## 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.

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## Notes

See, e.g., Leijonhufvud (2006), Colander et al. (2008), LeBaron and Tesfatsion (2008), Buchanan (2009), Farmer and Foley (2009), Gatti et al. (2010), Fagiolo and Roventini (2012), Dosi et al. (2013), Lengnick (2013), Assenza et al. (2015), Dosi et al. (2015), Lorentz (2015), and Caiani et al. (2016). See also the recent review in Fagiolo and Roventini (2017), and other papers in this issue.

The index for sector

*n*is suppressed because we represent both the final good and the capital good sectors.Caiani et al. (2016) propose an interesting, simplified, static version of the firm hierarchical structure, introducing heterogeneous wages within each tier. For simplicity, in our model, we assume that all workers in a given level earn the same wage.

The index for sector

*n*is suppressed because we represent both the final good and the capital good sectors.The aim of this paper is not to explain the rise in executives’ compensation. However, the proposed wage and bonus structure mean that the model conforms to a stylised representation of the evidence on firms’ compensation structures, and on the recent increases in executive pay. There is some evidence that suggests that the rise in CEO pay is linked mainly to stock options (Frydman and Jenter 2010). However, there is other evidence suggesting that the main components of the increased incomes of the top 1% are salaries and bonuses (Atkinson et al. 2011). The crucial aspect that we highlight here, is the exponential wage increases with an organisation’s tiers, and the use of profits to amplify this difference. Dividends, which can be thought of as stock options, also augment the income of the wealthiest classes relative to the less wealthy, as discussed below. Whether they come from savings or from firm compensation is not critical in this model.

*P*(*t*) is the weighted average of the final good firms’ prices:$$P(t)=\sum\limits_{n = 1}^N\sum\limits_{f = 1}^{F(t)}\frac{Y_{f}(t)}{{\sum}_{n = 1}^N{\sum}_{f = 1}^{F(t)}Y_{f}(t)}p_{f}(t-1) $$Aggregate productivity is the ratio between aggregate output and employment:

$$A(t)=\sum\limits_{n = 1}^N\sum\limits_{f = 1}^{F(t)}\frac{Y_{n,f}(t)}{{\sum}_{n = 1}^N{\sum}_{f = 1}^{F(t)}Y_{n,f}(t)}A_{n,f}(t-1) $$The actual savings can differ from the desired share in the case of sudden changes in income: accumulated when income increases, and used when income reduces.

See Eq. 41.

\(\hat {{\Pi }}_{f}(t)=\hat {{\Pi }_{f}(t-1)} a + (1-a) {\Pi }_{f}(t)\).

See Eq. 41.

See Appendix A.3.3.

Labour costs are computed only with respect to the shop-floor workers.

If successful, no more trials are used in that period, and the firm must wait Ξ periods before the next investment in R&D.

100 runs when investigating the model properties and empirical validation, and 25 runs when investigating the regimes.

For some of the behavioural parameters we were unable to find any evidence and were forced to rely on qualitative evidence.

The initial adjustment is due to small differences in consumer preferences, productivity and labour market adjustments, introduced to reflect parameter values that are closer to those observed in a modern system with respect to those observed in a pre-take-off economy: the first class of wage earners are less selective with respect to price; innovation efforts are more successful, and wages more closely follow changes in prices and in productivity. These changes cause an initial minor downturn in the economy as prices, firms’ market shares and concentration (exit and entry) adjust to the new system.

Regulation theory discusses two other relevant dimensions: finance and the role of the state. Both are crucial, but for the sake of clarity we leave their analysis to further research.

As noted, in our model we do not consider any redistributive mechanism. We study income distribution as an outcome of the structure of production and demand

See Eq. 62 in the Appendix

*Cœteris paribus*, by this we mean the benchmark configuration (Table 6).See the effect of selectivity and entry probability on market concentration in Appendix Table 20.

Results not shown here are available from the authors.

Note that firms with high backlogs in our model also have an incentive to increase mark-up.

In the benchmark configuration the first (least wealthy) class is populated by approximately 66% of the total population and the second class by approximately 22% of the total population. Their respective shares of total consumption are approximately 47% and 25%.

In the benchmark scenario.

The Beveridge curve constant is set to 1 because the values in Börsch-Supan (1991) range between -5 and 4.

Initialised with a value generating vacancy rates corresponding to the empirical evidence.

We assume equal savings for all the consumers in a class.

The constant assumption is corroborated by numerous empirical studies, starting with Kaldor (1957). The investment decision in in new capital vintages ensures that capital stock intensity remains fixed over time.

Given that, in our model, consumption occurs at the level of the class and goods are then distributed to consumers, there is no rationing at the consumer level. We assume that, although all consumers make a demand for all goods in all periods, only consumers who have not purchased the good in previous periods will need it. In other words, backlogs is a simplifying assumption to provide firms with market signals about future demand and allow a class of consumers to consume the same good in different time periods.

When completely depreciated, capital vintages are disposed of at no cost.

