Structural changes and growth regimes

Abstract

We study the relation between income distribution and growth, mediated by structural changes on the demand and supply sides. Using the results from a multi-sector growth model, we compare two growth regimes that differ in three aspects: labour relations, competition and consumption patterns. Regime one, similar to Fordism, is assumed to be relatively less unequal, more competitive and to have more homogeneous consumers than regime two, which is similar to post-Fordism. We analyse the parameters that define the two regimes to study the role of the economy’s exogenous institutional features and endogenous structural features on output growth, income distribution, and their relation. We find that regime one exhibits significantly lower inequality, higher output and productivity and lower unemployment compared to regime two, and that both institutional and structural features explain these differences. Most prominent amongst the first group are wage differences, accompanied by capital income and the distribution of bonuses to top managers. The concentration of production magnifies the effect of wage differences on income distribution and output growth, suggesting the relevance of competition norms. Amongst structural determinants, firm organisation and the structure of demand are particularly relevant. The way that final demand is distributed across sectors influences competition and overall market concentration; demand from the least wealthy classes is especially important. We show also the tight linking between institutional and structural determinants. Based on this linking, we conclude by discussing a number of policy implications that emerge from our model.

Appendices

Appendix A: Remaining components of the model

In this appendix we report on the components of the model not described in the main text, which only discusses the elements of the model most relevant to the definition of growth regimes. The components described below are important for an understanding of the functioning and outcomes of the model, although they are not connected directly to the discussion on growth regimes.

1.1 A.1 Macroeconomic dynamics

The macroeconomic dynamics of the model is the result of aggregating microeconomic behaviour except the minimum wage which depends on aggregate changes in unemployment, productivity and inflation. Therefore, wages link the supply and demand sides of the model via income distribution, and mediate the feedback between the macro- and micro-dynamics. The remainder of this section presents the computations for the main macroeconomic variables and how they define the minimum wage.

1.1.1 A.1.1 Aggregate unemployment

We estimate the level of unemployment using the well-established Beveridge Curve, the negative relation between the rate of unemployment and the rate of vacancies, which are endogenously determined at firm level in our model. In this respect, without explicitly modelling the dynamics of the labour market, we assume that it mimics a matching model (Petrongolo and Pissarides 2001; Yashiv 2007). We adopt a hyperbolic form for the Beveridge curve as estimated in Börsch-Supan (1991):

$$ u(t) = 1+\frac{\beta}{v(t)+{\Upsilon}} $$

(24)

where u(t) is the unemployment rate at time t, ϒ is a constant, β defines the relation between the vacancy rate v(t) and unemployment.³⁶

For every tier of worker i in every firm k ∈ {f,g}, we estimate the number of vacancies V_i,k(t). We assume that the vacancies in a given tier i are proportional to the vacancies on the shop-floor:

$$ V_{i,k}(t)=\nu^{1-i}V_{1,k}(t) $$

(25)

Therefore, the total number of vacancies for firm k can be expressed as a multiple of the vacancies for first-tier workers:

$$ V_{k}(t)=\sum\limits_{i = 1}^{{\Lambda}_{k}(t)}\nu^{1-i}V_{1,k}(t) $$

(26)

The vacancies at the shop-floor level are computed as the difference between the number of shop-floor workers required to produce the planned output, and the number of workers hired (matched). Formally, for the final good sectors and the capital good sector respectively:

$$ \begin{array}{l} V_{1,n,f}(t)=max\left\{0\text{ ; }(1+\upsilon)\frac{min\{{Q^{d}_{f}}(t);\bar{B}K_{f}(t-1)\}}{A_{f}(t-1)}- L_{1,n,f}(t-1)\right\}\\ V_{1,g}(t)=max\left\{0\text{ ; }(1+\upsilon_{g})\frac{{K^{d}_{g}}(t)}{A_{f}(t-1)}- L_{1,g}(t-1)\right\} \end{array} $$

(27)

The mismatch between firms’ labour demand and hiring depends on the parameter 𝜖 in Eq. 3.³⁷ and is due to the assumed frictions in the labour market, which are equal in both regimes

The vacancy rate for firm k then is the ratio between vacancies and the overall labour demand:

$$ v_{n,k}(t)=\frac{V_{k}(t)}{L_{k}(t)+V_{k}(t)} $$

(28)

The vacancy rate for the whole economy is computed as the average of firms’ vacancy rates weighted by their contribution to total employment (L(t)):

$$L(t)=\sum\limits_{n = 1}^N\sum\limits_{f = 1}^{F(t)}\sum\limits_{i = 1}^{{\Lambda}_f(t)}L_{i,f}(t-1)+\sum\limits_{g = 1}^G\left( L_g(t)+L_{0,g}(t)\right) $$

$$ v(t)=\sum\limits_{n = 1}^{N}\sum\limits_{f = 1}^{F}\frac{L_{n,f}(t)}{L(t)}v_{n,f}(t)+\sum\limits_{g = 1}^{G}\frac{L_{g}(t)}{L(t)}v_{g}(t) $$

