Policy Research Working Paper 11127 Beyond Aggregates A Sector-Specific Framework for Long-Term Growth Modeling Charl Jooste Remzi Baris Tercioglu Economic Policy Global Department May 2025 Policy Research Working Paper 11127 Abstract This paper develops a bottom-up, sector-specific approach the framework, wedges in sectoral factor prices, substi- to modeling potential output that overcomes limitations of tution elasticities, and productivity differentials describe traditional top-down estimates for long-term projections the contribution of between-effects to aggregate produc- and policy analysis. The model disaggregates total-factor tivity. Although the approach here can be applied to any productivity (TFP) growth into within-sector productivity macro-structural model, its benefits are illustrated by intro- effects and between-sector reallocations. Such endogenous ducing it into the World Bank’s semi-structural models for between effects capture structural transformation, notably Ghana and the Kyrgyz Republic to showcase its potential to the shift from low-productivity sectors like agriculture to enhance the analysis of long-run growth dynamics through higher-productivity industrial and service sectors—a key structural change. driver of growth in developing countries. At the heart of This paper is a product of the Economic Policy Global Department. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at cjooste@worldbank.org and rtercioglu@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Beyond Aggregates: A Sector-Specific Framework for Long-Term Growth Modeling∗ Charl Jooste1 , Remzi Baris Tercioglu1 1Economic Policy Global Practice, World Bank, Washington, DC JEL Classification: C10, C50, Q43 Keywords: Economic modeling, structural change ∗ We thank Andrew Burns, Florent McIsaac, Andreas Eberhard, Frederico Gil Sander, James Sampi, Ekaterina Vostroknutova, Lazar Milivojevic, Stefano Curto, Chung Gu Chee and Ragchaasuren Galindev for helpful comments. The authors thank the multi-donor Climate Support Facility for funding this work. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/- World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. 1 Introduction Semi-structural macroeconomic models are extensively used by central banks (Angelini et al., 2019; Lemoine et al., 2018; Brayton et al., 1997; Cusbert, Kendall, et al., 2018), ministries of finance (Bullen et al., 2021; Welfe, 2011; Dam, 1987; Dudek et al., 2012), international insti- tutions (Burns et al., 2019) and the private sector for macroeconomic projections and policy simulations. While some versions of these models include detailed sectoral representations, these models tend to focus on aggregate quantities and often feature stylized representations of general equilibrium. They typically focus on short- to medium-term developments and are content to leave much of the determinants of growth to an aggregate potential output function. The models tend to have limited sectoral detail and limited endogenous long-run dynamics. For more granular growth and sectoral studies, planning ministries and central banks frequently rely on a range of modeling tools. For instance, they may rely on Dynamic Recursive Computable General Equilibrium (DRCGE) (Robinson and Roland-Holst, 1988; Callonnec et al., 2013; Robinson, 1991; Davis, 2004; Roson and Mensbrugghe, 2018), and growth models are preferred for analyzing long-term trends (Rebelo, 1991; Srinivasan, 1995; Hevia and Loayza, 2012; Devadas and Pennings, 2018). Some exciting new work over the past few years has built out network structures to account for sectoral spill-overs - the framework of Baqaee and Farhi (2020) achieves this objective and can be used to map onto dynamic structural macroeconomic models. Other macroeconomic models provide a more detailed representation of the economy by incorporating richer sectoral details and productivity variations. For instance, Boppart et al. (2023) demonstrate that capital and intermediate input costs decline more rapidly than labor costs across most sectors, with the exception of agriculture. This differential explains the labor productivity gap between wealthy and developing countries. By integrating sectoral details, these models offer a more nuanced understanding of how shocks impact the economy and allow for the endogenous modeling of structural changes. This paper enriches the supply side of a standard macro-structural model (in this case, the World Bank’s MFMod framework) by building up aggregate potential output from the bottom up, incorporating sector-specific productivity, capital and employment into sectoral supply functions. The revised model yields a more nuanced understanding of how sector- specific shocks impact the economy and allows for an endogenous modeling of structural change and aggregate TFP. Specifically, our goal is to highlight two critical features discussed in the literature: (i) the process of structural change through endogenous factor allocation 1 (Matsuyama, 2008; Rodrik et al., 2013; Bondarev and Greiner, 2019; Rajhi, 2014), and (ii) the decline in labor shares of output, despite stable or rising growth (Autor et al., 2017), affecting total factor productivity. The remainder of the paper outlines the modifications made to the MFMod. In Section 2, we detail the changes to the core equations. Section 3 describes the types of shocks introduced in the model. Section 3.2 presents the simulation results for various policies. Finally, Section 4 concludes with our final observations. 2 Endogenizing structural change in a semi-structural macroe- conomic model Endogenous structural change modeling takes the form of modifying the production block of the model. Instead of modeling a single production function, we have production functions at the sector level, where labor and capital can move between sectors. Ultimately, the framework we propose is a general model to account for how the between-effects of TFP materialize, accounting for preferences, wedges and innovations within sectors. We retain the core features of a standard model involving aggregate labor demand and capital demand. Later we show that aggregate production technology is a special case of the sectoral variant. The expenditure side of the model is retained, but now sensitive to the weights in sectoral prices. The allocation of production factors to sectors is based on economic theory. For instance, aggregate employment is estimated similarly to the standard MFMod model (see Burns et al., 2019), while the distribution across sectors is done according to relative wage differentials and preferences. A similar approach is applied to capital: the total capital stock is determined using the capital accumulation equation (law of motion of capital) and investment estimates, after which this aggregate capital stock is allocated to sectors according to their respective capital costs. In this allocation process, capital flows from sectors with higher capital costs to those with lower costs. At the heart of this analysis, is that the