameters by matching the model’s predictions of labour compensation at the sectoral level with the compensations observed in the data. To calibrate α, we use elasticity of labour supply at the sectoral level. Below, we provide details of our estimation strategy.

In order to evaluate labour compensation by sector as a function of parameters, we can use equation (6): (20) Njwj=sjwjαΣkskwkαα−1α {N_j}{w_j} = \left( {{{{{\left( {{s_j}{w_j}} \right)}^\alpha }} \over {{{\left( {{\Sigma _k}{{\left( {{s_k}{w_k}} \right)}^\alpha }} \right)}^{{{\alpha - 1} \over \alpha }}}}}} \right) Note that the calibration is performed for the base year and in that year μ=1. The total compensation can then immediately be evaluated as: (21) ∑jNjwj=∑kwkskα1/α \sum\nolimits_j {{N_j}{w_j}} = {\left( {\sum\nolimits_k {{{\left( {{w_k}{s_k}} \right)}^\alpha }} } \right)^{1/\alpha }} Using these two expressions as well as information on labour compensation in each sector j, w_jN_j, we can uniquely determine s_jw_j (22) NjwjΣjNjwj1/α∑jNjwj=sjwj {\left( {{{{N_j}{w_j}} \over {{\Sigma _j}{N_j}{w_j}}}} \right)^{1/\alpha }}\sum\nolimits_j {{N_j}{w_j}} = {s_j}{w_j} Note that in the model we can normalise every sectoral wage at the base year to unity by choosing appropriate values for s_j.

We calibrate the parameter α by using the elasticity of labour at the sectoral level. Before we proceed with mapping the predictions of the model with observed empirical results, we must highlight two important distinctions. First, note that the theoretical framework in Section 3 distinguishes between two supply curves: the relation between wages and supply of labour expressed in physical units (equation 2), and the relation between wages and supply of labour expressed in efficiency units (equation 6). In the CGE framework, the production choices of firms depend only on the amount of available efficiency units. Thus, it is most convenient to express quantity of labour in the CGE model using efficiency units and integrate the model with the efficiency unit supply curves. However, for calibration purposes, we need to use the physical labour supply curve since employment in terms of efficiency units is not observable. The mathematical derivations in Section 3.1 show that the slopes of the two curves are different, but the relation between the two can easily be determined.

The second distinction concerns the interpretation of variable w_j. In our model, w_j represents the wage per efficiency unit in sector j. In the model, we assume that every worker can offer a constant amount of efficiency units in sector j (but for a given worker, the potential amount of efficiency units could differ between sectors, and in a given sector different workers can have different amounts of efficiency units). This implies that the change in wage w_j is proportional to the observed change in wage received by a worker in sector j. Note that this change will be different from the change in observed average wage in the sector since after the change in wj, the composition of workers changes – those with the lowest amounts of efficiency units are leaving the sector. We need to keep this distinction in mind when calibrating the parameter α. In particular, we need to use empirical supply curves obtained from analysing the response of number of workers to exogenous wage shocks; that is, that which is observed in survey data that trace the wage and sector of employment of the same individual over time. The supply curve obtained from estimating a simple relationship between labour supply and average wage observed in the sector is not appropriate for our calibration.

The elasticity parameter α in equation 2 can be approximated with empirically observed elasticity of labour supply as long as each sector is relatively small. When this condition is satisfied, a change in sectoral wage has only a small impact on the aggregate wage index (the denominator on the right-hand side of equation 2) and therefore elasticity of labour supply predicted by equation 2 is close to α. For instance, in the case of mining sector, which employs 0.4% of workers in the Polish economy, according to equation 2, elasticity of labour would be equal to 2.489 when α =2.5.

We obtain the value of the elasticity of labour supply at the sectoral level from literature that estimates the elasticity of separation rate ⁽⁵⁾ with respect to wages at the firm level. Manning (2003) shows that the long-run elasticity of labour supply should be exactly twice the elasticity of separation rate. The empirical literature was reviewed by Ashenfelter (2010), who finds the values of elasticity of labour supply in the range 1.5–4. However, there is also evidence in the literature of much lower values: the elasticity for monopsonies was estimated using survey data by Booth and Katic (2011) who obtained estimates in the range 0.71–0.75.