(29)

1.1.2 A.1.2 Aggregate consumption

The selection procedure described in Section 2.2.2 is replicated H times per consumer class, representing a distribution of H random draws of perceived price and quality. To establish the aggregate expenditures directed at firm f in sector n, we sum the H replicates for selected firm f,n in the subset of selected goods:

$$ y_{n,f}=X_{n}(t) \frac{h_{n,f}}{H} $$

(30)

where X_n(t) are the consumer class n expenditures and h_n,f are the number of times that the selection procedure selected the product of firm f. Finally, the number of units sold is derived by dividing the revenue by the unit price:

$$ x_{n,f}=\frac{y_{n,f}}{p_{f}} $$

(31)

Because consumers and firms are partially myopic, there is a mismatch between the quantity demanded and the quantity produced. See Section A.3.1.

1.1.3 A.1.3 GDP, and total employment

Nominal GDP is the sum of the value of sales across sectors and firms, corresponding to final and intermediate goods:

$$ Y(t)=\sum\limits_{n = 1}^{N}\sum\limits_{f = 1}^{F(t)}p_{f}(t-1)Y_{f}(t)+\sum\limits_{g = 1}^{G}p_{g}(t-1)K_{g}(t) $$

(32)

where p_f(t − 1) and p_g(t − 1) are defined in Eqs. 20 and 52; Y_f(t) = min {Y_f(t); Q_f(t)}, respectively Eqs. 36 and 37; and K_g(t) is defined in Eq. 47.

Total employment is the sum of workers employed in all the tiers of all the firms in all sectors:

$$L(t)=\sum\limits_{n = 1}^N\sum\limits_{f = 1}^{F(t)}\sum\limits_{i = 1}^{{\Lambda}_f(t)}L_{i,f}(t-1)+\sum\limits_{g = 1}^G\left( L_g(t)+L_{0,g}(t)\right) $$

1.2 A.2 Consumer classes

1.2.1 A.2.1 Savings and rents

The level of savings (S_i(t)) of a class is the income that is left to that consumer class after expenditure:

$$ S_{i}(t)=D_{i}(t)-X_{i}(t)+D_{i}^{-}(t)=(1-(1-\gamma)(1-s_{i}))D_{i}(t)-\gamma X_{i}(t-1) + D_{i}^{-}(t) $$

(33)

where $\phantom {\dot {i}\!}D_{i}^{-}(t)$ are the returns from past demand that could not be met by firms (see Section A.3.1 for more details).

Households’ savings are invested in the financial sector in the form of a financial title – a “token” – issued by the financial sector, which provides access to future dividends. For simplicity, we do not consider any other transactions on the financial market (tokens cannot be traded amongst consumers or firms). Thus, the number of financial tokens pertaining to class i, U_i(t), is computed as:

$$ U_{i}(t)=U_{i}(t-1)+\frac{S_{i}(t)}{P_{u}(t)} $$

(34)

where P_u(t) is the current price of the token (See Eq 57).

In line with recent empirical evidence (Dynan et al. 2004), we assume that the saving rate s_i increases with income. Considering that classes are indexed according to increasing levels of income, the desired saving rate of two adjacent classes can be expressed as:³⁸

$$ s_{i}=s_{i-1}\left( 1-\sigma\right) + \sigma $$

(35)

where σ is the rate growth of savings from class i to the next one.

1.3 A.3 Final good firms

1.3.1 A.3.1 Output

The total demand of a final good firm is the sum of expenditures over all bootstraps over all classes, following the selection algorithm described in Section 2.2 and aggregated in Section A.1.2. If the demand exceeds a firm’s supply, the total units sold Y_f(t) correspond to its current production Q_f(t):

$$ Y_{f}(t)=\min\left\{\frac{1}{p_{f}(t-1)}\sum\limits_{z = 1}^{{\Lambda}_{t}}\sum\limits_{m = 1}^{H_{n},z}y_{f_{n},z,m,t}\frac{X_{z,t}}{H} ; Q_{f}(t) \right\} $$

(36)

where p_f(t − 1) is the price charged by the firm at time t.