between-part of TFP is primarily driven by wedges that account for wage and capital cost differentials. An attempt is made to describe some of the wedges (notably product and factor markups, taxes and subsidies and implicit cost of borrowing due to differences in sectoral risk premia). It should be noted from the onset that the modeling framework does not account for factors explaining within-productivity drivers - sectoral level within-TFPs are exogenous. We employ a constant elasticity of transformation (CET) functional form to distribute 2 employment and constant elasticity of substitution (CES) to distribute capital, ensuring the preservation of additivity.1 Once sectoral employment and capital stocks are established, we model sectoral potential output using Cobb-Douglas production functions. Aggregate potential output is then defined as the sum of these sectoral potentials, with actual sectoral output converging towards their potential levels. As GDP is determined from the expenditure side, we close the model by treating the demand for service output as a residual component. 2.1 Theoretical overview of the firm side - a sectoral approach We assume a representative firm for each sector. Each sector has a final goods producer, while the intermediate good producer is a monopolistically competitive firm. Potential gross value-added in each sector i is produced using capital (K ) and labor (N ). A Cobb-Douglas production technology is assumed. Sectoral employment that enters the production function refers to structural employment, while each sector’s TFP (A∗ ) is a smoothed (2-sided HP filter) Solow residual.2 As such, the Cobb-Douglas production function represents potential output, but now at the sector level. Prices act on final demand to ensure that actual GDP converges to potential GDP, while also influencing how resources are allocated across sectors. Aggregate potential GDP is then just the sum of the sectoral potential GDP’s. 2.2 Modeling MFMod sectoral production functions The representative firm in each sector is cost-minimizing. Sectoral production is defined as: ∗ ∗ αi 1−αi Yi,t = A∗ i,t Ni,t Ki,t−1 (1) ∗ where Yi,t is the sectoral potential output in sector i and in period t. Structural employ- ment (N ∗ ) and capital stock (K ) are defined below. The data construction is discussed in the case studies. Aggregate potential GDP is the sum of the sectoral potential outputs accounting for net indirect taxes Ttind . 1 See Mensbrugghe and Peters, 2016 for a broader treatment of additivity. 2 The type of filter is not important for the functional forms in this model. Of course it is important to account for end-point problems and employ an appropriate filter in the data preparation stage (see Phillips and Shi, 2021). 3 Yt∗ = ∗ Yi,t + Ttind (2) i∈ I 2.2.1 Sectoral labor block Aggregate structural employment is derived as in MFMod and depends on the non-accelerating wage rate of unemployment (N AW RU ), the labor force participation rate (P R) and the working age population (W P OP ): Ni∗ = (1 − N AW RUt ) · P Rt · W P OPt (3) We assume that the long-run labor supply curve is vertical, meaning that the structural employment is exogenous and independent of wages.3 In contrast, the short-run labor supply curve is positively related to the real wage. The N AW RU is estimated empirically. Wages are a central feature in determining shifts in labor across sectors. In the long run sectoral wages follow sectoral labor productivity, but in the short run are influenced by the cyclical unemployment in a classical Phillips curve specification (note that lower case letters represent logs of the variables, and ∆ is the first difference operator). ∗ ∆wi,t = θi (wi,t−1 − (log(Yi,t −1 /Ni,t−1 ) − pi,t−1 − log(βi,1 )) (4) ∗ +βi,10 ∆wi,t−1 + (1 − βi,10 )(∆log(Yi,t /Ni,t ) + ∆pi,t ) − βi,11 (ut − u∗ t) This equation indicates that shocks to sectoral labor productivity influence relative wages. Labor productivity differentials, therefore, drive structural transformation. Also note that two wedges creep up in the specification; log(βi,1 ) captures the wage markup directly, while the product markup is implicitly embedded in the sectoral price deflator pi,t . Higher sectoral wages resulting from increased productivity have two main effects: they raise both sectoral output and real wages. However, there’s a counteracting effect where increased productivity (i.e., TFP) reduces marginal costs, thereby lowering prices. The net impact on sectoral wages depends on the estimated elasticities in the wage equation and the degree of price stickiness. Following MFMod, we estimate aggregate employment using an Error Correction Model 3 This assumption can be relaxed when estimating labor supply elasticities. 4 (ECM) of the following form: Wt Y∗ ∆nt = β2 (nt−1 − n∗ ∗ t−1 ) + ∆nt − β1 ∆log − ∆log αt t∗ (5) Pt Nt Actual employment (Nt ) converges to structural employment (Nt∗ ) in the long-run. In the short-run, real wage (Wt /Pt ) growth over potential productivity (Yt∗ /Nt∗ ) negatively affects the demand for employment. This is in line with the marginal productivity of labor in linearized form. This equation ensures that actual unemployment converges to the N AW RU , but still allows for cyclical unemployment when differences between real wages and the marginal productivity materializes as a consequence of shocks. Long-run unemployment will go up if the N AW RU increases with changes in real frictions.4 The reallocation of labor is driven by changes in productivity across sectors and is the principal driver of structural change in the model. Since wages are linked to labor pro- ductivity, this implies a sectoral shift from low to high productivity sectors, the between component of structural change (Rodrik et al., 2013). Aggregate available labor (Nt ) is derived as described above, while the allocation of labor across industries is determined by relative wages, resembling the standard labor mobility mechanism found in CGE models.5 In MFMod, there is an assumed continuum of labor varieties aggregated into a final labor input used in production, utilizing a sector specific Dixit-Stiglitz aggregator. Firms allocate available labor supply across sectors based on prices in product markets, relative wages and firms’ profit maximization. An additive CET function allows for sector specificity in labor use, with the extent to which relative wages must change to induce a given reallocation is determined econometrically by the estimated elasticity of employment, 4 An empirical estimate of the N AW RU is obtained from a simple Phillips Curve equation ∆wt = γ0 − β1 ut + β2 ∆at + β3 πt + εt where ut is the unemployment rate and at is aggregate TFP. Dropping time subscripts we have U ¯ = γ0 . If we take the conditional expectations of the wage equation we get an expression for the β1 1 N AW RU as a function of TFP and the inflation target: E(N AW RUt ) = U ¯ + β2 − 1−α ∆at + (β3 −1 π β1 β1 ¯t . 