3.A. Details on the restricted demand model
3.A.1. Restricted model. To quantify the importance of demand accumulation for ag-gregate growth, we consider a restricted version of the model in which the customer base is time-invariant. We keep all the structural parameters at the values they hold in the baseline model and simply replace firm-level demand with the type-specific averages observed in the stationary distribution of the baseline model. These averages areb=20 for small firms and b=1350 for large firms.
Table 3.7 compares moments targeted in the estimation procedure in the data, baseline model, and restricted model. Since the customer base does not accumulate, firms tend to be smaller, less innovative, and more likely to exit.
TABLE3.7. Externally calibrated parameters, the restricted model
Model
Target Data Baseline Restricted
labor prod. annual growth 1.5% 1.5% 0.7%
exit rate of firms<6 years 15.2% 11.0% 10.7%
mean exit rate 8.6% 9.1% 11.4%
mean R&D/sales 10.4% 9.7% 3.8%
firm share, small non-R&D 36.6% 36.0% 45.8%
firm share, large non-R&D 0.1% 0.1% 0.2%
firm share, small R&D 62.5% 63.6% 53.2%
Notes: Starting from the leftmost column, the table presents the symbol of a parameter, its meaning, assigned value, and the calibration target.
Figure 3.5 presents the lifecycle profile of firm size in the restricted model and in the BDS data. Since the incentives to innovation are severely diminished, and the customer base does not accumulate, firms tend to get smaller as they age.
3.A.2. Re-calibrated restricted model. We also study a version of the baseline model that in addition to constraining the customer base to time-invariant valuesb=20 for small firms andb=1350 for large firms, we re-calibrate the remaining parameters to match the same moments in the data as in the case of baseline calibration. Table 3.8 presents calibrated parameters, Table 3.9 estimated parameters and targeted moments in the data followed by corresponding values in the model. Figure 3.6 presents age profile of the average firm size.
FIGURE 3.5. Firm lifecycle profile of average employment: data and re-stricted model
Notes: Average firm size by age, where 0 refers to startups, 6 refers to 6-10 year old firms, 11 refers to 11-15 year old firms and 16 refers to 16-20 year old firms. Data is taken from the BDS.
TABLE3.8. Externally calibrated parameters, the re-calibrated restricted model
Parameter Value Target
I. households
β discount factor 0.97 real interest of 3%
ν labor disutility 1 normalizationWt=1
η elasticity of demand 4 static markup of 33%
II. firms
δ customer base depreciation 0.2 Gourio and Rudanko (2014) ψ elasticity of innovation prob. 2 Akcigit and Kerr (2018)
ρ exogenous exit rate 0 normalization
φe entry cost 0.7 normalizationCt=1
be initial customer base for entrants 1 normalization
Notes: Starting from the leftmost column, the table presents the symbol of a parameter, its meaning, assigned value, and the calibration target.
TABLE3.9. Internally estimated parameters, the re-calibrated restricted model
Parameter Value Target Data Model
I. common to all firms
λ innovation rate 2.90% labor prod. annual growth 1.5% 1.5%
µH mean operational costs 1.75 exit rate of firms<6 years 15.2% 11.0%
σH std. operational costs 16.07 mean exit rate 8.6% 9.1%
x0 technology diffusion 50.00% firm lifecycle dynamics see Fig. 3.1
II. idiosyncratic
γ R&D efficacy 0.3 mean R&D/sales 10.4% 9.7%
α1 customer base elas., large 1.00 fixed demand
α0 demand scaling 1.00 fixed demand
PαS entry share, small non-R&D 0.33 respective firm share 36.6% 36.0%
PαL entry share, large non-R&D 0.0003 respective firm share 0.1% 0.1%
PαS,γ entry share, small R&D 0.66 respective firm share 62.5% 63.6%
Notes: Starting from the leftmost column, the table presents the symbol of a parameter, its meaning, estimated value, description of the targeted moment, followed by value of the targeted moment in the data, and the corre-sponding value in the model.
FIGURE3.6. Firm lifecycle profile of average employment: data and the re-calibrated restricted model
Notes: Average firm size by age, where 0 refers to startups, 6 refers to 6-10 year old firms, 11 refers to 11-15 year old firms and 16 refers to 16-20 year old firms. Data is taken from the BDS.
3.B. Numerical appendix We can simplify and re-write the firm’s problem as follows
V(b,q)=max
b0,x Φ(b,q)×nW q C
"
µ b0 (1−δ)−b
¶1−1η³ q W
´
bα/η− b0 (1−δ)+b
#
−W µxψ
γ +φˆ(b,q)
¶ +β˜¡
xV(b0,q+)+(1−x)V(b0,q)¢o s.t.
x∈[0, 1−x0], (1−δ)b−b0≤0 (3.29)
121
where primes0denote next period variables and we leave implicit the dependence of all vari-ables on firm type j. There are two state variables of the firm problem: productivity q and customer baseb. We parameterize the grid for productivity using 41 points. Values of pro-ductivityq are normalized such thatq=1 corresponds to average productivity in the econ-omy. The grid for customer base uses 61 points. Both grids are equally-spaced. Moreover, both grids are type-specific. The upper and lower bounds of the grids corresponding to each type are choses as follows. The upper bound of productivity grid for non-R&D firm isq=1, since these firms enter at the average productivityq=1 and afterwards slide downwards as the economy grows. The lower bound is set such that it takes a non-R&D firm 61 years to reach the lower bound of the grid,q=0.475. Theq grid for R&D firms uses the same lower bound. The upper bound is set at three times the lower bound value, in line with evidence on productivity dispersion Bartelsman and Doms (2000). The lower bound on grid of customer base corresponds tobe=1. The upper bound is set such that the largest of small firms has 250 employees, and the largest of large firms 10000 employees.
We solve the firm problem using value function iteration. Towards this end, we begin with an initial guess forV(b,q) for each state (b,q) and solve firm problem (3.29) given the current guess for V. The maximization problem yields policyb0(b,q),x(b,q). This allows us to derive the survival probabilitiesΦ(b,q) using equation (3.17) as well as pricing policy
µ(b,q)= µ b0
(1−δ)−b
¶−1/η³q W
´ bα/η.
The optimal policy, in turn, allows as to calculate the implied new value functionV(b,q) for each state (b,q). We iterate this procedure until convergence.
To approximate the stationary firm distribution, we simulate 75 thousand firms. We fol-low each firm until it decides to exit, but no longer than 61 years. Given the approximated stationary firm distribution, we compute all aggregate variables.