In the short-run, firms produce using a fixed coefficient technology. The level of output produced Q_f(t) is constrained by the availability of production factors:

$$ Q_{f}(t)=\min\left\{{Q^{d}_{f}}(t); A_{f}(t-1)L_{1,f}(t-1);\bar{B}K_{f}(t-1)\right\} $$

(37)

where A_f(t − 1) is the level of labour productivity L_1,f(t − 1) embodied in the firms’ capital stock K_f(t − 1), and $\phantom {\dot {i}\!}\frac {1}{\bar {B}}$ is a constant capital stock intensity.³⁹

Firms decide on a desired output level $\phantom {\dot {i}\!}{Q^{d}_{f}}(t)$ to match their expectations about sales $\phantom {\dot {i}\!}{Y^{e}_{f}}(t)$, which are formed on the basis of past inventories (I_f(t − 1) > 0) or unfulfilled orders (I_f(t − 1) < 0):

$$ {Q^{d}_{f}}(t)=\left( 1+\phi\right){Y^{e}_{f}}(t)-I_{f}(t-1) $$

(38)

In order to cover unexpected changes in demand, firms maintain an inventory level $\phantom {\dot {i}\!}\phi Y_{f,t}^{e}$ – where ϕ is a fixed ratio. Firms form their sales expectations ($\phantom {\dot {i}\!}Y^{e}_{f,t}$) in an adaptive way to smooth short term volatility

$$ {Y^{e}_{f}}(t)=\alpha {Y^{e}_{f}}(t-1)+(1-\alpha)Y_{f}(t-1) $$

(39)

where (1 − α) is the rate at which expectations on demand converge to the current value of demand, and Y_f is total demand.

The difference between planned production $\phantom {\dot {i}\!}{Q^{d}_{f}}(t)$ and actual output Q_f,t determines the inventory level I_f(t − 1):

$$ I_{f}(t)={Q^{d}_{f}}(t)-Q_{f}(t) $$

(40)

When demand exceeds output, firms increase the value of backlogs (negative inventories) to be fulfilled with future output and increase the mark-up. Consumer classes failing to access demanded goods because of insufficient production retain their unspent money as forced savings whilst waiting for a delivery in the future. These resources are employed either as extra-consumption when the firm is able to fulfil the order or remain as permanent savings in the case the firm cannot fulfil the order. In other words, we assume that at each time step, backlogs are either fulfilled – delivering past unfulfilled sales – or reduced by a fixed ratio – representing orders cancelled by consumers. The value of cancelled goods is returned to the consumer class that purchased them in the past, contributing to its saving and, therefore, its future consumption.⁴⁰

Assuming that consumers prefer to buy goods from firms that can deliver immediately, backlogs negatively affect firms’ visibility ($\hat {\upsilon }_{f,t}$). Visibility is computed as a moving average of the ratio of the difference between expected sales and backlogs, and expected sales:

$$ \hat{\upsilon}_{f}(t)=\hat{\upsilon}_{f}(t-1) \alpha_{\hat{\upsilon}} + \frac{max\{{Y^{e}_{f}}(t) - BL_{f}(t),0.001\}}{{Y^{e}_{f}}(t)} (1-\alpha_{\hat{\upsilon}}) $$

(41)

where $\phantom {\dot {i}\!}\alpha _{\hat {\upsilon }}$ is the pace at which visibility adapts through time.

1.3.2 A.3.2 Production capacity and productivity

Following Amendola and Gaffard (1998) and Llerena and Lorentz (2004), the accumulation of capital stock is a pre-condition for producing, and a determinant of labour productivity. A firm’s f capital stock K_f(t) is the sum of capital vintages k_f,g(τ) purchased from capital good firm g in time τ and cumulated through time:

$$ K_{f}(t)=\sum\limits_{\tau = 1}^{t} k_{f,g}(\tau)(1-\delta)^{t-\tau} $$

(42)

where δ is the depreciation rate. The level of productivity embodied in the capital stock is computed as the average productivity across all the vintages available:

$$ A_{f}(t)=\frac{1}{K_{f}(t)}\sum\limits_{\tau = 1}^{t} k_{f,g}(\tau)(1-\delta)^{t-\tau}a_{g}(\tau) $$

(43)

where a_g(τ) is the productivity embodied in the h vintage.⁴¹

1.3.3 A.3.3 Investment in capital stock

Firms’ investment in a new a vintage ($\phantom {\dot {i}\!}{k^{d}_{f}}(t)$) is a function of expected sales $\phantom {\dot {i}\!}{Y^{e}_{f}}(t)$, the level of production capacity given the capital stock and labour force currently available, respectively $\phantom {\dot {i}\!}{Y^{K}_{f}}(t)$ and $\phantom {\dot {i}\!}{Y^{L}_{f}}(t)$, and the current amount of backlog sales, BL_f(t).

$$ {k_{f}^{d}}(t)=max \{ min\{{Y^{L}_{f}}(t) \alpha_{k}; \left( {Y^{e}_{f}}(t)+ BL_{f}(t) \beta_{k} \right) (1+\upsilon) - {Y^{K}_{f}}(t)\}; 0 \} \bar{B} $$

(44)

where α_k is a multiplier expanding the increased capital stock to a multiple of the available labour force, in order to avoid capital stock bottlenecks in the short period (in line with the assumption that capital stock investment is lumpy); β_k is a coefficient indicating a share of the backlog sales that the firms would like to absorb with the new investment; υ is the share of desired unused (capital stock) capacity; $\phantom {\dot {i}\!}\bar {B}$ is the intensity of capital stock, translating production into units of capital.