The choice of the value of the elasticity is further complicated for three reasons. First, our application requires the use of the elasticity of labour at the sectoral level, whereas the available literature provides evidence on the elasticity at the firm level. Since switching between firms is likely to be easier than switching between sectors, the elasticity at the sectoral level should be smaller than at the firm level. Second, in our framework, any resistance by workers to leaving their current job is attributed to the expected loss in productivity. In reality, however, this resistance could be at least partly due to other factors, such as workers’ preferences (if they dislike other jobs). Third, the empirical studies do not take into account natural attrition: in the very long time horizon we consider in our simulations, the older cohorts of workers retire and are replaced by younger cohorts who have few constraints on their choice of sector. For the latter two reasons, the elasticity relevant to our study should be larger than the one reported in the previous paragraph.

For our study we chose a value of 2.5 for the elasticity of labour in our base simulation. We also ran sensitivities for values of 1.2, 1.5, 2.0, 3.0 and 5.0. We have also attempted to run the simulations for the values lower than 1.2, but in these cases the numerical model was not able to find a solution, which may indicate that ambitious emission reductions are not feasible when the elasticity of transformation is very low.

In this section we present the numerical predictions of the model for the decarbonisation scenario, which assumes 76% reduction in GHG emissions by 2040 (relative to 1990). We estimate the resulting costs of the structural change for workers by comparing the macroeconomic outputs of the model in the simulations with and without frictional transition of labour across sectors. In the supplementary material to this paper (see footnote 4), we also present the main projections on structural change, fuel demand and detailed results for output and employment by sector in the simulation with labour market frictions.

In order to compute the loss of earnings for workers due to frictional transition between sectors, we compare two simulations. In the first simulation, we assume that the flow of workers is smooth; all workers in all sectors have exactly the same wage. Sectoral employment is determined by the intersection of horizontal labour supply and downward-sloping labour demand in each sector. Every worker who leaves, for instance, the mining sector and moves to another sector receives exactly the same payoff as in the old sector. Thus, the shift leaves productivity of the workers unaffected. This simulation is intended to provide a projection for a hypothetical scenario in which there were no transition costs for workers. We call this scenario the ‘no frictions scenario’. In the second simulation we take into account the loss of productivity due to the effects described in Section 3. This simulation represents a scenario with frictions.

Inspection of the results under the two simulations suggest that the structural changes in both are almost identical. We observe very similar changes in output, in demand for electricity and fuels. Therefore, we will focus on the comparison of the key macroeconomic variables: GDP and compensation of labour.

The differences in compensation of employees and the GDP are depicted in Figure 1. In 2030, accounting for the loss of labour productivity due to frictional transition reduces the total compensation of labour by 0.38% relative to the case when the loss is ignored. In 2040, the loss increases to 0.44%. The decline of productivity due to frictional transition also implies a reduction of GDP. In 2030, this reduction is rather small (0.16%). By 2040, however, the difference in GDP between the two simulations attains 0.41%.

Figure 1.

GDP and labour compensation loss in Poland due to frictional transition of labour between sectors (percentage difference between the frictionless decarbonisation simulation and the simulation with frictions throughout the transition)

The monetary value of the potential loss for workers due to frictional transition is presented in Figure 2. The potential loss of earnings is computed as a difference in total labour compensation between the two simulations. In 2020, the loss is US$0.24 billion (at 2011 prices) per annum. In 2030, the annual loss is at the level of $0.69 billion, and in 2040 it reaches $0.97 billion.

Figure 2.

Labour compensation loss for all workers and for miners in Poland due to frictional transition of .labour between sectors (absolute difference between frictionless decarbonisation simulation and the simulation with frictions throughout the transition)

We have also performed a sensitivity analysis by running the simulations under the alternative values of three parameters: (i) elasticity of transformation of labour (parameter α in the framework in Section 3), which governs how easy it is for workers to switch between sectors, (ii) elasticity of substitution between electricity and fossil fuels, which governs how easy it for firms to replace use of fossils with electrified production, and (iii) elasticity of substitution between capital-labour composite and energy, which governs how easy it is for firms to switch to more energy-saving (but more capital-intensive) production.

