Finance and Economic Growth in a Dynamic Game

Abstract

We investigate how the relaxation of financial constraints affects economic growth in a dynamic game of the tragedy of the commons by introducing an imperfect financial market into Tornell and Velasco’s (J Polit Econ 100(6):1208–1231, 1992) model. It is shown that whereas the relaxation of financial constraints enhances economic growth if agents have access only to a common asset whose property rights are not secure, the relaxation of financial constraints reduces economic growth if agents can have access not only to a common asset but also to a private asset whose property rights are secure.

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Notes

  1. 1.

    Dockner et al. [5] and Dockner and Nishimura [6,7,8], among others.

  2. 2.

    The interest rate, \(r_\tau \), is endogenously determined, as will be explained later [refer to equation (16)].

  3. 3.

    The proofs of the main results carry through even if a capital depreciation rate is introduced. For simplicity of exposition, we assume that capital fully depreciates in one period.

  4. 4.

    The i.i.d. assumption for the productivity shocks can be interpreted in various ways. For instance, defective products, illness among workers, or a fire may accidentally reduce production.

  5. 5.

    For the setting of financial constraints, see also [2] and [3].

  6. 6.

    Lamoreaux [12] examines insider lending in post-Revolution New England in the early nineteenth century and reports that insider lending resulted in the formation of investment clubs that pressured financial intermediaries to lend funds. Insider lending could promote economic growth resolving the asymmetric information between lenders and borrowers in a malfunctioning financial market. This implies that the value of \(\lambda \) might increase as the number of the interest groups increases. The result of the current paper will hold also for that case as in the working paper version of the paper [11]. In the current paper, we assume \(\lambda \) to be constant for simplicity.

  7. 7.

    The derivation of an optimal portfolio allocation of firm h’s savings follows [10].

  8. 8.

    Note that each firm optimally allocates its savings to \(k_{\tau }^h\) and \(b_{\tau }^h\) in each period. In this sense, one can consider that the return, \(R_\tau ^h\), is determined by nature in this dynamic game.

  9. 9.

    \(\alpha N=\alpha (N)N\) is smaller than 1 and increasing in N.

  10. 10.

    We assume the uniform distribution for simplicity of exposition, as it significantly simplifies our analysis. We can investigate the growth rates in this and the next sections by applying more general distributions for productivity shocks (especially in the next section) without changing the main results.

  11. 11.

    One notes from (13) that as the number of interest groups increases, the appropriation rate, \(\alpha \), decreases, and thus, given \(I_t^h\), each individual interest group’s consumption decreases. However, the aggregate appropriation rate, which is given by \(\alpha N=(1-\tilde{\beta })\), increases as the number of interest groups increases. Therefore, given the total product \(Y_t\), the total consumption becomes larger and the total saving becomes smaller as the number of interest groups increases, as seen in (15).

  12. 12.

    It is assumed that agents cannot borrow from abroad, and thus, \(f_\tau ^{i,h}\ge 0\).

  13. 13.

    In addition to Remark 3, for \(\rho \) to exist, the extent of financial constraints, \(\mu \), must be somewhat large. More concretely, it must hold that \(\mu >1/(N-1)\). Therefore, if \(\mu >1/2\), \(\rho \) satisfying Assumption 2 can exist for all \(N\ge 3\).

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Correspondence to Kazuo Nishimura.

Additional information

This work is financially supported by the Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research (Nos. 15H05729, 16H02026, and 16H03598). We are grateful to anonymous referees and a guest editor whose comments and suggestions helped improve and clarify this manuscript. Of course, all remaining errors, if any, are ours.