All capital investment is financed with loans, without discriminating amongst firms (in our model, selection is done by consumers).⁴² The financial institution grants the loan to any firm with a probability proportional to the ratio between the cash available in the institution (Γ(t)) and the total value of the resources in the financial sector (Θ(t)) (see Eqs. 54 and 55). Rejected loans are resubmitted in the following time steps until they are accepted.⁴³

When investing in a new vintage, firms f select one of the capital good producers g ∈{1; ...; G} and place an order $\phantom {\dot {i}\!}k_{g,f}^{d}(t)$ for the desired amount of capital goods. A capital good producer is selected with a probability that depends positively on the vintage’s productivity a_g(t − 1), and negatively on its price p_g(t − 1) and on g’s delivery time. Hence, capital good producers with big order books may be rejected, despite producing the best capital vintage, because delays in acquiring a new vintage may cause large losses for f.

After a capital producing firm receives an order, it places it in its order book, using its production capacity to complete all the orders according to the sequence in which they arrive.

1.4 A.4 Capital good firms

The capital good sector is populated by g ∈{1, 2,…,G} capital suppliers that produce one type of capital good with an embodied productivity a_g(t). Firms in the capital good sector can sell to firms from any of the final good sectors on receipt of an order $\phantom {\dot {i}\!}k_{f,g}^{d}(t)$. Capital goods are produced on a first in, first out basis and the time needed to produce each of them depends on the firm’s capacity and the number of orders.

1.4.1 A.4.1 Production

We assume that the production of capital goods is just-in-time, with no expectation formation or accumulation of inventories. The total demand $\phantom {\dot {i}\!}{K^{d}_{g}}(t)$ for a capital supplier g at t is the sum of the current order and earlier unfinished orders (I_g(t − 1)):

$$ {K^{d}_{g}}(t) = \sum\limits_{f = 1}^{F(t)} k_{g,f}^{d}(t) + I_{g}(t-1) $$

(45)

We assume, for simplicity, that capital good firms employ only labour, with constant productivity:

$$ Q_{g}(t)=L_{1,g}(t-1) $$

(46)

where L_1,g(t − 1) are the shop-floor workers. Then, the amount produced is the minimum between a firm’s capacity and demand:

$$ K_{g}(t)=min\{Q_{g}(t); {K^{d}_{g}}(t)\} $$

(47)

and unfinished orders are the difference between current production and the sum of unfinished orders in t − 1:

$$ I_{g}(t)= \sum\limits_{\tau= 1}^{t} {K^{d}_{g}}(\tau) - \sum\limits_{\tau= 1}^{t}K_{g}{\tau} $$

(48)

The total number of workers in a firm can be computed as:

$$ L_{g}(t)={L^{1}_{g}}(t) +... + L^{{\Lambda}_{g}(t)}_{g}(t) = L_{1,g}(t)\sum\limits_{i = 1}^{{\Lambda}_{g}(t)}\nu^{1-i} + \rho_{g} {L^{1}_{g}}(t) $$

(49)

where ρ_g is the share of engineers per shop-floor worker.

1.4.2 A.4.2 Process innovation

Capital good producers improve the productivity embodied in capital vintages a_g(t) by means of their R&D department staffed by L_0,g(t) engineers. The number of engineers is a constant share ρ_g of the total number of the firm’s employees. In the tradition of Schumpeterian growth models (Silverberg and Verspagen 2005), the outcome of R&D is stochastic and the probability of an increase in productivity (Φ_g(t)) depends on the amount of financial resources invested to increase the total number of engineers (L_0,g(t − 1)):

$$ {\Phi}_{g}(t)= 1-e^{-\zeta L_{0,g}(t-1)} $$

(50)

where ζ is the effectiveness of R&D investment.

If the R&D is successful, the productivity of the new capital vintage is drawn randomly from a normal distribution with average a_g(t − 1) and variance σ^a representing the speed of technological change:

$$ a_{g}(t)=a_{g}(t-1)\left( 1+max\{\varepsilon_{g}(t);0\}\right) $$

(51)

where ε_g(t) ∼ N(0; σ^a).

1.4.3 A.4.3 Production costs, pricing, and financial account

Wages follow the same hierarchical structure as firms in the final good sectors (Eq. 6). The wage of engineers working in the R&D department is a multiple ω₀ of the minimum wage.

The price of capital goods p_g(t) is a fixed mark-up $\phantom {\dot {i}\!}\bar {m}_{g}$ over variable costs: shop-floor workers, executives and engineers, divided by the level of output Q_g,t:

$$ p_{g}(t)=(1+\bar{m}_{g})\left( \frac{{\sum}^{{\Lambda}_{g}(t)}_{i = 1}w_{i,g}(t)L_{i,g}(t-1)+w_{0,g}(t)L_{0,g}(t-1)}{Q_{g}(t)} \right) $$

(52)

where w_0,g(t) is the wage of engineers.