5 In DSGE models with simple unemployment (e.g., Erceg et al., 2000), a union maximizes utility by setting wages for its members, considering the firm’s labor demand schedule, while the wage that the firm faces is taken as given. In MFMod and standard DSGE models, it is typically assumed that workers will always supply (Ni,t ). More realistic formulations, such as labor supply constraints, can be found in Huo and Ríos-Rull, 2020. However, DSGE models often detrend variables to explain business cycles, achieving a balanced growth path without studying structural changes (i.e., between-sector effects). 5 −σ N N Wt∗ Ni,t = ωi Nt (6) Wi,t N where ωi is CET share of sectoral employment, σ N is the elasticity of transformation of 1 N employment (and also a proxy for labor mobility), and Wt∗ = N σ σ N i∈I ωi Wi,t . However, with Wt∗ , the sectoral wage bills will not add up to the aggregate wage bill. That is why we define the revenue additive aggregate wage rate (Mensbrugghe and Peters, 2016): σ N +1 N i∈ I ωi Wi,t Wt = N (7) Wt∗ σ Sectoral structural employment follows actual employment. ∗ log Ni,t = αi,1 + log(Ni,t ) (8) This formulation implies that the long run unemployment rate is consistent with the structural unemployment rate.6 2.2.2 Sectoral capital block We start our discussion of capital stock at the aggregate level and then describe how capital is allocated across sectors. Investment choices (the flow) pins down aggregate capital. Ag- gregate investment in MFMod is a function of expected long-run returns. This implies that the investment to capital ratio converges to the long-run growth rate of the economy plus capital depreciation. Taking the aggregate production function and rewriting the first order condition in terms of capital we get: Yt Kt = (1 − αt ) (9) Rt If we assume that capital evolves according to a perpetual inventory method then capital 6 To keep aggregate structural employment unchanged, we treat service sector structural employment as the residual. 6 accumulation can be written as Kt = (1 − δ )Kt−1 + It (10) Divide the equation above by Kt−1 and express in terms of investment (using equation 9): Kt It − 1 +δ = (11) Kt−1 Kt−1 SS gt SS SS Yt−1 It = (gt + δ )Kt−1 = (gt + δ )(1 − αt ) (12) Rt−1 From this condition we express our investment equation as:    SS Yt∗−1   Rt (gt + δ )(1 − ∆it = θ(it−1 − log  αt )   Rt−1  + β3 ∆yt − β4 ∆log (13) p Pt PtI −1 SS where Rt is the aggregate cost of capital and PtI is the investment deflator. gt is the long-run growth rate of the economy, which depends on aggregate TFP (to be defined below and population growth, which is exogenous). Investment adjusts to capital stock in the long-run. In the short-run, GDP growth has a positive impact, while increases in the real cost of capital has a negative impact on investment growth. The aggregate capital stock is allocated to sectors as a function of the relative rates of return to capital in each sector with the ease with which capital flows between sectors determined by the statistical estimates of the elasticity of substitution: ∗ σK K Rt Ki,t = ωi Kt (14) Ri,t which is then used to pin down sectoral investments: Ii,t = Ki,t − (1 − δ )Ki,t−1 (15) 7 K where ωi is CES share of sectoral capital stock (or a proxy for capital mobility), σ K is −1 ∗ K −σ K σK the elasticity of substitution of capital, and =Rt . Note that for small i∈I ωi Ri,t K σ we get very little capital mobility with changes to the relative rental rates. The aggregate cost of capital is then the weighted sum of the sectoral cost of capital. K K 1−σ i∈I ωi Ri,t Rt = ∗ −σ K (16) Rt Sectoral costs of capital are identities p Ri,t = pI t (Γi,t + δi,t + Rt − πt ) (17) p where Γi,t are sectoral risk premia, Rt is the monetary policy rate which follows a standard Taylor rule, and πt is the inflation rate. Sectoral risk premia are ECMs of the aggregate risk premium. Γi,t = βi,2 (Γi,t−1 − Γt−1 − βi,1 ) (18) with βi,2 being an estimate of how quickly risk premia converge to equilibrium, inclusive of sector specific risk βi,1 . 2.2.3 Marginal costs and closure assumptions Having defined sectoral wages and rental rates, we proceed to writing marginals costs of each sector. Sectoral nominal marginal costs are derived using the shadow price in the first-order conditions of cost minimization. Importantly, an increase in sectoral TFP reduces marginal costs, which feed into both price and wage determination. αi 1−αi ∗ Wi,t Ri,t Pi,t = ∗ αi (19) Ai,t αi (1 − αi )1−αi We account for stickiness in agriculture and industry prices empirically, where prices converge to the nominal marginal costs plus an estimated constant sectoral markup. 8 πi,t = θi (pi,t−1 − p∗ ∗ i,t−1 − log(βi,1 )) + πi,t (20) 2.3 Aggregate TFP and marginal costs From sectoral output, structural employment and capital stock, we calculate sectoral TFP by inverting the Cobb-Douglas production function. Yi,t Ai,t = (21) Ni,t Ki,t−1 1−αi ∗ αi Aggregate TFP is a weighted average of sectoral TFPs. We assume that aggregate production in the economy has the same functional form as the sectoral production functions (e.g., Cobb-Douglas).7 If aggregate production is the sum of sectoral production we have: Yt∗ = Yi,t ⇒ At Nt∗ αt Kt−1 1−αt = ∗ αi Ai,t Ni,t Ki,t−1 1−αi (22) i∈I i∈I Then it follows that aggregate TFP is ∗ αi Ni,t Yt∗ ∗ αi Ni,t Ki,t−1 1−αi Ki,t−1 Ki,t−1 At = = Ai,t ∗ αt = Ai,t αt (23) Nt∗ αt Kt−1 1−αt i∈I Nt Kt−1 1−αt i∈I Nt Kt−1 Kt−1 We can write aggregate TFP growth then as a within component and a between compo- nent: ∆ at = ∆ai,t λi,t−1 + ∆λi,t ai,t−1 + cross terms (24) i∈I i∈I ∗ αi K Ni,t 1−αi i,t−1 where λi = Nt∗ αt K t−1 1−αt and cross terms includes all the cross terms. The second 7 Note that this may not generally hold. If we assume that we can combine value added from each sector 1 ρ ρ as a CES (Yt = i∈I θi Yi,t ) and that ρ = 0 then the aggregate production function, under constant shares is a Cobb-Douglas. 9 expression after the right-hand side of the equality is the between effect - or structural transformation.8 Equation (23) illustrates how aggregate TFP is influenced by inter-sectoral structural change. The within-sector component (Ai,t ) is exogenous. Aggregate labor and capital and total factor productivity in this framework are endogenous, allowing for a comparison with the within and between components of structural change in Rodrik et al., 2013. The allocation of capital and labor plays a crucial role in explaining between-sector contributions to structural change, driven by frictions in factor markets (see Boppart et al., 2023 for a detailed discussion). Several key observations emerge. If αt < αi the labor-capital ratio per sector is typically higher than the aggregate labor-capital ratio. As σ → 0 then αt → αi , which will also bring the relative labor-capital ratios equal to each other. Thus, higher sectoral elasticities of sub- stitution lead to higher aggregate labor shares, implying a larger aggregate TFP (assuming relative TFP between sectors equals 1). A higher aggregate labor share is positively related to aggregate TFP, corresponding to the positive empirical relationship between output per worker and labor share (Sturgill, 2012) but may differ when considering "superstar" firms (Stiel and Schiersch, 2022).9 An important implication of Equation (23) is that sectoral TFP levels and sectoral la- bor and capital shares are crucial in shaping the aggregate TFP response to shocks. As we demonstrate, the extent to which sectoral productivity shocks transmit to aggregate pro- ductivity depends on the elasticity of transformation/substitution of labor and capital, with higher elasticities being associated with larger aggregate productivity effects – because the between effects are larger. Finally, one can use this methodology to decompose the standard growth accounting 8 This resembles the standard output per worker decomposition, which equals a within, between and a covariance component or ∆yt = si,t−1 ∆yi,t + ∆si,t yi,t−1 +cross terms. The difference is that we are i∈I i∈ I within between weighting the TFPs by sectoral output as opposed to weighting labor productivity by sectoral employment shares. 