Appendix

Appendix

Proof of Lemma 2

We set up the Lagrangian as follows:

$$\begin{aligned} L_t^{i,h}:&=E_t\left[ \Sigma _{\tau =t}^\infty \beta ^{\tau -t} \ln c_\tau ^{i,h}\right] \nonumber \\&\quad +E_t\left[ \Sigma _{\tau =t}^\infty \beta ^{\tau -t}\psi _\tau ^{i,h}\left( (1-\theta _\tau (N-1))R_\tau ^h s_{\tau -1}^h-d_\tau ^{i,h}-s_\tau ^h\right) \right] \nonumber \\&\quad +E_t\left[ \Sigma _{\tau =t}^\infty \beta ^{\tau -t}\zeta _\tau ^{i,h}\left( \rho f_{\tau -1}+d_\tau ^{i,h}-f_\tau ^{i,h}-c_\tau ^{i,h}\right) \right] . \end{aligned}$$
(B.1)

The first-order conditions are given by

$$\begin{aligned} \frac{1}{c_t^{i,h}}-\zeta _t^{i,h}= & {} 0, \end{aligned}$$
(B.2)
$$\begin{aligned} -\psi _t^{i,h}+\beta (1-\theta _{t+1}(N-1))R_{t+1}^hE_t[\psi _{t+1}^{i,h}]= & {} 0, \end{aligned}$$
(B.3)
$$\begin{aligned} -\psi _t^{i,h}+\zeta _t^{i,h}= & {} 0, \end{aligned}$$
(B.4)

and

$$\begin{aligned} -\zeta _t^{i,h}+\beta \rho E_t[\zeta _{t+1}^{i,h}]=0. \end{aligned}$$
(B.5)

Under the Markov perfect equilibrium, the optimal control path, \(\{\tilde{d}_t^{i,h}, c_t^{i,h}\}\), exists if and only if \(d_t^{i,h}=\tilde{d}_t^{i,h}=\theta _t R_t^hs_{t-1}^h\) in (21) and (23), and the first-order conditions, (B.2)-(B.5) and the transversality conditions, \(\lim _{s\rightarrow \infty }\beta ^sE_t\left[ s_{t+s}^h/c_{t+s}^{i,h}\right] =0\) and \(\lim _{s\rightarrow \infty }\beta ^sE_t\left[ f_{t+s}^{i,h}/c_{t+s}^{i,h}\right] =0\), are satisfied.

From (B.3)-(B.5), we have

$$\begin{aligned} (1-\theta _{t+1}(N-1))R_{t+1}^h=\rho \end{aligned}$$
(B.6)

or, equivalently,

$$\begin{aligned} \theta _{t+1}=\frac{R_{t+1}^h-\rho }{R_{t+1}^h(N-1)}. \end{aligned}$$
(B.7)

From (B.7), we obtain

$$\begin{aligned} \tilde{d}_t^{i,h}(I_t^h)=\frac{R_t^h-\rho }{R_t^h(N-1)}I_t^h=:\theta _t I_t^h. \end{aligned}$$
(B.8)

This is equation (25). From (23), with \(d_t^{i,h}=\theta _{t}I_t^{i,h}=\theta _{t}R_t^hs_{t-1}^h\), it follows that \(s_t^h=(1-\theta _t N)R_t^hs_{t-1}^h\). From this equation and (B.7), we obtain

$$\begin{aligned} s_t^h=\frac{\rho N-R_t^h}{R_t^h(N-1)}I_t^h. \end{aligned}$$
(B.9)

This is equation (26). We must verify that \(\tilde{d}_t^{i,h}(I_t^h)>0\) and \(s_t^h>0\). Because \(R_{t}^h-\rho \ge r_{t}-\rho =A\mu -\rho >0\), from Assumption 2, we obtain \(\tilde{d}_t^{i,h}(I_t^h)>0\). Moreover, from Assumption 1, it follows that \(R_t^h\le (A-r_t \mu )/(1-\mu )=A(1+\mu )\). Therefore, we obtain \(\rho N-R_t^h\ge \rho N-A(1+\mu )>0\) from Assumption 2. Finally, we derive agent i’s consumption schedule in equilibrium. From (21), (23), and (B.6), it follows that

$$\begin{aligned} \frac{f_{t+1}^{i,h}+s_{t+1}^h}{c_{t+1}^{i,h}}+1=\rho \frac{f_{t}^{i,h}+s_{t}^h}{c_{t+1}^{i,h}}. \end{aligned}$$
(B.10)