Profits are computed as the difference between revenues and labour costs:

$$ {\Pi}_{g}(t) = p_{g}(t)K_{g}(t) - \sum\limits^{{\Lambda}_{g}(t)}_{i = 1}w_{i,g}(t)L_{i,g}(t-1)-w_{0,g}(t)L_{0,g}(t-1) $$

(53)

If profits are positive, a share π is distributed as premia to the firm’s managers in proportion to their share of the payroll (Eq. 9), and a share ρ_g is invested in R&D. The remaining profits (1 − π − ρ_g) are pooled with those from all firms and distributed as dividends to households, in proportion to the number of tokens owned by each class and, therefore, to their cumulated savings.

1.5 A.5 Financial sector

The financial sector is an institution dealing with all the financial aspects of firms and households. The model adopts a very stylised representation of the financial relations amongst the actors. Essentially, consumers (separately for each class) invest their savings purchasing a number of financial tokens that provide access to firms’ profits and, if necessary, can be sold later to sustain their expenditures if these exceed available income. Tokens cannot be traded and their price is unique and endogenously determined at each time step.

Firms requiring credit (to purchase capital goods or to cover losses) access the financial system receiving cash as a sort of infinitely termed loan. In summary, the model represents the financial system as a single, large investment fund with liabilities composed by the tokens owned by consumers and its capital, the financial sector assets composed of cash (savings used to buy tokens), and loans to operating firms.

The price of the tokens varies reflecting the total value of the fund’s capital and the number of tokens in circulation. Finally, firms’ net profits (after bonuses and R&D expenditures) are distributed to the consumer classes proportional to their share (number) of existing tokens, acting, essentially, as dividends contributing to the income of consumers.

The financial institution rests on a fundamental identity: the value of all the financial tokens owned by households must be identical to the value of the capital of the financial institution, composed of the cash provided consumers and the loans to firms.

Formally, the total value of the financial sector is expressed as:⁴⁴

$$ {\Theta}(t)= {\Gamma}(t)+ \sum\limits_{k = 1}^{F+G} \hat{K}_{k}(t) $$

(54)

where k ∈ {f,g}. The value of the stock of cash in the financial sector (Γ(t)) increases with new households’ savings, and decreases with the loans granted to firms:

$$ {\Gamma}(t)={\Gamma}(t-1)+\sum\limits_{i = 1}^{{\Lambda}} S_{i}(t) - \sum\limits_{k = 1}^{F+G}{J^{l}_{k}}(t) $$

(55)

where S_i(t) are consumer class i savings; $\phantom {\dot {i}\!}{J^{l}_{k}}$ is the loan received by firm k ∈ {f,g}. The value of the outstanding loans consists of the sum of all past loans to firms minus the debt owned by firms that went bankrupt and exited the market:

$$ \sum\limits_{k = 1}^{F+G} \hat{K}_{k}(t)= \sum\limits_{k = 1}^{F+G} \hat{K}_{k}(t-1)+ \sum\limits_{k = 1}^{F+G}{J^{l}_{k}}(t) - \sum\limits_{k \in W(t)} \hat{K}_{k}(t) $$

(56)

where W(t) is the set of firms that went out of business at time t. We assume that society bears the cost of bankruptcy.

As shown in Eq. 34, consumer classes use their savings to purchase a unique form of financial title, the tokens (U_i(t)), issued by the financial sector. The price of a token, P_u(t), is determined by the ratio between the total value of the financial sector’s capital Θ(t − 1) and the number of financial tokens owned collectively by households in t − 1:⁴⁵

$$ P_{u}(t)=\frac{{\Theta}(t-1)}{{\sum}_{i = 1}^{{\Lambda}} U_{i}(t-1)} $$

(57)

The dividends received by household class i (E_i(t)) is computed as the share of distributed profits generated by all firms at time t proportional to the share of cumulated savings represented by the share of tokens owned by the class:

$$ E_{i}(t)=(1-\pi-\rho) \sum\limits_{i=f}^{F} {\Pi}_{f} \frac{U_{i}(t)}{{\sum}_{j = 1}^{{\Lambda}(t)} U_{j}} + (1-\pi-\rho_{g}) \sum\limits_{i=g}^{G} {\Pi}_{g} \frac{U_{i}(t)}{{\sum}_{j = 1}^{{\Lambda}(t)} U_{j}} $$

(58)

Appendix B: Indices

We discuss the computation of the indexes used in the results sections.