9 Alternatively one can generate a relationship between TFP and labor shares using a standard cost minimization procedure where capital and labor are substitutes. Starting with markups equalling prices σKN −1 divided by marginal costs and deriving the first order conditions for labor we have W N α P Y = µ AN Y σKN . The labor share will increase with a higher output elasticity (α), and decreases in the markup (µ). If we σKN −1 Y σKN have a Cobb-Douglas then AN = 1 then the relationship between the labor share and TFP is non- existent, i.e., the labor share is constant and does not shift with TFP. However, if σKN > 1, then an increase in TFP will reduce the labor share in income. This just means that TFP increases will lead to capital deepening. However, if σKN < 1 then we have an increase in the labor share. 10 formula into finer segments, other than just capital, labor and TFP. Importantly, it also allows to decompose TFP’s contribution into the impact of sectoral productivity (within effects) and the effects of structural change (between effects), and then, how much of the between movements are due to labor versus capital. At − At−1 = [λi,t−1 Ai,t − λi,t−1 Ai,t−1 + λi,t Ai,t−1 − λi,t−1 Ai,t−1 ] (25) i Within Between where changes in λ captures the between shifts. N∗ Ki,t−1 Noting that ωn,i,t = Ni,t ∗ and ωk,i,t = K t−1 , we can write the between part as: t Nt∗ αi Kt−1 1−αi λi,t = (ωn,i,t )αi (ωk,i,t )1−αi . (26) Nt∗ αt Kt−1 1−αt sector’s fraction of N,K scale factor The scale factor will equal 1 if the labor shares in each sector is equal to the aggregate labor share. One can split the decomposition into an additional layer by noting that in Equation 26, the labor and capital reallocation is directly related to the elasticity of transformation, and the initial wedges of factors costs between sectors. Section 2.4 is devoted to describing how the labor and capital reallocation effects materialize. Simply put, λi,t can be expressed as a function of relative factor prices. Deviations of sectoral prices from aggregate prices are due to labor heterogeneity (e.g., skills), preferences, elasticities of transformation (e.g., labor mobility) and distortions: αi  1−αi σK  σN ω N Wi,t ω R Rt Nt∗ αi Kt−1 1−αi λi,t = i  i  . ∗ αt (27) Wt Ri,t Nt Kt−1 1−αt allocation due to wedges scale factor Now we can decompose the between part into the labor and capital reallocation as: ∂λi,t ∂λi,t Ai,t−1 ∆λi,t = Ai,t−1 |t ∆ωn,i,t + Ai,t−1 |t ∆ωk,i,t +cross terms (28) i i ∂ωn,i,t i ∂ωk,i,t Between Labor reallocation Capital reallocation 11 Finally, after having defined aggregate TFP, we are able to discern the sectoral impact on aggregate marginal costs of the economy too: αt 1−αt 1 Wt Rt M Ct = (29) At αt 1 − αt Sectoral wages and capital costs, accounting for rigidities and elasticities of substitution, determine factor mobility and the underlying causes of cost changes. 2.4 The role of the elasticity of transformation, initial productivity differ- entials and employment allocations The degree of structural transformation depends on the within and between components. These two effects can move in opposite directions for different policy choices, and is sen- sitive to initial shares and the initial relative productivity differentials. The allocation of employment (or labor mobility) can be illustrated as a function of elasticity of transforma- tion, price, and productivity differentials. Consider a two-sector economy where the optimal labor supply for each sector is given by: −σ W N1 = ω1 N W1 −σ W N2 = ω2 N W2 Expressing these as ratios, the allocation of labor to sector 1 becomes an increasing function of the relative wages in sector 1 compared to sector 2. −σ N1 ω1 W2 = N2 ω2 W1 Taking logs n1 − n2 = log(ω1 ) − log(ω2 ) − σ (w2 − w1) 12 and differentiating, we can see that the growth for labor allocation to sector 1 is sensitive to the elasticity of transformation and the relative wages, which in steady state grows at the rate of sectoral inflation and productivity: ∆n1 − ∆n2 = −σ ((π2 + ∆a2 /ε2 ) − (π1 + ∆a1 /ε1 )) From this equation, it is evident that differential growth of labor allocation between sectors is not particularly sensitive to sectoral productivities when σ is small. Even if sector 1 was able to grow rapidly due to an increase in productivity this would not necessarily change the allocation of labor in sector 1. Graphically, the degree of transformation from sector 2 to sector 1 is depicted in Figure 1. An increase in relative wages leads to a greater degree of transformation if the elasticity of transformation between different labor types is high. Conversely, low levels of elasticity dampen the impact of relative wage differences on employment allocations. To better un- derstand this, consider σ < 1. An increase in wages in sector 1 relative to sector 2 results in a modest increase in the allocation of labor towards sector 1. Productivity shocks in this scenario may not necessarily lead to significant structural transformation. On the other hand, when σ > 1 , an increase in the wage in sector 1 relative to sector 2 will result in a stronger allocation of labor to sector 1. In this simplified case, labor allocation over time is highly sensitive to both the degree of labor mobility and differences in productivity growth across sectors. 