We assume that all profits are distributed to households as dividends.

In the configurations adopted in this paper, this form of rationing is rare and, when it happens, lasts a maximum of a few time steps.

We adopt the convention that the nominal value of a firm’s debt is constantly equal to the loans received as long as the firm remains active, and drops to zero in the case the firm exits the market.

In general, the price of financial titles, such as companies’ stock, is determined by trade and, consequently, the market value of a company is computed by multiplying the price by the number of outstanding stocks. In our model, the tokens are not traded and we use the same identity to compute their price, determined by the ratio of the total value of the financial sector (cash plus debt) and the number of tokens. This ensures that the total value of the tokens owned by households equals the current value of the financial sector’s capital.

As we observed earlier, the feedbacks are slightly more complex at the micro level, depending on firm growth and the competition, which depends on firms’ investment in product and process innovation and on consumer preferences, which depend on firm growth and industry dynamics.

We use the default STATA value of 1,600, following the Ravn-Uhlig rule, after transforming the series from weekly to quarterly by estimating a moving average.

We estimated the following equation: \(\phantom {\dot {i}\!}\ln \bar {w}^{r}_{t} = \alpha ^{w} + \beta ^{w}\ln {u^{r}_{t}} + {\gamma ^{w}_{0}} cp{i^{r}_{t}} + {\gamma ^{w}_{1}} p{i^{r}_{t}} + {\epsilon ^{r}_{t}}\); where \(\phantom {\dot {i}\!}\bar {w}\) is the average wage across classes,

*cpi*is the consumer price index and*pi*is the productivity index, respectively the ratio of price and productivity in*t*> 0 to price and productivity in*t*= 0.The Shapiro-Wilk and the skewness and kurtosis tests for normality reject the normality hypothesis.

Mean= 0.00996, standard deviation= 0.012, kurtosis= 3.76. Skewness is larger in our simulations and equals 0.39.

We obtained results similar to the empirical evidence also for the fortnightly growth rates.

11.3 for quarterly growth rate of output and 7.8 for quarterly growth of employees. For both series the skewness/kurtosis tests for normality and the Shapiro-Wilk test reject the null hypothesis of a normal distribution.

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## Acknowledgments

We are grateful for comments and suggestions from participants in: EURKIND (Valencia, 2016), Schumpeter Society (Montreal, 2016), SPRU50 (Sussex, 2016) and EAEPE (Manchester, 2016) conferences and seminars at ECLAC, ECLAC summer school and Curitiba. The paper has benefited greatly from comments and suggestions from Robert Blecker, Francesco Lamperti, Carolina Pan, Gabriel Porcile, Anna Salomons, Engelbert Stockhammer, Ariel Wirkierman, an anonymous reviewer of the SPRU Working Paper Series and two anonymous referees of this journal. We acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 649186 – Project ISIGrowth.

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This paper received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 649186 – Project ISIGrowth.

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**Conflict of interests**

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## 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):

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.^{Footnote 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:

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

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:

The mismatch between firms’ labour demand and hiring depends on the parameter *𝜖* in Eq. 3.^{Footnote 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:

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*)):

#### 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:

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:

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:

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:

### 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:

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:

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:^{Footnote 38}

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*):

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:

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.^{Footnote 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):

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

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):

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.^{Footnote 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:

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:

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:

where *a*_{g}(*τ*) is the productivity embodied in the *h* vintage.^{Footnote 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, *B**L*_{f}(*t*).

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).^{Footnote 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.^{Footnote 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)):

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

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:

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

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

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)):

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:

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 *ω*_{0} 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}:

where *w*_{0,g}(*t*) is the wage of engineers.

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

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:^{Footnote 44}

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:

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:

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:^{Footnote 45}

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:

### 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:

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:

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

*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*:

*Capital-labour ratio – degree of mechanisation*

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

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 \(\phantom {\dot {i}\!}\mathcal {E}(t)\):

The remaining share corresponds to the rents on savings.

### Appendix C: Initialisation

### Appendix D: Empirical validation

The feedbacks between technological and demand dynamics generate business fluctuations.^{Footnote 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.^{Footnote 47} To make the fluctuations comparable, the cyclical component was 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.

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.

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).

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).

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.^{Footnote 48} Table 18 shows that the results are remarkably close to the empirical evidence across countries (Nijkamp and Poot 2005).

We estimated the distribution of quarterly output growth rates and found them not to be normally distributed,^{Footnote 49} and to have moment values quite similar to those estimated by Fagiolo et al. (2008) for US data^{Footnote 50} Fig. 8b plots the skewed distribution.^{Footnote 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.^{Footnote 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).

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).

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.

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

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Ciarli, T., Lorentz, A., Valente, M. *et al.* Structural changes and growth regimes.
*J Evol Econ* **29**, 119–176 (2019). https://doi.org/10.1007/s00191-018-0574-4

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DOI: https://doi.org/10.1007/s00191-018-0574-4