Atkinson inequality index

Income inequality is measured using the Atkinson index $\phantom {\dot {i}\!}\mathcal {A}_{ind}(t)$ computed as follows:

$$ \mathcal{A}_{ind}(t)= 1-\frac{1}{{\sum}_{i = 1}^{{\Lambda}(t)}\frac{D_{i}(t)}{L(t)}}\left[\frac{1}{L(t)}\sum\limits_{i = 1}^{{\Lambda}(t)}L_{i}(t)\left( \frac{D_{i}(t)}{L_{i}(t)}\right)^{1-\rho}\right]^{\frac{1}{1-\rho}} $$

(59)

where D_i(t) is the total income for consumer class i, L_i(t) is the total number of workers in class i, and ρ is the measure of inequality aversion.

Concentration of output and employment across sectors

We measure the degree of concentration of production in terms of output and employment using an inverse Herfindahl index:

$$\begin{array}{@{}rcl@{}} \mathcal{H}_{Y}(t)=\left[\sum\limits_{n = 1}^{N}\sum\limits_{f}^{F(t)}\left( \frac{p_{n,f}(t-1)Y_{n,f}(t)}{Y(t)}\right)^{2}+{\sum\limits_{g}^{G}}\left( \frac{p_{g}(t-1)K_{g}(t)}{Y(t)}\right)^{2}\right]^{-1} \end{array} $$

(60)

$$\begin{array}{@{}rcl@{}} \mathcal{H}_{L}(t)=\left[\sum\limits_{j}\left( \frac{\mathcal{L}_{j}(t)}{{\sum}_{j = 1}\mathcal{L}_{j}(t)}\right)^{2}\right]^{-1} \end{array} $$

(61)

We measure the degree of concentration in sales in the final good sector using an inverse Herfindahl index:

$$ \mathcal{I}(t)=\left[\sum\limits_{n = 1}^{N}\sum\limits_{f}^{F(t)}\left( \frac{p_{n,f}(t-1)Y_{n,f}(t)}{{\sum}_{n = 1}^{N}{\sum}_{f}^{F(t)}p_{n,f}(t-1)Y_{n,f}(t)}\right)^{2}\right]^{-1} $$

(62)

Value added, output, and employment sectoral shares

We measure the contribution of the value added of each sector to GDP $\phantom {\dot {i}\!}\mathcal {Y}_{j}(t)$, and the respective shares of output $\phantom {\dot {i}\!}\mathcal {Q}_{j}(t)$ and employment $\phantom {\dot {i}\!}\mathcal {L}_{j}(t)$ for each final good sector and the capital good sector j:

$$\begin{array}{@{}rcl@{}} \mathcal{Y}_{j}(t)\,=\,\frac{p_{j}(t-1)Y_{j}(t)}{Y(t)}\text{ with }Y_{j}(t)\,=\,\sum\limits_{f = 1}^{F(t)}Y_{j,f}(t)\forall n\in{1,...N}\text{ or }Y_{j}(t)\,=\,\sum\limits_{g = 1}^{G}K_{g}(t) \end{array} $$

(63)

$$\begin{array}{@{}rcl@{}} \mathcal{L}_{j}(t)\,=\,\frac{L_{j}(t)}{L(t)}\text{ with }L_{j}(t)\,=\,\sum\limits_{f = 1}^{F(t)}L_{j,f}(t)\forall n\in{1,...N}\text{ or }L_{j}(t)\,=\,\sum\limits_{g = 1}^{G}L_{j,f}(t) \end{array} $$

(64)

Capital-labour ratio – degree of mechanisation

We measure the degree of mechanisation of the economy $\phantom {\dot {i}\!}\mathcal {M}(t)$ as follows:

$$\begin{array}{@{}rcl@{}} \mathcal{M}(t)=\frac{{\sum}_{j = 1}^{N}{\sum}_{f = 1}^{F(t)}K_{j,f}(t)}{L(t)} \end{array} $$

(65)

In doing so we consider the changes in the factor composition of the production.

Households’ income composition

To account for changes in the structure of households’ income, we measure the contributions of wages and profits as their respective share of wage income in total income $\phantom {\dot {i}\!}\mathcal {W}(t)$, share of premia in total income $\phantom {\dot {i}\!}\mathcal {P}(t)$, and share of returns on savings in total income $\phantom {\dot {i}\!}\mathcal {E}(t)$:

$$\begin{array}{@{}rcl@{}} \mathcal{W}(t)=\frac{{\sum}_{i = 1}^{{\Lambda}(t)}W_{i}(t)}{D(t)} \end{array} $$

(66)

$$\begin{array}{@{}rcl@{}} \mathcal{P}(t)=\frac{{\sum}_{i = 1}^{{\Lambda}(t)}{\Psi}_{i}(t)}{D(t)} \end{array} $$

(67)

$$\begin{array}{@{}rcl@{}} \mathcal{E}(t)=\frac{{\sum}_{i = 1}^{{\Lambda}(t)}E_{i}(t)}{D(t)} \end{array} $$

(68)

The remaining share corresponds to the rents on savings.

