Financial Frictions at Entry, Average Firm Size, and Productivity

A Other Measures of Financial Development

Here we show the cross-country relationships between external finance ratios and several outcomes of interest are robust to replacing external finance ratios with two alternative proxies for financial development – bank deposits and equity, both measured relative to GDP (Figures 7 and 8).^[16] As in Section 2, the elasticity of average firm size with respect to each proxy seems to differ across sectors in Figures 9 and 10. But the ratio of average size in manufacturing relative to that in services has no statistically significant relationship to either proxy (as is the case with respect to external finance ratios), suggesting that the differences between manufacturing and services in Figures 9 and 10 are simply due to different samples of countries. While the employment share of large firms is higher in economies with more equity, Figure 11a shows no systematic relationship between the large firm employment share and bank deposits.

Figure 7:

Aggregate TFP, bank deposits, and equity. (a) Bank deposits / GDP. (b) Equity / GDP. Notes: All variables are shown in log scale. See the text for the definition of variables and sources. The coefficients (standard errors) from OLS regressions are (a) 0.36 (0.06) and (b) 0.14 (0.02), based on 86 and 66 observations.

Figure 8:

R&D Intensity, bank deposits, and equity. (a) Bank deposits / GDP. (b) Equity / GDP. Notes: All variables are shown in log scale. See the text for the definition of variables and sources. The coefficients (standard errors) from OLS regressions are (a) 0.86 (0.14) and (b) 0.38 (0.06), based on 100 and 71 observations.

Figure 9:

Establishment size and bank deposits / GDP. (a) Manufacturing. (b) Services. Notes: All variables are shown in log scale. See the text for the definition of variables and sources. The coefficients (standard errors) from OLS regressions are (a) 0.41 (0.08) and (b) 0.28 (0.06), based on 112 and 105 observations.

Figure 10:

Establishment size and equity / GDP. (a) Manufacturing. (b) Services. Notes: All variables are shown in log scale. See the text for the definition of variables and sources. The coefficients (standard errors) from OLS regressions are (a) 0.16 (0.04) and (b) 0.13 (0.03), based on 73 and 72 observations.

Figure 11:

Employment share of large firms, bank deposits, and equity. (a) Bank deposits / GDP. (b) Equity / GDP. Notes: All variables are shown in log scale. See the text for the definition of variables and sources. The coefficients (standard errors) from OLS regressions are (a) 0.10 (0.20) and (b) 0.16 (0.06), based on 31 and 29 observations.

B Detailed Derivations

This section provides details of derivations involved in Section 3.

Life-cycle Investment: Taking for granted that w and A ̄ remain constant in a stationary equilibrium, each period a firm with productivity A chooses x to maximize the following value function;

where x′ denotes next period’s choice of x. Note that V(A|x) can be expanded in the following way;

In the next period the firm chooses x′ to maximize V′(A|x′);

Since V′(A|x′) is just V(A|x) with every term scaled up by the previous choice of x, a firm effectively faces the same problem each period. The first-order condition for this problem is;

And given that the firm faces the same problem each period, it is clear that the firm chooses the same x each period and x = x′. As a result, optimal x can be characterized by equation (5);

Entrant Productivity Investment: We assume that when selling equity at entry, entrants can commit to using all proceeds on initial productivity investment. Entrants therefore choose a fraction χ ̂ of the value of the firm to maximize its retained value;

where V _E (A ₀) is equal to;

This is equivalent to the following problem;

The derivative of the above objective function is;

which is decreasing in A, and positive so long as;

or

Clearly, unconstrained optimal investment is equal to a fraction 1/ϕ of V _E (A ₀)(A/A ₀). For a constrained entrant choosing investment, optimal χ ̂ = χ , so constrained optimal investment satisfies;

as represented by equation (6).

Equilibrium conditions: To obtain equation (12), first note that using labor market clearing in equation (9), profit can be written as π ( A 0 ) = A 0 w 1 − α α ⋅ L − N c p N A ̄ . Substituting this expression for profit together with the expression for V _E (A ₀) gives equation (12).

An alternate way to find the equilibrium productivity increase at entry is to use equation (6) and substitute π ( A 0 ) = A 0 w 1 − α α ⋅ L − N c p N A ̄ . After re-arranging terms substitute equation (13) for L/N to obtain the equilibrium productivity increase in equation (14).

For the equilibrium average productivity increase over the life-cycle, we use equation (5) and substitute π ( A 0 ) = A 0 w 1 − α α ⋅ L − N c p N A ̄ to obtain equation (15).

