A Two-Sector Growth Model with Credit Market Imperfections and Production Externalities

Abstract

A two-sector dynamic general equilibrium model with financial constraints and production externalities is studied. Agents face idiosyncratic productivity shocks in each period. Agents who draw high productivity borrow resources in the financial market and become capital producers, whereas agents who draw low productivity become lenders. We analyze how the interaction between the extent of financial constraints and sector-specific production externalities affects the characterization of equilibria in a two-sector economy.

JEL Classification: E32, E44, O41

Mathematics Subject Classification (2010): 37N40, 39A30

Appendix

1.1 Proof of Proposition 3.1

It follows from Lemma 3.1 that

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \int_{\Xi_t}\int_{\Omega^t}s_t(\omega^{t-1})dP^t(\omega^{t-1})dP(\omega_t) \\ &\displaystyle &\displaystyle \qquad \qquad \qquad -\frac{\mu}{1-\mu}\int_{\Omega\setminus \Xi_t}\int_{\Omega^t}s_t(\omega^{t-1})dP^t(\omega^{t-1})dP(\omega_t)=0. \\ {} \end{array} \end{aligned} $$

(34)

where Ξ_t = {ω _t ∈ Ω| Φ_t(ω _t) ≤ ϕ _t}. (34) can be rewritten as

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \int_{\Omega^t}s_t(\omega^{t-1})dP^t(\omega^{t-1})\int_0^{\phi_t}dG(\Phi) \\ &\displaystyle &\displaystyle \qquad \qquad \qquad -\frac{\mu}{1-\mu}\int_{\Omega^t}s_t(\omega^{t-1})dP^t(\omega^{t-1})\int_{\phi_t}^\eta dG(\Phi)=0, \\ \end{array} \end{aligned} $$

which is computed as

$$\displaystyle \begin{aligned} \int_{\Omega^t}s_t(\omega^{t-1})dP^t(\omega^{t-1})\left[G(\phi_t)-\frac{\mu}{1-\mu}(1-G(\phi_t))\right]=0. \end{aligned}$$

Solving the last part, we obtain G(ϕ _t) = μ. $\square$

1.2 Proof of Lemma 3.2

Because $R_t(\omega _{t-1})=\max \{r_t, (q_t\Phi _{t-1}(\omega _{t-1})-r_t\mu )/(1-\mu )\}$, it follows that

$$\displaystyle \begin{aligned} \int_{\Omega^t}R_t(\omega_{t-1})s_{t-1}(\omega^{t-2})dP^t(\omega^{t-1})&=\int_{\Omega^t}\max\left\{r_t, \frac{q_t\Phi_{t-1}(\omega_{t-1})-r_t\mu}{1-\mu}\right\} \\ &\times s_{t-1}(\omega^{t-2})dP^t(\omega^{t-1})=:I_t \end{aligned} $$

Define Ξ_t−1 = {ω _t−1 ∈ Ω| Φ_t−1(ω _t−1) ≤ ϕ _t−1}, as in the proof of Proposition 3.1. Because ϕ _t−1 = r _t∕q _t, I _t can be computed as follows:

$$\displaystyle \begin{aligned} I_t&=\int_{\Omega^{t-1}\times \Xi_{t-1}}r_ts_{t-1}(\omega^{t-2})dP^t(\omega^{t-1}) \notag \\ &+\int_{\Omega^{t-1}\times(\Omega\setminus \Xi_{t-1})}\frac{q_t\Phi_{t-1}(\omega_{t-1})-r_t\mu}{1-\mu}s_{t-1}(\omega^{t-2})dP^t(\omega^{t-1}) \notag \\ &=\int_{\Omega^{t-1}\times(\Omega\setminus \Xi_{t-1})}\frac{q_t\Phi_{t-1}(\omega_{t-1})}{1-\mu}s_{t-1}(\omega^{t-2})dP^t(\omega^{t-1}), {} \end{aligned} $$