13 Figure 1: Between component of structural transformation Employment allocations are derived from both substitution and income effects. An in- crease in the relative productivity (marginal product of labor) should increase employment demand in that sector (substitution effect) until wages equal its productivity. Using our CET framework we can illustrate this as follows: ∂Yt ∂Yt ∂Nt =0: =0 ∂Ni,t ∂Nt ∂Ni,t σ ∂Yt ∂Nt Pt Yt Nt = αt ωi ∂Nt ∂Ni,t Nt Ni,t   Wi,t log(Ni,t ) = log(Nt ) + σ log  P t Yt  αt ωi Nt An increase in sectoral MPL will lead to an increase in sectoral employment allocated to that sector. An increase in aggregate output (which is the sum of the weighted sectoral outputs) will increase employment demand. Assume we have a decrease in TFP in sector 1, then this will lead to a decrease in MPL 14 in that sector and hence a lower labor supply to that sector. However, nominal value-added (Pt Yt ) depends on the sectoral reallocation and the response of that sector’s value-added to a decrease in TFP. There are thus cases when a decrease of a sector’s TFP may result in an increase in labor allocated to that sector. This depends critically on the price elasticity of demand for certain goods. Assume that demand for agricultural goods are inelastic to price changes, then a negative TFP shocks can lead to a reduction in output, but an increase in employment allocated to that sector. Equation 4 captures this idea. Given that real wages in equilibrium should equal the MPL, the nominal wage is indexed to the sectoral price. If demand for goods are inelastic then the wage equation will reflect that. 2.5 The dynamics of the labor share and structural change The modeling framework outlined above enables us to describe how aggregate labor shares evolve over time. Shifts in labor shares can be attributed to structural change (Moreira, 2022), changes in market power (Autor et al., 2017), and sensitivity to the elasticity of substitution (Grossman and Oberfield, 2022). Each of these factors is incorporated into the macroeconomic model discussed. It is important to note that the aggregate labor share also plays an important role in describing structural transformation as discussed in Section 2.3. 10 The labor share in output corresponds with the output elasticity of labor, represented as αt = WYt Nt t . For a cost-minimizing firm, the marginal revenue product of labor equates Yt to the wage Wt = εE,t N t , and εE,t = αt under perfect competition. We can now write out the sectoral parts by substituting the numerator as the sum of the sectoral wage bills i∈I Wi,t Ni,t αt = (30) Yt multiply and divide by sectoral output: i∈ I Wi,t Ni,t Yi,t αt = (31) Yt Yi,t 10 An important point to note here is that TFP in this setup is closely related to TFPQ, which focuses on the physical quantity of output produced per unit of input, while TFPR considers the revenue generated per unit of input, which includes price variations in the market. Essentially, TFPQ reflects pure production efficiency while TFPR incorporates both production efficiency and price effects via adjustments in the labor share for markups. See Hall, 2018 for a discussion on the bias of estimating productivity without accounting for markups in estimating the output elasticity. 15 Simplify YA,t YI,t YS,t αt = αA + αI + αS (32) Yt Yt Yt In the case where price equals a markup (µi ) over marginal costs (εE,i = αi /µi ) αi Yi,t αt = (33) i∈ I µi Ytf cst where µi = βi,1 , the sectoral mark-up estimated from equation (20). Declining labor shares as functions of markups, substitutes and relative factor costs Output from each sector is either a complement or a substitute in the consumption of that good: 1 ρ Y f cst = [ωA YA + ωI YIρ + ωS YSρ ρ ] (34) The demand for goods in each sector is simply inversely related to prices: σ ∂Y 1−ρ P ρ = [ ω A YA + ωI YIρ + ωS YSρ ρ ] ωi ρYiρ−1 ⇒ Yi = ωi σ Y f cst (35) ∂Yi Pi which if we plug into our aggregate labor share equation we have σ αi σ P αt = ωi (36) i∈ I µi Pi Aggregate labor shares decline when (i) sectoral markups increase, and (ii) relative price differentials exist with σ ̸= 0. While price stickiness can temporarily cause shifts in labor share, such effects dissipate as prices across sectors eventually equalize. According to Equation (36), aggregate labor share movements are primarily driven by three factors: changes in the sectoral composition of the economy, variations in markups, and capital deepening within sectors. The observed decline in labor shares is a well-established 16 trend in the global economy, but several explanations for the change in labor shares fall outside the scope of this paper (see for Bergholt et al., 2022,Grossman and Oberfield, 2022). 3 Application to two countries: Ghana and the Kyrgyz Re- public We selected two countries, Ghana (GHA) and the Kyrgyz Republic (KGZ), with different levels of data availability to illustrate the application of sectoral extensions. For Ghana, the available data comprise sectoral employment, a single year of input-output data, and stan- dard national accounts. In contrast, for the Kyrgyz Republic, we have more comprehensive data, including sectoral investment and wage information in addition to the data available for Ghana. We begin by outlining the data preparation process for each model. Following this, we compare the effects of an agricultural drought and a risk premium shock within the sectoral framework against those observed in the standard MFMod framework. 3.1 Constructing data An important part of this work is the data generation process. The framework described in this paper is hard to apply on countries with limited data at the sectoral level. We describe how our data are constructed and which simplifying assumptions are made. The modeling requires times series data for sectoral value-added, sectoral employment, sectoral capital stocks and the revenue share of labor. These data do not exist across all countries. Most countries will have data for sectoral value-added and sectoral employment (mostly using ILO estimates). In the absence of sectoral wage bill data, we approximate the revenue share from input-output tables using GTAP, with the assumption that these shares are constant over time. This allows one to construct an average wage per sector. We estimate a proxy for time series sectoral nominal wages the constant revenue elasticity (αi ): f cst f cst Pi,t Yi,t Wi,t = αi Ni,t For capital, we first define aggregate cost of capital using aggregate labor share 17 Ptf cst Ytf cst Rt = (1 − α) Kt−1 and define the aggregate risk premium as the difference between the marginal product of capital and the elements of the cost of capital which includes taxes, short run interest rates, expected inflation and the capital depreciation rate: Rt p Γt = I (1 − τ CIT ) − (δ + Rt − πt ) Pt where τ CIT is the corporate income tax rate. If we do not have sectoral investment flows (as in the case of GHA), we assume that the cost of capital and risk premia are the same across sectors and equal to the aggregate estimate. Ri,t = Rt ; Γi,t = Γt Finally we can back out sectoral capital stocks Ptf cst Ytf cst Ki,t−1 = (1 − αi ) Ri,t If we have investment by sector (KGZ case), we build sectoral capital stocks using perpet- ual inventory method (under the assumption that sectoral depreciation rates are the same) and get sectoral costs of capital. Ptf cst Ytf cst Ri,t = (1 − αi ) Ki,t−1 An important data construction step is the estimation of the elasticity of substitution between different labor and capital types and the initial shares of labor and capital types in aggregate labor and capital. Estimation of elasticities of substitution is possible if we have 18 data for sectoral factor costs.11 For KGZ, we estimate σ N = 1.5 and σ K = 0.8 over 1999-2022. For GHA, we calibrate those parameters (σ N = 1.2 and σ K = 0.8) and