Appendix C: Initialisation

Table 17 Parameters setting

Fig. 3

Source: Own elaboration using UK FES

Expenditure shares: initial (c_i,n, p10) and asymptotic ($\bar {c}_{n}$, p99). The distribution of the asymptotic level of shares corresponds to the shares of expenditures for the higher percentile of UK consumers in 2005-6. The distribution of the level of shares of the first class corresponds to the shares of expenditures for the bottom decile of UK consumers in 2005-6. We thank Alessio Moneta for sharing the data.

Appendix D: Empirical validation

The feedbacks between technological and demand dynamics generate business fluctuations.⁴⁶ Figure 4 plots business cycles for output (Fig. 4a), investment (Fig. 4b), consumption (Fig. 4c), and unemployment (Fig. 4d) computed using the Hodrick-Prescott high-pass filter.⁴⁷ To make the fluctuations comparable, the cyclical component was normalised by the series trend.

Fig. 4

Cyclical component of the main macro variables. Notes. The four panels exhibits the cyclical components of output (4a), investment (4b), consumption (4c), and unemployment (4d). To separate the trend from the cyclical component we employ a Hodrick-Prescott high-pass filter. The cyclical component is normalised by the series trend

All series exhibit fluctuations that are qualitatively similar to those observed in the data (Assenza et al. 2015; Caiani et al. 2016; Dosi et al. 2010; Dosi et al. 2015). The volatility of employment and investment is significantly higher than the volatility of consumption and output, and consumption is less volatile than output. In contrast to the observed time series, in our model investment is more volatile than employment. This is related to the lumpiness of capital stock investment, which, in our model, is constrained by the choice of capital good producers and their production backlog (we do not model entry of new firms in the capital good sector).

Figure 5 plots the autocorrelation structure for de-trended real output (Fig. 5a), investment (Fig. 5b), consumption (Fig. 5c), and unemployment (Fig. 5d) for 20 lags. The simulated series are quite similar to real series (Assenza et al. 2015). The first lag autocorrelation of real series estimated by Assenza et al. (2015) for output, investment, consumption and unemployment are, respectively, 0.8485, 0.7952, 0.8176, 0.6454. For our simulated series, the first lag autocorrelations are 0.8492, 0.8169, 0.9577, and 0.6826.

Fig. 5

Autocorrelation of the main macro variables: output, investment, consumption and unemployment. Notes. The four panels show the autocorrelation graphs for de-trended real output (5a), investment (5b), consumption (5c) and unemployment (5d) for 20 lags. The autocorrelations are computed with pointwise confidence intervals (light blue lines) based on Bartlett’s formula for moving average time series of order 20 (MA(20)). The horizontal axis shows the number of lags and the vertical axis the autocorrelation

Figure 6 plots the cross-correlation between the cyclical component of real output and the cyclical components of, respectively, real output (Fig. 6a), investment (Fig. 6b), consumption (Fig. 6c) and unemployment (Fig. 6d) for 10 lags. Investment is pro-cyclical and coincident, consumption follows with a couple of lags, as in Caiani et al. (2016), and short term unemployment is countercyclical and coincident.

Fig. 6

Cross-correlation between the cyclical component of output and the main macro variables: output, investment, consumption and unemploymnet. Notes. The four panels exhibits cross-correlation plots between the cyclical component of real output, and the cyclical components of real output (6a), investment (6b), consumption (6c) and unemployment (6d) for 10 lags. The horizontal axis shows the number of lags and the vertical axis the cross-correlation between the cyclical components of the two series at a given lag

The model replicates a number of other macro stylised facts (Caiani et al. 2016; Dosi et al. 2010; Dosi et al. 2015). Figure 7 plots the cross-correlation between the cyclical component of real output and a number of other aggregate dynamics. In line with the literature, inventory growth is pro-cyclical and increases sharply (Fig. 7a); the ratio between inventories and sales is counter cyclical (Fig. 7b); average wages are pro-cyclical, but lagged (Fig. 7c); and average mark-ups are counter-cyclical (Fig. 7d).

Fig. 7

Cross-correlation between the cyclical component of output and other aggregate variables: inventories, wages, prices and mark-up. Notes. The four panels show the cross-correlation plots for the cyclical component of real output and the cyclical component of inventory growth (7a), inventory/sales ratio (7b), average wages (7c) and average mark-up (7d) for 10 lags. The horizontal axis shows the number of lags and the vertical axis the cross-correlation between the cyclical components of the two series at a given lag

Labour market regularities also emerge in our model. Figure 8a plots the Beveridge curve. We estimated the relation between the vacancy rate and the unemployment rate for the 100 simulation replicates, and for the whole sample. The light grey series are the single run curves obtained from plotting the residuals of the following polynomial regression of order two: $u_{tr} = \alpha ^{b} + {\alpha ^{b}_{1}}v_{tr} + {\alpha ^{b}_{2}}v^{2}_{tr} + \iota ^{b} + {\alpha ^{b}_{3}}v_{tr}\iota ^{b} + {\epsilon ^{b}_{t}}$, where r is a simulation iteration, ι^b is a run fixed effect and 𝜖^b the residual. The black series is the regression fit of the data pooled from the different series, and the red bands represent the confidence interval. Overall, the curve is quite close to that found for several countries (Nickell et al. 2002).

