Profitability and the lifecycle of firms

Missaka Warusawitharana

doi:10.1515/bejm-2017-0124

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Profitability and the lifecycle of firms

Missaka Warusawitharana

Published/Copyright: June 8, 2018

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From the journal The B.E. Journal of Macroeconomics Volume 18 Issue 2

Abstract

Using data on private and public firms, this study documents that profitability follows a hump shape over the lifecycle of a firm. Profitability rises with age for young firms, remains elevated, and then declines slowly for mature firms. A dynamic lifecycle model captures the observed age profile of profitability. Investment in product development generates profitability increases for young firms while wage pressures from more productive entrants lead to profitability declines for mature firms. The model generates the lifecycle behavior of financing and growth documented in the literature, even though it contains no financial frictions. It also implies greater sensitivity of financing and growth to age for young firms, a prediction supported by empirical tests. Taken together, these findings indicate that profitability dynamics influence the financing and growth of firms over the lifecycle.

Keywords: financial frictions; firm lifecycles; profitability; quality ladder

JEL Classification: D92; G31; E22; L20

Acknowledgement

I thank two anonymous referees, Viral Acharya, Marco Cagetti, Kevin Fox, Joao Gomes, Geng Li, Jeffrey Lin, Robin Lumsdaine, Ralf Meisenzahl, Jay Ritter, Michel Robe, Michael Roberts, Jonathan Wright, the editor and seminar participants at American University, Bureau of Economic Analysis, University of Dallas – Texas, University of Florida, the Federal Reserve Board, the 2010 NBER Summer Institute, the 2011 Federal Reserve System Applied-Micro conference, the conference on Economics of Innovation and Firm Survival, the 2015 Fall Midwest Macro meetings, and the 2016 International Industrial Organization conference for helpful comments. I thank Molly Shatto for excellent research assistance. The views expressed in this paper do not reflect the views of the Board of Governors of the Federal Reserve System or its staff.

Appendix

A Trend growth rates

Consider a steady state with an invariant distribution of firms with ages a_j and quality levels q_j(n), and assume that the vintage productivity term μ_t grows at a constant rate g. This section shows that aggregate capital, consumption and wages also grow at rate g.

Differentiate (1) with respect to K_j to obtain the following:

∂Yj∂Kj=(Kjμt−ajLj)1−α.

Given a constant marginal product of capital, one obtains that the capital stock will be proportional to the effective labor input. I.e.

Kj∝μt−ajLj.

Integrating over all firms, one obtains that

∫jKjdj=s0∫jμt−ajLjdj,

(11)⇒∫jKjdj=s0μt∫j(1+g)−ajLjdj,

where s₀ is an integration constant and the second equation follows substituting in the vintage productivity terms of firms with the current vintage productivity term μ_t while adjusting for its constant growth rate. As the aggregate labor force is a constant, the distribution of labor remains invariant in the steady state. Thus, the integral on the L.H.S. of equation (11) equals a constant, implying that the aggregate capital stock grows at the same rate, g, as μ_t.

A similar derivation shows that the consumption aggregator C_t also grows at the constant rate g. Applying the definition of the consumption aggregator and simplifying, one obtains the following:

Ct=(∫jqj(n)Yj1−νdj)11−ν,

=(∫jqj(n)(μt−aj1−αzj~KjαLj1−α)1−νdj)11−ν,

=(∫jqj(n)(μt1−α(1+g)−ajzj~KjαLj1−α)1−νdj)11−ν.

In the steady state, the distribution of product quality, firm age, and labor inputs will remain invariant. As such, aggregate consumption will growth at the same rate as ∫jμt1−αKjαdj, which grows at rate g.

The growth rate of wages obtains from the first order condition for labor:

wtLj=qj(n)Ctν(1−α)(1−ν)Yj1−ν.

Integrating over all firms in the economy, one obtains that

wt∫jLjdj=(1−α)(1−ν)Ctν∫jqj(n)Yj1−νdj,

⇒wtL=(1−α)(1−ν)Ct,

implying that aggregate wages grow at the same rate, g, as aggregate consumption.

B Optimal product development

Proposition 2

The first order condition for product development implies that

marginalcostoffunds=marginalbenefitofproductdevelopmentexpenses

⇒1=β(1+g)Ez[v(k′,q(n+1),a+1,z~′)−v(k′,q(n),a+1,z~′)q(n)]b(1−p(r)).

Proof.

Taking first order conditions from the Bellman equation (10), one obtains that,

marginal cost of funds=marginal benefit of product development expenses.

In the absence of financial frictions, the marginal cost of funds (L.H.S of the above equation) equals one,

L.H.S=1.

The marginal benefit of product development expenses (R.H.S. of the above equation) is given by:

R.H.S.=∂∂r(β(1+g)Ez[p(r)vc(k′,q(n+1),a+1,z~′)+(1−p(r))vc(k′,q(n),a+1,z~′)])

=β(1+g)Ez[v(k′,q(n+1),a+1,z~′)−v(k′,q(n),a+1,z~′)]∂p(r)∂r.

Some algebra yields that

∂p(r)∂r=bq(n)(1−p(r)).

Substituting this into the previous expressions completes the proof. ∎

C Model solution and simulation

The optimal policies of the firm are obtained using value function iteration to solve the Bellman equation given in Equation (10). This process employs the optimal product development expense given in Proposition 1. At each step, the solution for physical investment is carried out numerically over a grid of values for capital. The numerical solution is obtained using the following grid sizes: a profitability shock grid with 5 values, a quality grid with values from 1 to 15, a capital grid with 120 values, and an age grid from 1 to 80. The simulated data sample is constructed using the value function solution and the associated optimal policy functions for financing and growth.

The simulated data set is obtained by simulating the model economy with 1000 firms over a period of 200 years. Observations in the first 100 years are discarded as a burn-in sample. An examination of the cross-sectional moments indicate that the simulations reach their steady state well before 100 years. Only a very small fraction of firms reach the maximum age level in the simulation. This simulated sample provides a steady state cross-section of firms that can be employed to further investigate firm policies in the model.

Firms exit endogenously in the simulation when their exit value exceeds the continuation value. Each firm that exits is replaced with a new firm of age 1 with capital stock near the bottom of its grid and quality index n = 1. New entrants are assigned a random profitability shock level drawn from its unconditional distribution.

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Published Online: 2018-06-08

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https://doi.org/10.1515/bejm-2017-0124

Keywords for this article

financial frictions; firm lifecycles; profitability; quality ladder