do sensitivity analysis to capture possible range of results conditional on different elasticities of substitution.12 Using sectoral employ- ment, wage rates, capital stock, costs of capital and elasticities of substitution, we define time series of CET shares of employment and CES shares capital which are plotted for GHA and KGZ in Figure 2. Sector shares are mostly constant for GHA, but a strong decline in KGZ agriculture shares is observed. Figure 2: Sectoral CET shares of employment and CES shares of capital The sectoral TFP’s are derived as a residual (equation 23). Aggregate TFP is the weighted sum of the within TFPs. The shifts in the weights are the between effects. The weighted sectoral TFPs that sum up to total TFP for GHA and KGZ are displayed in Figure 3 in the top panel.13 The contribution of services to aggregate TFP is the larges in both KGZ and GHA. The weighted agricultural contribution to TFP shows a strong decline over time for KGZ while industry and service’s contributions are flat This is in spite of rising unweighted agricultral TFP rising (bottom panel rising). Take note that from 2000 to 2023, KGZ agriculture’s share in GDP declined from 49% to 33%. The employment share also declined from 53% to 18%. During this period, agriculture share in nominal investment fluctuated between 1% and 4% - implying a decline in agriculture capital stock as well. This clearly shows that the weights, which are functions of labor and capital movements over time, can generate large shifts in how aggregate TFP is measured. 11 We estimate a joint-system where the first-order conditions for labor and capital types are linearized. 12 As we explain further below, initial shares will depend on elasticities of substitution. 13 Note that we decompose aggregate TFP to sectoral sources, i.e., there are two factors affecting the levels in the charts: Sectoral TFPs and sectoral factor shares. 19 For GHA, aggregate TFP has increased in the service and agriculture sectors, while the weighted industry TFP leveling off beginning in 2018. Agriculture GDP as a share of total GDP decreased from about 28% in 2006 to 22% in 2023. At the same time, agricultural employment as a share of total employment decreased by 15 percentage points. Services GDP share in total increase by 4 percentage points from 2006 to 2023, while employment shares increased by about 10 percentage points. Figure 3: Decomposition of aggregate TFP to sectoral sources We use Equation 28 to plot the decomposition of aggregate TFP in the economy. A few interesting features emerge from these decompositions for KGZ - the first is that most of what explains negative TFP growth since 2010 is primarily attributable to between-effects for KGZ. These between-effects are dominated by shifts in labor reallocation. For GHA, we observe an increase in aggregate TFP growth that is primarily driven by an increase in the within component. The between component has also contributed to positive TFP growth, mainly driven by labor reallocation. 3.2 Agricultural drought shocks We compare the sectoral-enhanced version of the model with the World Bank’s standard MFMod, which lacks the labor and capital sectoral allocation mechanisms. The first shock introduced is a permanent reduction in agricultural crop yields due to droughts, which reduces agricultural productivity by 10% from 2024 to 2050. In the standard model, this shock is applied to aggregate TFP, weighted by the share of the agriculture sector in total gross value-added, and simultaneously to agricultural output. agr Y2023 A′t = At 1 − ∆agr f cst Y2023 20 Figure 4: KGZ TFP decomposition Figure 5: GHA TFP decomposition 21 where A′t is aggregate TFP in the scenario, At is TFP in the baseline. ∆agr is the shock, Y agr which is the percent of agricultural output loss relative to the baseline and Y 2023 f cst is the 2023 share of agriculture output in total value-added. Note that the shock is relative to 2023. A key limitation of this approach is that the share of crops in agriculture remain unchanged over the simulation period. This description of the model is acceptable under certain conditions. However, once we allow for the elasticity of transformation between different labor types to vary, we generate responses that are different to the standard model. Implementing the shock in the sectoral version of the model is more straightforward, as we have agricultural TFP as a distinct variable. We can directly apply the shock to obtain the impulse responses: A′agr,t = (1 − ∆agr )Aagr,t The transmission mechanism in the sectoral model differs from the standard model. In the standard model without sectoral factors of production, a reduction in TFP leads to an aggregate decrease in the firm’s wage but still an overall increase in total marginal costs, as wages are weighted by output elasticity. While this mechanism also exists in the sectoral model, the net wage effect now depends on sectoral wage responses. A reduction in agricultural TFP lowers agricultural labor productivity, leading to a labor outflow of agriculture compared to other sectors. Depending on the initial conditions and how sectoral shares in output evolve over time, the net effect on aggregate TFP and employment may differ from the standard model. The impact on agriculture employment, is however, sensitive to the demand for agricultural goods. If demand is inelastic then it is possible that a negative shock to agriculture productivity generates an increase in employment relative to the baseline (which may already be declining).14 14 Wi,t Y The MPL is Pi,t = αi Ni,t i,t , which simply states that the real wage must increase with a rise in labor Pi −σ productivity. However, note that demand for good i is just Yi = ω −σ P Y . Plugging this into the MPL Pi −σ Wi,t ω −σ P Y equation yields Pi,t = αi Ni,t . Thus if demand is inelastic (σ = 0) then the MPL may rise when a negative TFP hits. According to the CET function for labor allocation, this will then induce an increase in labor to that sector. 