(35)

where the second equality of (35) is obtained from Lemma 3.1. Agents who draw Φ_t−1(ω _t−1) > ϕ _t−1 invest x _t−1(ω ^t−1) = s _t−1(ω ^t−2)∕(1 − μ), and thus, (35) becomes

$$\displaystyle \begin{aligned} I_t&=\int_{\Omega^{t-1}\times(\Omega\setminus \Xi_{t-1})}\frac{q_t\Phi_{t-1}(\omega_{t-1})}{1-\mu}(1-\mu)x_{t-1}(\omega^{t-1})dP^t(\omega^{t-1}) \notag \\ &=q_t\int_{\Omega^{t-1}\times(\Omega\setminus \Xi_{t-1})}\Phi_{t-1}x_{t-1}(\omega^{t-1})dP^t(\omega^{t-1}). {} \end{aligned} $$

(36)

Because $k_t=\int _{\Omega ^{t-1}\times (\Omega \setminus \Xi _{t-1})}\Phi _{t-1}x_{t-1}(\omega ^{t-1})dP^t(\omega ^{t-1})$, (36) becomes

$$\displaystyle \begin{aligned} I_t=q_tk_t.~~\square \end{aligned} $$

1.3 Proof of Lemma 3.3

By using Lemma 3.2 and because $\int _{\Omega ^t}\pi _tdP^t(\omega ^{t-1})=\Pi _t$, the aggregation of (6) across all agents is obtained as follows:

$$\displaystyle \begin{aligned} \int_{\Omega^t}s_t(\omega^{t-1})dP^t(\omega^{t-1})&=\int_{\Omega^t}[R_t(\omega_{t-1})s_{t-1}(\omega^{t-2})+w_t+\pi_t \notag \\ &-p_tc_t(\omega^{t-1})]dP^t(\omega^{t-1}) \notag \\ &=q_tk_t+w_t+\Pi_t-p_tC_t. {} \end{aligned} $$

(37)

From (8), we have $F^1(l_t^1, k_t^1)+p_tF^2(l_t^2, k_t^2)+(1-\delta )k_t=q_tk_t+w_t+\Pi _t$. Additionally, the consumption goods market-clearing condition leads to $p_tF^2(l_t^2, k_t^2)=p_tC_t$. Therefore, (37) is transformed into

$$\displaystyle \begin{aligned} \int_{\Omega^t}s_t(\omega^{t-1})dP^t(\omega^{t-1})=F^1(l_t^1, k_t^1)+(1-\delta)k_t.~~~\square {} \end{aligned} $$

(38)

1.4 Proof of Proposition 3.2

Because k _t+1 is produced by capital producers who draw an individual-specific productivity, Φ_t(ω _t), that is greater than ϕ _t, Lemma 3.3 and the i.i.d. assumption compute k _t+1 as follows:

$$\displaystyle \begin{aligned} k_{t+1}&=\int_{\Omega^{t}\times (\Omega\setminus \Xi_t)}\Phi_t(\omega_t)x_t(\omega^t)dP^{t+1}(\omega^t) \notag \\ &=\int_{\Omega\setminus \Xi_t}\int_{\Omega^{t}}\Phi_t(\omega_t)\frac{s_t(\omega^{t-1})}{1-\mu}dP^t(\omega^{t-1})dP(\omega_t) \notag \\ &=\int_{\phi_t}^\eta\frac{\Phi_t(\omega_t)}{1-\mu}dG(\Phi)\int_{\Omega^{t}}s_t(\omega^{t-1})dP(\omega^{t-1}) \notag \\ &=\frac{H(\phi^*)}{1-\mu}(F^1(l_t^1,k_t^1)+(1-\delta)k_t), \notag \end{aligned} $$

where $H(\phi ^*)=\int _{\phi ^*}^\eta \Phi _t(\omega _t) dG(\Phi )$ because ϕ _t = ϕ ^∗ in equilibrium. $\square$.