Taking the expectation operator \(E_t(.)\) for both sides of (B.10) yields

$$\begin{aligned} E_t\left[ \frac{f_{t+1}^{i,h}+s_{t+1}^h}{c_{t+1}^{i,h}}\right] +1=(f_{t}^{i,h}+s_{t}^h)\rho E_t\left[ \frac{1}{c_{t+1}^{i,h}}\right] . \end{aligned}$$
(B.11)

By applying (B.2) and (B.5) to the right-hand side of (B.11), we obtain

$$\begin{aligned} \frac{f_{t}^{i,h}+s_{t}^h}{c_{t}^{i,h}}=\beta +\beta E_t\left[ \frac{f_{t+1}^{i,h}+s_{t+1}^h}{c_{t+1}^{i,h}}\right] . \end{aligned}$$
(B.12)

From (B.12), it follows that

$$\begin{aligned} \frac{f_{t}^{i,h}+s_{t}^h}{c_{t}^{i,h}}&=\beta +\beta ^2+\cdots +\beta ^s+\beta ^s E_t\left[ \frac{f_{t+s}^{i,h}+s_{t+s}^h}{c_{t+s}^{i,h}}\right] \nonumber \\&=\frac{\beta }{1-\beta }+\lim _{s\rightarrow \infty }\beta ^s E_t\left[ \frac{f_{t+s}^{i,h}+s_{t+s}^h}{c_{t+s}^{i,h}}\right] \end{aligned}$$
(B.13)

From the transversality conditions, we have \(\lim _{s\rightarrow \infty }\beta ^s E_t\left[ (f_{t+s}^{i,h}+s_{t+s}^h)/c_{t+s}^{i,h}\right] =0\), and thus, \(f_{t}^{i,h}+s_{t}^h=\beta c_{t}^{i,h}/(1-\beta )\). This equation and (B.10) yield

$$\begin{aligned} c_t^{i,h}=\rho (1-\beta )(f_{t-1}^{i,h}+s_{t-1}^h). \end{aligned}$$
(B.14)

This is agent i’s consumption schedule given by (27). Conversely, we verify that the transversality conditions hold. Suppose that agent i consumes following the schedule given by (B.14). Then, from (B.10) and (B.14), we obtain \(f_{t}^{i,h}+s_{t}^h=\beta c_{t}^{i,h}/(1-\beta )\). From the last equation and (B.13), it follows that

$$\begin{aligned} \lim _{s\rightarrow \infty }\beta ^s E_t\left[ \frac{f_{t+s}^{i,h}+s_{t+s}^h}{c_{t+s}^{i,h}}\right] =0. \end{aligned}$$
(B.15)

In (B.15), because \(f_{t+s}^{i,h}\ge 0\) and \(s_{t+s}^h>0\), it holds that \(\lim _{s\rightarrow \infty }\beta ^sE_t\left[ s_{t+s}^h/c_{t+s}^{i,h}\right] =0\) and \(\lim _{s\rightarrow \infty }\beta ^sE_t\left[ f_{t+s}^{i,h}/c_{t+s}^{i,h}\right] =0\). \(\square \)

Proof of Theorem 2

The second claim is obvious. For the first claim, define a function such that \(f(x)=(2\rho x-A(1+\mu ))/(x-1)\) in \(x\in (0, \infty )\). Then, it follows from Assumption 2 that \({{\mathrm{sign}}}(f^\prime (x))={{\mathrm{sign}}}(A(1+\mu )-2\rho )>0\). The first claim follows from the last inequality. \(\square \)

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Kunieda, T., Nishimura, K. Finance and Economic Growth in a Dynamic Game. Dyn Games Appl 8, 588–600 (2018). https://doi.org/10.1007/s13235-018-0249-7

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Keywords

  • Dynamic game
  • Interest groups
  • Property rights
  • Financial market imperfections
  • Economic growth

JEL Classification

  • C73
  • D92
  • O43