22 Figure 6: Sectoral wages, employment, potential output and prices in response to 10% agricultural drought shock in sectoral MFMod - KGZ 3.2.1 KGZ Figure 6 illustrates the sectoral decomposition of the responses of the sectoral model for KGZ to an agricultural drought shock. When the demand for agricultural goods is elastic, the shock leads to a decline in agricultural employment, driven by a drop in agricultural wages relative to other sectors. As agricultural potential falls, so does aggregate potential. Rising agricultural prices push up industry prices due to increased wage costs. In contrast, service prices decline, partially offsetting the overall increase in prices across the economy These responses assume that the demand for agricultural goods are elastic to prices. If we make demand inelastic, then the demand curve becomes more vertical. As shown in Figure 7, a reduction in TFP results in a reduction in the goods supply curve, but now instead of a reduction in employment flowing out of agriculture we have an increase. We still have lower output in agriculture, but less so than when demand is elastic. Note that aggregate TFP and GDP responses are less negative when agriculture demand is inelastic as labor inflows to the sector with the highest level of TFP and reducing the negative shock. Figure 8 presents the responses of the standard and sectoral models. In the standard model, aggregate TFP experiences a permanent decline of approximately 3.5%. In the sectoral model, aggregate TFP declines by 4.5% initially and then converges to 3.5%. This is due to labor moving out of agriculture. The TFP shock affects both potential and actual GDP. Inflation is more pronounced in the sectoral model, where the higher labor costs lead to higher marginal costs. In the standard model, nominal wages fall in line with the reduction in potential output more than the increase in inflation. However, in the sectoral model, aggregate nominal wages initially rise due to labor mobility toward the industry and services sectors (these sectors have rising wages after the shock). Over time, wages moderate as sectoral potentials go 23 Figure 7: KGZ agriculture TFP shock when demand is elastic vs. inelastic down. Despite attracting labor inflows, industry and service outputs contract due to the overall decline in aggregate potential, which reduces investment and capital in each sector. Aggregate employment falls more sharply in the sectoral model compared to the standard model. 3.2.2 Ghana Figure 9 presents the responses of the standard and sectoral models for Ghana (GHA). Unlike KGZ, the sectoral model for GHA predicts a smaller contraction compared to the standard model. This outcome is primarily due to the labor flow from agriculture to ser- vices, which increases service sector potential more than it decreases agricultural potential. This dynamic drives aggregate potential GDP and, consequently, the TFP response. Here structural transformation moves labor to a more productive sector and hence dampens the negative agriculture shock on the economy. Additionally, industry and service outputs rise as the inflow of labor outweighs the decline in capital, which is relatively limited, as shown by the modest potential output response. 3.3 Industry cost of capital shock This section presents the investment responses of the model to cost of capital shocks. We introduce a permanent 1 percentage-point increase to the industry sector’s cost of capital 24 Figure 8: 10% agricultural drought shock to the standard and sectoral MFMod - KGZ (e.g., sector risks or regulatory costs). In the standard model, we apply the shock to the aggregate risk cost of capital and scale it according to the share of industry capital within the total capital stock, as derived from the sectoral model. In the standard model, this shock increases the aggregate cost of capital, leading to a re- duction in aggregate investment. The decrease in investment gradually diminishes aggregate capital stock and, consequently, potential output. As investment demand declines, GDP also falls. Additionally, the higher cost of capital pushes inflation upward. The sectoral responses are primarily driven by the overall contraction in GDP. In the sectoral model, the shock directly affects the capital stock in the industry sector, as depicted in Figure 10. This induces a decline in investment in the industry sector and reduced potential output. As capital re-allocates from industry to the other sectors the marginal product of capital in those sectors declines. The increase in potential in those sectors (due to increased capital) increases aggregate potential initially but does not fully offset the declines in the industry sector so overall potential GDP is lower in the long run. Industry prices rise, while agriculture prices initially increase but subsequently decline as the cost of capital adjusts. Service prices adjust to balance value-added accounts to aggregate price dynamics. As shown in Figure 11, the sectoral model forecasts a lower aggregate impact for KGZ as capital reallocates from industry to more productive sectors, such as services and agriculture. This reallocation leads to a stronger TFP response and only a modest decline 25 Figure 9: 10% agricultural drought shock to the standard and sectoral MFMod - GHA in GDP, following an initial increase due to the rapid movement of capital. In the case of GHA, the results are the opposite: the sectoral model predicts a sharper rise in the cost of capital, leading to a more pronounced contraction in the capital stock. Although the reallocation of capital toward agriculture and services initially boosts their potential output and softens the initial decline in TFP, over time, the drop in industry potential dominates the aggregate response. This occurs because the capital shares of agriculture and services are relatively low, and the TFP differences across sectors are small, meaning that incoming capital does not generate the same aggregate potential increase as observed in KGZ. The divergence between GHA and KGZ is driven by the relative TFP levels across sectors. In KGZ, the industry’s TFP is so low that it outweighs labor share dynamics, which are more influential in GHA, where TFP differences between sectors are comparatively smaller. 3.4 Interesting results based on the sensitivity of elasticities of transfor- mation and initial shares Our sensitivity analysis uses different values for the elasticity of transformation for labor and the elasticity of substitution for capital in both KGZ and GHA. Overall, as the elasticities decrease, the results of the sectoral model converge with those of the standard model, re- gardless of the country and the shock. This outcome is anticipated, as the standard model does not allow for substitution between sectors. The standard treatment without elasticities 26 Figure 10: Sectoral cost of capital, capital, potential output and prices in response to a 1 percentage-point shock to industry risk premium in sectoral MFMod - KGZ of substitution and transformation of factors across sectors will by design mean that the between contribution to growth is zero. This analysis also underscores the flexibility of the sectoral model, which not only includes the standard model as a special case but also permits a wide range of outcomes depending on the elasticity values. The standard treatment misses the influence of structural change on overall productivity growth in the medium to long-term. 3.4.1 Sectoral TFP shocks We run two additional sets of simulations with σ N = 1.01 (Cobb-Douglas) and σ N = 0.01 (complements). The key finding from Figures 13 and 14 is that higher elasticities can either mitigate or amplify the initial shock, depending on the sectoral productivity differentials of the country, particularly the levels of sectoral TFPs at the time when the shock materializes. While a higher elasticity generally amplifies the impact of a drought shock on the agri- cultural sector, the effect on aggregate potential and TFP varies. In KGZ, this leads to more pronounced macroeconomic impacts. In