For the investment ratio in equation (18), note that investment by entrants is;

and investment by incumbent firms is;

Aggregate output is Y = N ⋅ y ( A ̄ ) . The investment ratio is the sum of investment by entrants and incumbents, all relative to aggregate output.

Large-Firm Employment Share: Total employment at a firm with productivity A is ℓ(A), if we exclude entrepreneurial services from the owner. To see how the share of employment in large firms is affected by χ, note that labor-market clearing combined with equation (2) imply firm-level employment in equilibrium as a function of firm-level productivity is equal to;

The above expression is used to derive the threshold ability A ²⁵⁰ at which a firm employs 250 employees shown in equations (16) and (17).

The share of all employees working in firms with at least 250 employees (‘large’ firms) can be expressed as;^[17]

where T is defined as the maximum t such that A 0 * ( A / A 0 ) x t ≤ A 250 ;

and the following holds when A ₀ is Pareto distributed;

The large-firm employee share (equation (17)) can now be expressed as;

C Allowing for Multiple Sectors

Here we extend the model developed in Section 3 to allow for two sectors - manufacturing (M) and services (S). We allow these sectors to differ with respect to operating costs c _p,i , investment costs c _A,i and c _x,i , exit rates λ _i , and exogenous productivity (common across firms within a sector) z _i , i ∈ {M, S}. We make the simplifying assumption that some fraction of the population γ can potentially operate a manufacturing firm, while the rest of the population can potentially operate a service-sector firm. Each person knows their entrepreneurial ability, which is drawn from a Pareto distribution as in Section 3.^[18] As in the benchmark model, whether a potential entrepreneur decides to start a firm depends in part on their ability.

Aggregate output in the economy is produced by a representative final-good firm that uses output produced in the manufacturing and service sector;

Profit maximization for the final-good firm implies;

and we continue to assume the aggregate good serves as the numéraire.

Producers in each sector maximize operating profits in each period, resulting in the following analogues to equations (2) through (4);

Within sectors, optimal life-cycle investment and initial investment are still represented by equations (5) and (6), but with sector-specific values for c _x , c _A , A ̄ , A ̄ 0 , π(A), and V(A ₀).

Labor-market clearing within each sector implies;

where L _i and N _i denote the supply of labor and the number of producers in sector i. A ̄ i refers to the average A across producers in sector i, equal to;

where λ, A 0 * , (A/A ₀), and x are now sector-specific.

Using the above labor-market clearing condition, we can obtain the same expressions for equilibrium average firm size, threshold productivity A 0 * , initial productivity (A/A ₀), average productivity growth x, and sector-specific TFP as in Section 3, adjusted for differences in sector-specific parameters. These expressions make clear that changes in the financial constraint χ can have a different impact across sectors if and only if λ or c _p /c _x differ across sectors. The BDS data for 2007 suggest almost identical exit rates (λ) across sectors, so it is appropriate here to assume λ _M = λ _S = 0.14 (our benchmark value). The BDS data for 2006 to 2007 suggests surviving manufacturing firms grew faster than service-sector firms, with an employment growth rate (relative to growth in average size across all firms) about 3.36 times higher than that for surviving service-sector firms. Using x = 1.05 for manufacturing firms (Hsieh and Klenow 2014), this implies x = 1.015 for service-sector firms. Using sector-specific versions of equation (15), we obtain the following values for c _p,i /c _x,i by matching the above sector-specific growth rates for firm-level productivity;

Intuitively, c _p must be higher for manufacturing firms to rationalize their greater average size (21.6 vs 5). The fact that they grow faster than service-sector firms implies that their c _x must be lower, relative to c _p .

Tables 3 and 4 show the quantitative impact on sectoral outcomes as χ decreases from its benchmark value. We note that results for manufacturing are identical to those reported in Table 1, where our one-sector model is calibrated using manufacturing data. The impact of χ on service-sector outcomes are effectively quantitatively identical, though the proportional impact on average firm size, x, and sectoral TFP are slightly dampened. Why are the impacts of χ not quantitatively different across sectors? We already discuss above how differences in average size driven by differences in c _p do not affect the proportional impact of χ. The only remaining difference across sectors here is the difference in c _x /c _p , which drives differences in x (average firm growth rates). But although growth rates (x − 1) differ substantially across manufacturing and services, x does not. Even drastic differences in x − 1, say 5 vs 1 %, only generate slight differences in x (1.05 vs 1.01). For the purposes of comparing the impact of χ on sectoral outcomes, our two sectors are effectively almost identical.