1.5 Proof of Lemma 3.4

Obviously, M(μ) is continuous in [0, 1). The inverse function theorem implies

$$\displaystyle \begin{aligned} \frac{\partial M(\mu)}{\partial \mu}=\frac{\partial}{\partial \mu}\left(\frac{H(\phi^*)}{1-\mu}\right)&=\frac{-(1-\mu)\phi^*G^\prime(\phi^*)(\partial \phi^*/\partial \mu)+H(\phi^*)}{(1-\mu)^2} \notag \\ &=\frac{\int_{\phi^*}^h\Phi_t(\omega_t)dG(\Phi)-\phi^*(1-G(\phi^*))}{(1-\mu)^2}>0. \notag \end{aligned} $$

Therefore, M(μ) is an increasing function with respect to μ in [0, 1). It is straightforward to verify that M(0) = H(0) is the mean of the idiosyncratic productivity shocks. By applying L’Hôpital’s rule, we obtain lim_μ→1 M(μ) =lim_μ→1 G ⁻¹(μ)G ^{−1 ′}(μ)G ^′(ϕ ^∗) = η. For the last equality, we have used the inverse function theorem again. $\square$

1.6 Proof of Proposition 3.3

Because ϕ _t = r _t+1∕q(p _t+1) and ϕ _t = ϕ ^∗, it follows that

To derive the last equality, Proposition 3.1 is applied. $\square$

1.7 Proof of Lemma 3.5

From Assumption 3.2, it follows that (1 − δ)M(μ) < 1. Then, under Assumption 3.3, from (12), (28), and (29), it follows that $sign\{k^*-\alpha _K^1w(p^*)/(\alpha _L^1v(p^*))\}=sign\{1-(1-\delta )\beta M(\mu )-\alpha _K^1 \beta (1-(1-\delta )M(\mu ))\}$. Because $1-(1-\delta )\beta M(\mu )-\alpha _K^1 \beta (1-(1-\delta )M(\mu ))>(1-\alpha _K^1\beta )(1-(1-\delta )M(\mu ))>0$, it follows that $sign\{k^*-\alpha _K^1w(p^*)/(\alpha _L^1v(p^*))\}>0$. Additionally, it follows that $sign\{\alpha _K^2w(p^*)/\\ (\alpha _L^2v(p^*))-k^*\}=sign\{1-(1-\delta )M(\mu )\}>0$. $\square$

1.8 Proof of Theorem 4.1

To prove Theorem 4.1, two lemmata are prepared.

Lemma A.1

Under Assumptions 2.1 , 3.2 , and 3.3 , it holds that 0 < κ ₂ < 1.

Proof

It follows from Assumption 3.2 that 0 < (1 − δ)βM(μ) < 1, which leads to (θ ₂ − θ ₁)∕(θ ₂ − θ ₁(1 − δ)βM(μ)) < 1 from Assumptions 2.1, 3.2, and 3.3. Additionally, it follows from Assumption 3.3 that (θ ₂ − θ ₁)∕(θ ₂ − θ ₁(1 − δ)βM(μ)) > 0 □

Lemma A.2

Under Assumptions 2.1 , 3.2 , 3.3 , and 4.4 , suppose that the mean of the stochastic productivity shocks, M(0), is smaller than M ₁ and that the maximum, η, is greater than M ₂ . Then, as the value of μ increases from 0 to 1, the value of κ ₁ increases in the ranges, as in the following.

• As μ increases in [0, μ ₁), κ ₁ increases in $[\tilde {\kappa }, -1)$.

• As μ increases in [μ ₁, μ ₂), κ ₁ increases in [−1, 0).

• As μ increases in [μ ₂, 1), κ ₁ increases in $[0, \bar {\kappa })$ ,

where $\tilde {\kappa }:=[(\Delta +\alpha _L^2)(1-\delta )\beta M(0)-\alpha _L^2]/(\beta \Delta )\in (-\infty , -1)$ and $\bar {\kappa }< 1-(1-\beta )\alpha _L^2/(\beta \Delta )\in (0, 1)$ , which is given when M(μ) = η.

Proof

M(μ) is an increasing function with respect to μ. Then, Fig. 2 and (32) prove the claims. □

Proof of Theorem 4.1

From Lemma A.1, we have |κ ₂| < 1. From Lemma A.2, if μ ∈ [0, μ ₁), |κ ₁| > 1, and if μ ∈ (μ ₁, 1), |κ ₁| < 1. Therefore, the steady state is a saddle point if μ ∈ [0, μ ₁), and the steady state is totally stable if μ ∈ (μ ₁, 1). □