GHA, labor mobility from agriculture to services increases with higher elasticities of transformation. As service potential increases, so do aggregate potential and TFP, compared to the initial decline. A second simulation includes a 10% reduction in service TFP. As demonstrated in Figures 27 Figure 11: 1 percentage points industry risk premium shock to the standard and sectoral MFMod - KGZ Figure 12: 1 percentage points industry risk premium shock to the standard and sectoral MFMod - GHA 15 and 16, the results are the exact opposite of the drought shock. We observe similar changes in sectoral responses to different elasticities in both countries: As the elasticity of transformation rises, employment flows out of services, which increases the negative effect on services. At the same time, a higher elasticity results in a more pronounced increase in agriculture, as employment shifts to agriculture. In KGZ, this helps to moderate the shock’s 28 Figure 13: Sensitivity analysis with elasticities of substitution for labor of 1.5, 1.01 and 0.01 - KGZ impact, while in GHA, this ultimately leads to a deeper contraction in both GDP and TFP. This again underscores the importance of relative TFP levels at the time of the shock. It is important to note that these impacts are closely tied to the employment shares of sectors in each economy. Since the employment share of the service sector is higher in KGZ than in GHA, the elasticity of transformation plays a more significant role in KGZ. 3.4.2 Sectoral risk premium shocks We also test the sensitivity of risk premium shocks to different values of the elasticity of substitution for capital, with σ K = 1.5 and σ K = 0.01. The simulation results are displayed in Figures 17 and 18. The outcomes for KGZ and GHA shift in different directions as capital becomes more responsive to relative costs. In both cases, a higher elasticity leads to a deeper contraction in the industry sector due to increased capital outflows. Simultaneously, capital inflows into agriculture and services boost their potential outputs. In KGZ, where the initial TFP differentials are large, this increase in agriculture and services outweighs the decline in industry potential. The total decline in capital stock is most significant when the elasticity is lowest, as the rise in industry capital costs directly translates into a higher 29 Figure 14: Sensitivity analysis with elasticities of substitution for labor of 1.5, 1.01 and 0.01 - GHA aggregate cost of capital. In scenarios with higher substitution elasticity, the production structure becomes more flexible, absorbing the initial shock by reallocating capital to lower- cost sectors, thereby resulting in a smaller rise in aggregate capital costs. In GHA, however, due to the low labor share in industry, the opposite occurs: TFP declines, and the contraction deepens as elasticity increases. While these simulations illustrate how factor reallocates due to sector-specific shocks, one may add several additional rigidities to the model. An important one is the degree of rigidity observed in capital reallocation. Capital, once installed, is perhaps very sticky and costly to reallocate, while new capital will have a higher elasticity of substitution. Some CGE models (e.g., MANAGE) treats vintage capital allocation with low elasticities of substitution. In the end, the degree of capital reallocation in the model could thus be more subdued, which is ultimately an empirical question. 30 Figure 15: Sensitivity analysis for SRV shock with elasticities of substitution for labor of 1.5, 1.01 and 0.01 - KGZ 3.5 Initial shares Another important implication of the elasticities of transformation and substitution relates to the share parameters. In Figure 19, we plot the CET share parameter for agriculture in KGZ, conditional on different values for labor elasticity. As the elasticity decreases, the historical CET share of agriculture tends to decrease. This trend also affects the results shown in Figure 13. We observe not only lower mobility of labor in the simulation period under a low elasticity of transformation but also a lower historical preference for agricultural employment. Since we assume that CET shares remain constant at their last observed values over the forecast horizon, this implies that we will have lower employment in agriculture, which is less mobile. 15 Consequently, an adverse shock to agriculture will be more significantly transmitted to the overall economy. 15 Alternative to CET estimates is a variable elasticity of transformation that changes with the shares over time. 31 Figure 16: Sensitivity analysis for SRV shock with elasticities of substitution for labor of 1.5, 1.01 and 0.01 - GHA 32 Figure 17: Sensitivity analysis with elasticities of substitution for capital of 1.5, 0.8 and 0.01 33 Figure 18: Sensitivity analysis with elasticities of substitution for capital of 1.5, 0.8 and 0.01 34 Figure 19: Agriculture CET employment shares with elasticities of substitution for labor of 1.5, 1.01 and 0.01 35 4 Concluding remarks and the way forward This paper presents a straightforward approach to incorporate structural change into a stan- dard macroeconomic model. Integrating sectoral details into a macroeconomic framework offers several benefits, including a deeper analysis of aggregate Total Factor Productivity (TFP) through shifts in factor allocation both between and within sectors. The framework can also be used to examine changes in aggregate labor shares resulting from sectoral markup dispersion, as well as the effects of sectoral output shifts and variations in factor elasticities of transformation. The endogenous structural change framework allows researchers to trace how the between- effects of TFP change with respect to both labor and capital allocations. These between shifts are driven by relative prices, within productivity differentials and preferences. The extent to which structural change materializes depends on the wedges in the economy, and the sensitivity of labor and capital supply to relative price shifts. Positive TFP contributions to structural change is limited if labor and capital across sectors are not substitutable. We compare the impact of sectoral shocks in two models: one where sectoral factors of production are explicitly endogenous and another where these distinctions are absent in an aggregate model (i.e., exogenous aggregate TFP). Our analysis focuses on the sensitivity of macroeconomic variables to sectoral shocks, revealing that under certain conditions, the standard model without sectoral distinctions can be seen as a special case of the sectoral variant. The factor elasticities of transformation play a pivotal role in illustrating how standard climate and macroeconomic shocks propagate through the economy, sometimes in meaningful, but unintuitive ways. Several extensions to the existing framework may include endogenizing the within-component to research and development, learning by doing and catching up to the frontier. 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