D Modeling Backward- and Forward-Looking Borrowing Constraints

In the model of Section 3, new entrepreneurs can finance initial productivity investment only by selling equity, and we assume they are constrained to sell a fraction of at most χ of their firm’s value. Here we show how our modeled equity constraint can be interpreted as equivalent to combinations of an equity constraint and a backward- or forward-looking borrowing constraint, allowing us to map χ to external finance ratios in the data. Our approach is motivated by Brooks and Dovis (2020), and we emphasize that all financing constraints continue to bind only at entry, as optimal life-cycle investment can be financed each period through current profit.

E Alternative Values for θ

As we discuss in Section 4.2, the sign of the impacts of equity constraints on average firm size and life-cycle growth depend on the value of θ, an inverted measure of how sensitive growth is to life-cycle investment. When choosing a value for θ we quite naturally target the elasticity of output with respect to life-cycle investment in intangible capital documented in the empirical literature. Here we reevaluate the impact of dropping χ when we use alternative values for θ. For each alternative value we recalibrate the values for c _x and c _p using the same targets as in Section 4.1. In the first row of Table 5, we again report the targeted values for average firm size L/N, average life-cycle growth x − 1, and TFP. We then show how dropping χ from the benchmark 0.72 down to 0.2 affects these outcomes when we consider alternative values of θ. For each value of θ, we also report the corresponding implied elasticity of firm output with respect to investment (our target for θ in Section 4.1).

Using a higher θ of 150, or a lower value of 50, does not significantly affect the resulting impacts of decreasing χ. But using a much lower value of θ, lower than 10.3, significantly changes the impact of χ. In particular, a decrease in χ leads to higher average firm size and life-cycle growth, and as a result, has a dampened impact on aggregate TFP. But these much lower values of θ imply counter-factually high elasticities of firm output with respect to intangible investment, over 20 times the elasticity estimated by Bontempi and Mairesse (2015).

F Alternative Calibration: Mexico

Here we use an alternative strategy to calibrate the model of Section 3 – we choose parameter values to match data moments from Mexico, a poorer economy with a relatively under-developed financial market. For r, α, ϕ, and θ, we continue to use the same values, as these parameter values are either standard in the literature or chosen to match international estimates. We choose λ = 0.16 to match an exit rate for manufacturing establishments in Mexico (Hsieh and Klenow 2014). We choose the Pareto shape parameter κ to match the share of employment in new manufacturing firms with at least 50 employees, relative to the employment share of new firms with at least 10 workers (similar to our target for the U.S.), obtaining a value of κ = 1.56. We choose c _x /c _p to match an average growth rate of employment at surviving firms (relative to average employment across all producers) of 2.5 %, again from Hsieh and Klenow (2014). And we choose c _p = 0.59 to match the average employment size of Mexican manufacturing establishments, equal to 8.6 persons per establishment.

Table 6 reports outcomes when χ varies from 0.01 to 0.72, using this alternative calibration, with χ = 0.11 now indicated as the (Mexican) benchmark. To better compare proportional outcomes to Table 1, we report in brackets outcomes for each χ relative to outcomes when χ = 0.72 (the U.S. level). The results are almost identical to those in Table 1, suggesting the inferred impacts of χ are very robust to this alternative calibration. To see why, note that the impact of χ only significantly interacts with κ, λ, and x. The value we use here for κ (the distribution of A ₀ across the population) is very similar to that in our benchmark calibration, implying a similar relationship between average size and A 0 * . Combined with our similar value for λ and the fact that x can not vary much (even while x − 1 can), the model using this alternative calibration generates very similar outcomes to our benchmark model.

G Assuming Exogenous Life-Cycle Growth

Here we show how the impact of financial constraints changes when productivity growth over the life-cycle of firms is exogenous. Departing from the model developed in Section 3, firms no longer invest in life-cycle growth, and productivity exogenously grows by a factor x each period on average after entry, conditional on survival. We continue to assume the same process for realized productivity growth. The value of a firm with productivity A ₀ in equation (7) now becomes;

As a result, we now characterize the entry-threshold A 0 * and average firm size as follows;

Average size now depends solely on χ, and it is clear that a tighter equity constraint (lower χ) now leads unambiguously to more firms and lower average size. Equations (14) and (20) still hold as characterizations of equilibrium (A/A ₀) and aggregate output per worker (TFP). Table 7 shows how average firm size, initial productivity, and TFP are impacted by χ, after calibrating the model to hit the same targets as in Section 4.1 and assuming exogenous x = 1.05.

Comparing the results here to those in Table 1, it is clear that allowing for endogenous life-cycle growth slightly magnifies the impact of χ on average firm size and aggregate TFP, consistent with the conclusions of Vereshchagina (2023).