The impact of firm size on dynamic incentives and investment

• Date: 01 March 2017
• Author: Chang‐Koo Chi,Kyoung Jin Choi
• Published date: 01 March 2017
• DOI: http://doi.org/10.1111/1756-2171.12171

RAND Journal of Economics

Vol.48, No. 1, Spring 2017

pp. 147–177

The impact of firm size on dynamic

incentives and investment

Chang-Koo Chi∗

and

Kyoung Jin Choi∗∗

Recent studies conclude that small firms have higher but more variable growth rates than large

firms. To explore how this empirical regularity affects moral hazard and investment, we develop

an agency model with a firm size process having two features: the drift is controlled by the agent’s

effort and the principal’s investment decision, and the volatility is proportional to the square root

of size. The firm improves on production efficiency as it grows, and wages are back-loaded when

size is small but front-loaded when it is large. Furthermore, there is underinvestment in a small

firm but overinvestment in a large firm.

1. Introduction

Recent empirical studies conclude that the dynamics of a firm are negatively associated

with size. It is now well documented that within an industry, small firms grow faster but have a

higher volatility of growth rates than large firms. Interpreting the volatility as corporate risk, this

empirical pattern—referred to as the size-dependence regularity by Cooleyand Quadrini (2001)—

implies that the degree of risk a corporation faces depends on its size. Then, in a dynamic agency

problem in which firm size changes over time, this regularity has important implications for

moral hazard and investment, as both are inextricably linked with the characteristic of shocks.

In this article, we propose a continuous-time principal-agent model in which firm size follows a

diffusion process with a diminishing volatility of the growth rate, which sheds light on the impact

of the regularity on dynamic incentives and investment.

There is a growing body of literature, initiated by He (2009), studying howtime-varying firm

size affects the structure of an optimal contract in a continuous-time framework. However, for the

sake of tractability,most of that literature assumes that fir m size evolves according to a geometric

∗Norwegian School of Economics and Aalto University; chang-koo.chi@nhh.no.

∗∗University of Calgary; kjchoi@ucalgary.ca.

Weare grateful to Felipe Aguerrevere, Hengjie Ai, Costas Azariadis, Bruno Biais, Alex David, Alex Edmans, Zhiguo He,

Hyeng Keun Koo, Christian Krestel, Rodolfo Manuelli, Alfred Lehar, Yuliy Sannikov, Jaeyoung Sung, Noah Williams,

John Zhu, and seminar participants in the 2013 Northern Finance Association Meeting and the 2014 Society of Financial

Studies Cavalcade for their helpful discussions and suggestions. We are particularly indebted to the Editor and three

anonymous referees for their valuable advice and comments.

C2017, The RAND Corporation. 147

148 / THE RAND JOURNAL OF ECONOMICS

Brownian motion which, in contrast with the regularity, entails a constant growth rate volatility.1

Our model features two main departures from the existing models, and they lead to a distinctive

firm-size process. First, to describe a firm’s growth path, we adopt a capital accumulation model

in which the principal can increase firm size through investment, and embed it into a dynamic

contracting framework. This gives rise to the drift of firm size controlled by the agent’s hidden

action and the principal’s investment decision. Second, to incorporate the regularity, we postulate

that the volatility of firm size is proportional to the square root of size.2The volatility thereby

increases with size, but their relationship diminishes as the firm grows. The model provides a

simple framework bywhich we can explore the impact of the regularity on both moral hazard and

investment and delivers qualitatively different predictions about the optimal contract, depending

on firm size.

Specifically, the model describes an environment in which a risk-averse principal delegates

the management of a firm to a risk-averse agent and offers the agent a long-term contract with full

commitment. The contract specifies a flow of compensation for the agent and a flow of dividends

for the principal. At each time t, the firm produces output or cash flow with two inputs: (i) the

firm’s capital stock kt, which also represents the current firm size, and (ii) the agent’s costly but

unobservable effort. The production technology is multiplicativewith respect to these two inputs,

so the agent’s effort has a bigger impact on the firm’s profitability in a large firm. To introduce a

moral hazard problem, we assume that production is exposed to shocks proportional to √kt, and

this is the only source of uncertainty in our model. The realized output can be used for paying

compensation and dividends and for investment toexpand fir m size. The investment plan, defined

as the remaining output after payments to both parties, determines the drift of ktand, moreover,

plays a role in transmitting the production shock to the ktprocess. As a result, the volatility of kt

is also proportional to √kt.

In this setup, we characterize an optimal long-term contract which maximizes the principal’s

expected lifetime payoffaccruing from dividends, subject to the standard individual rationality and

incentive-compatibility conditions. To this end, we first utilize the martingale method developed

by Sannikov (2008) to derive a stochastic representation for the agent’s continuation payoff qt.

As in the previous literature, qtplays the role of a state variable, and its volatility provides the

agent with an incentive for putting forth the necessary effort. Using a recursive definition of

the principal’s value function, we then formulate the dynamic contract problem into a Hamilton-

Jacobi-Bellman equation. However, as our model involves time-varying firm size, the principal’s

value function inevitably depends on the two state variables, ktand qt. Put differently, given

current size and promised value to the agent, the principal has to decide how to control the agent’s

effort and how to expand her own business.

In general, this two-dimensional problem gives rise to partial differential equations which

are often difficult to solve even numerically.3For the sake of tractability, we assume that both

contracting parties have utility exhibiting constant absolute risk aversion(CARA) `

alaHolmstr¨

om

and Milgrom (1987). As is well known, CARA utility allows us to abstract away from the wealth

effect on both sides.4The absence of wealth effects on the agent’s side implies that the agent’s

promised payoff qtdoes not affect his optimal choice of effort. A more important feature of

our framework is that on the principal’s side, the absence of wealth effects implies that qt

does not influence her investment decision. Taking advantage of these two implications, we can

1In He (2009), the agent’shidden action affects the drift of firm size, but not the volatility; the volatility is assumed

to be directly proportional to the current firm size.

2With the square-root volatility that givesa flavor of the Cox-Ingersoll-Ross (CIR) process in Cox et al. (1985), we

can easily derive sufficient conditions under which the process reaches zero. This is one advantageof working with the

square-root volatility rather than other general increasing concave ones. See the webAppendix for the details.

3To circumvent such difficulty, most literature (e.g., He, 2009; Biais et al., 2010; DeMarzo et al., 2012) exploits

the scale-invariance principle that stems from (i) the homogeneity of degree 1 of geometric Brownian motions and (ii)

the risk-neutrality of contracting parties.

4If the principal is risk-neutral, the √ktvolatility has no meaningful implications for dynamic investment.

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