Growth, innovation, credit constraints, and stock price bubbles

Abstract

We study the potential for rational bubbles in the innovation sector to affect long term economic growth. We show that stock market prices of research and development (R&D) firms can include a bubble component when credit constraints are present. Bubbles are self-sustained in equilibrium by a “liquidity” premium that originates when credit constraints are relaxed. Bubbles expand the borrowing and production capacities of R&D firms, stimulate innovation, and increase the growth rate. Small firms benefit more from bubbles than big firms. Bubbles are magnified by tighter credit constraints and scarce investment opportunities. Thus, financially underdeveloped countries benefit more from bubbles. Finally, we show that bubbles can create permanent reallocation effects that benefit the innovation sector over other sectors.

Appendices

Appendix 1: Proof of Proposition 6

Assume that the solution of R&D firm j’s problem is $V_{t}\left( K_{t} ^{j}\right) =a_{t}K_{t}^{j}+B_{t}$. Substituting (6)–(9), and the solution we guess into (32), we have the following:

$$\begin{aligned} a_{t}K_{t}^{j}+B_{t}&=\max _{K_{t+1}^{j},K_{It+1}^{j},I_{t}^{j},B_{t}^{j} }\left( 1-\theta \right) r_{t}K_{t}^{j}+\left( 1-\theta \right) q_{t}\left( 1-\delta \right) K_{t}^{j}\\&\quad +\,\left( 1-\theta \right) \rho _{t+1}BB_{t+1}+\left( 1-\theta \right) (1-\pi ) \left[ -q_{t}K_{t+1}^{j}+\rho _{t+1}a_{t+1}K_{t+1}^{j}\right] \\&\quad +\,\left( 1-\theta \right) \pi \left[ \begin{array}{c} \left( q_{t}-1\right) \left( r_{t}K_{t}^{j}+\rho _{t+1}a_{t+1}\xi \left( 1-\delta \right) K_{t}^{j}+\rho _{t+1}B_{t+1}\right) \\ -q_{t}K_{It+1}^{j}+\rho _{t+1}a_{t+1}K_{It+1}^{j} \end{array} \right] \\&\quad +\,\theta a_{t}^{\#}K_{t}^{j} \end{aligned}$$

By taking first order derivative of $K_{t+1}^{j}$, we have the following:

$$\begin{aligned} q_{t}=\rho _{t+1}a_{t+1} \end{aligned}$$

By comparing the left-hand and right-hand sides, we obtain the following:

$$\begin{aligned} a_{t}& {}= \left( 1-\theta \right) r_{t}+\left( 1-\theta \right) q_{t}\left( 1-\delta \right) +\left( 1-\theta \right) \pi \left( q_{t}-1\right) \left[ r_{t}+\rho _{t+1}a_{t+1}\xi \left( 1-\delta \right) \right] +\theta a_{t}^{\#}\\ B_{t}& {}= \left( 1-\theta \right) \left[ 1+\pi \left( q_{t}-1\right) \right] \rho _{t+1}B_{t+1} \end{aligned}$$

This is the proposition.

Appendix 2:Derivation of reallocation effects model

1.1 Households

The only difference is the budget constraints are now as follows:

$$\begin{aligned} C_{t}+ {\int } \left( V_{t}^{j}-D_{t}^{j}\right) \psi _{t+1}^{j}dj= {\int } V_{t}^{j}\psi _{t}^{j}dj+W_{t}\overset{\_}{L}. \end{aligned}$$

1.2 Final goods producer

The technology is represented as follows:

$$\begin{aligned} Y_{t}=A \int \nolimits _{n=0}^{N_{t}} \left( X_{t}^{n}\right) ^{\sigma }(L_{t}^{Y})^{1-\sigma }dn,0<\sigma <1 \end{aligned}$$

Then, the profit maximization problem of the final goods producer, subject to the production function, is expressed as follows:

$$\begin{aligned} \max _{X_{t}^{n},L_{t}^{Y}}A {\int \nolimits _{n=0}^{N_{t}}} \left( X_{t}^{n}\right) ^{\sigma }(L_{t}^{Y})^{1-\sigma }dn- {\int \nolimits _{n=1}^{N_{t}}} P_{t}^{n}X_{t}^{n}dn-W_{t}L_{t}^{Y} \end{aligned}$$