Figure 3 compares the results for the cases where the elasticity of labour supply at sectoral level is 1.2, 1.5, 2.0, 2.5, 3.0 and 5.0. We have also tried to run the simulations for elasticities equal to and lower than 1.1, but for this case the model is not able to find a numerical solution, suggesting that either ambitious emission reductions are not feasible when the elasticity of transformation is very low, or that our numerical tool is not suitable for analysing transformation when switching sectors is extremely costly for some workers. The results in Figure 3 show that a higher elasticity of transformation is associated with lower transition costs for workers, which is consistent with the predictions of the theory presented in Section 4: a high elasticity of transformation corresponds to a high elasticity of labour supply at the sectoral level, indicating that a small increase in wages leads to a large inflow of workers into a sector. According to the theory presented in 3.1, this implies that the worker who switches experiences only a small drop in productivity. Conversely, a small elasticity of transformation and a small elasticity of labour supply are associated with a large drop in productivity. In the context of our simulation, the switching workers are workers leaving the phasing-out sectors, such as mining. Forcing them to switch is associated with a large loss of productivity and hence wages for these workers when the elasticity is low.

Figure 3.

GDP (left panel) and labour compensation (right panel) loss in Poland due to frictional transition of labour between sectors (percentage difference between the frictionless decarbonisation simulation and the simulation with frictions throughout the transition) for the alternative assumptions regarding elasticity of labour supply at the sectoral level

Figures 4 and 5 show the results for the alternative values of the elasticities of substitution between electricity and fossil fuels, and between the energy composite and other inputs. For the former, changing the elasticities clearly does not affect our results much. In the latter case, GDP results are identical for all values of elasticities considered. Labour compensation in a high elasticity (we consider the value of 1.5) is associated with a 0.04 percentage point higher loss in 2035, compared to the low elasticity case (the value of 0.5). This is because the ease of substituting energy with capital leads to a greater reduction in the consumption of fuels, which in turn leads to a greater reduction in demand for the sectors producing these fuels, necessitating a greater movement of workers between sectors. However, the results clearly show that the effect is temporary and its magnitude is small.

Figure 4.

GDP (left panel) and labour compensation (right panel) loss in Poland due to frictional transition of labour between sectors (percentage difference between the frictionless decarbonisation simulation and the simulation with frictions throughout the transition) for the alternative assumptions regarding elasticity of substitution between fossils and electricity

Figure 5.

GDP (left panel) and labour compensation (right panel) loss in Poland due to frictional transition of labour between sectors (percentage difference between the frictionless decarbonisation simulation and the simulation with frictions throughout the transition) for the alternative assumptions regarding elasticity of substitution between energy composite and the capital-labour composite

We also estimate the financial loss of income for miners using the methodology developed in Section 3.4. We trace the labour compensation for a group of workers who were employed in the mining sector in 2015. This includes workers who left the sector after 2015 and those who decided to stay. The loss is computed as a difference between labour compensation for that group in a given year and the compensation that group received in 2015. In 2030, the loss of those workers is $0.57 billion (Figure 2) and in 2040 that loss grows to $0.81 billion.

The estimates of losses of earnings produced by our model are likely to be biased upwards due to the imperfect treatment of dynamics in our setup. In particular, in our derivations we have assumed that all workers belong to one cohort. In reality, there are different cohorts with different transition costs. For example, workers who belong to cohorts that enter the labour market after the announcement of climate policy will avoid investing in skills that are only relevant for dirty sectors, so their transition costs will be very low. If the climate policy is announced in the first year, the number of cohorts entering the labour market after the announcement of the climate policy will continue to increase over time. Therefore, for years in the distant horizon, especially for 2040, the average transition costs are likely to be lower than those reported by our model.