Fig. 8

Beveridge curve and output growth rate distribution. Notes. The left panel (8a) plots the estimation of the Beveridge curve for 60 runs (light grey series) and the estimation of the Beveridge curve for the pooled sample of the 100 series for 1000 time periods. The red band is the confidence interval of the aggregate curve. The horizontal axis shows the vacancy ratio (number of vacancies over employment) and the vertical axis the unemployment rate. All series are estimated with a polynomial regression of order 2. The right panel 8b exhibits the real output growth rate distribution (continuous line) against the normal distribution (dashed line) for the average growth rate across 100 runs

We also tested for the wage curve. Because, in our model, the wage curve shifts with price and productivity changes, plots are not particularly informative. We estimated the relation between the unemployment rate and wages using a panel estimator with fixed effects and robust standard errors clustered at the simulation run level, controlling for productivity and price indexes.⁴⁸ Table 18 shows that the results are remarkably close to the empirical evidence across countries (Nijkamp and Poot 2005).

Table 18 Wage curve

We estimated the distribution of quarterly output growth rates and found them not to be normally distributed,⁴⁹ and to have moment values quite similar to those estimated by Fagiolo et al. (2008) for US data⁵⁰ Fig. 8b plots the skewed distribution.⁵¹

The model also replicates some well-known meso and micro-stylised facts. Figure 9 plots the distribution of firm size measured by quantity and employees, averaged across periods and pooled across the 100 series. The plot shows the relation between the log size and the log rank, compared to a log normal distribution with the same average and standard deviations. Both measures show a striking similarity to the real data. The final distribution of firm size is related to the firms’ growth process, which, as expected, is also not normally distributed and has a high kurtosis.⁵² The distribution of firm growth emerging from our model is more akin to (but does not fit perfectly to) a Laplace distribution than a Gaussian distribution (Fig. 9c and d).

Fig. 9

Log-log plot of firm size distribution. Notes. The first two plots show the relation between the log of firm size (horizontal axis) and the log of the size rank (vertical axis). Size is measured as firm output (9a) and employment 9b. We pool all firms across the 100 time series and average size over the firm’s life span. Black circles represent the distribution of simulated firms and grey circles represent a log normal distribution. The last two plots show the distribution of quarterly firm size growth with respect to output (9c) and employees (9d)

Firms differ also with respect to their productivity and these differences build over time and tend to be persistent. We plot the time pattern of the productivity series for the “oldest” 14 firms surviving until the end of the simulation, in a random replication (Fig. 10c). All firms tend to maintain their relative position with respect to competitors. Figure 10a plots the average and standard deviation across all firms across all 100 simulation replicates. The average increases sharply as do the differences across firms (standard deviation).

Fig. 10

Firm productivity, capital, and size. Notes. The three panels plot micro-regularities across the firms. Panel 10a plots the average and the standard deviation of the productivity of all firms across all 100 simulation replicates. Panel 10b plots the average of the within-simulation average across 25 simulation runs (with confidence intervals) and the average standard deviation within 25 simulation runs. Panel 10c plots the series of the 14 “oldest” firms in the first simulation replication. Panel 10d plots the capital stock of the 14 “oldest” firms in the first simulation replication

We also studied the autocorrelation of firm productivity for all firms, for the first replication employing the Cumby-Huizinga test, controlling for heteroscedasticity for the possibility that the series may exhibit arbitrary autocorrelation (Baum and Schaffer 2013). Table 19 shows that there is strong and significant correlation at the micro level, looking at both the range between the first and the fifth lags, and each lag, controlling for autocorrelation in the previous lag.

Table 19 Autocorrelation of firm productivity

As a result of the vertical interaction between final good firms and capital good suppliers, our model also shows significant lumpiness in capital stock investment. Figure 10d plots the time pattern of the capital stock of the “oldest” 14 firms that survive until the end of the simulation in a random simulation replicate. Capital stock depreciates through time and investment in new stocks clearly is lumpy.

Finally, as discussed by (Poschke 2015), the average size of firms increased substantially in the last century, as did their dispersion – increasing the skewness of the size distribution. Figure 10b plots the average of both the within run average and standard deviation of firms across 25 replications. Following an initial decrease, both average firm size and dispersion increase substantially.

Appendix E: Extra tables

Table 20 Inverse Herfindahl Index for different levels of competition