It is easy to solve the profit maximization problem, and we have the demand function for intermediate goods n as follows:

$$\begin{aligned} P_{t}^{n}& {}= \sigma A\left( X_{t}^{n}\right) ^{\sigma -1}(L_{t}^{Y})^{1-\sigma }\\ W_{t}& {}= \left( 1-\sigma \right) A {\int \nolimits _{n=1}^{N_{t}}} \left( X_{t}^{n}\right) ^{\sigma }(L_{t}^{Y})^{-\sigma }dn. \end{aligned}$$

1.3 Intermediate goods producers

The intermediate goods producer n has the following profit:

$$\begin{aligned} \left( P_{t}^{n}-1\right) X_{t}^{n}=\sigma A\left( X_{t}^{n}\right) ^{\sigma }(L_{t}^{Y})^{1-\sigma }-X_{t}^{n} \end{aligned}$$

Since we already have the intermediate goods producer n’s demand function, we can find the price of goods that the intermediate goods producer n sets.

$$\begin{aligned} P_{t}^{n}=\frac{1}{\sigma } \end{aligned}$$

The amount the producer produces is expressed as follows:

$$\begin{aligned} X_{t}^{n}=\sigma ^{\frac{2}{1-\sigma }}A^{\frac{1}{1-\sigma }}L_{t} ^{Y} \end{aligned}$$

(35)

Then, the profit of producing goods for producer n in every period is as follows:

$$\begin{aligned} \left( \frac{1-\sigma }{\sigma }\right) \sigma ^{\frac{2}{1-\sigma }}A^{\frac{1}{1-\sigma }}L_{t}^{Y} \end{aligned}$$

The discounted total profits from selling goods n must be equal to the license fee to produce goods n. This implies,

$$\begin{aligned} \sum _{s=t}^{\infty }\rho \left( s,t\right) \left( \frac{1-\sigma }{\sigma }\right) \sigma ^{\frac{2}{1-\sigma }}A^{\frac{1}{1-\sigma }}L_{t}^{Y}=\eta _{n}. \end{aligned}$$

Here, $\rho \left( s,t\right) = {\prod \nolimits _{v=t+1}^{s}} \left( \rho _{v+1}\right)$ if $s\ne t$, $\rho \left( s,t\right) =1$ if $s=t$. Since only variables in $\eta _{n}$ are time varying, patents created in the same period have the same price. This result gives us the following equation:

$$\begin{aligned} \sum _{s=t}^{\infty }\rho \left( s,t\right) \left( \frac{1-\sigma }{\sigma }\right) \sigma ^{\frac{2}{1-\sigma }}A^{\frac{1}{1-\sigma }}L_{t}^{Y}=\eta _{n}=\eta _{t}. \end{aligned}$$

(36)

1.4 R&D sector

Same as the baseline model.

1.5 Competitive equilibrium

Definition 7

A competitive equilibrium is defined as allocations.

$\left\{ Y_{t},K_{t},C_{t},I_{t},N_{t},E_{t}^{j},T_{t},L_{t}^{j},L_{t} ^{Y},I_{t}^{j},K_{t}^{j},T_{t}^{j},Y_{t}^{i},\psi _{t}^{j},X_{t}^{n}\right\}$ and prices

$\left\{ w_{t},P_{n}^{t},R_{t}^{j},q_{t},\eta _{t},r_{t},V_{t}^{j}\right\}$ such that a household maximizes its utility, firms in all three sectors maximize their profits, and market clearing conditions are satisfied as follows: stock market is clearing $\psi _{t}^{j}=1$, labor market is clearing $\int _{0}^{1}L_{t}^{j}dj+L_{t}^{Y}=\overset{\_}{L}$, debt market is clearing $\int _{0}^{1}E_{t}^{j}dj=0$, capital market is clearing $K_{t+1}=\left( 1-\delta \right) K_{t}+I_{t}$, goods market is clearing $C_{t}+\int _{n=0}^{N_{t}} {\int _{0}^{1}} X_{t}^{n}dn+I_{t}=Y_{t}$, and the amount of patent follows $N_{t+1} =N_{t}+T_{t}$.

1.6 Detrended dynamic system

The detrended dynamic system now becomes:

$$\begin{aligned}&X_{t}=\sigma ^{\frac{2}{1-\sigma }}A^{\frac{1}{1-\sigma }}L_{t}^{Y}\\&\sum _{s=t}^{\infty }\rho \left( s,t\right) \left( \frac{1-\sigma }{\sigma }\right) \sigma ^{\frac{2}{1-\sigma }}A^{\frac{1}{1-\sigma }}L_{t}^{Y}=\eta _{n}=\eta _{t}\\&\rho _{t+1}=\beta \frac{c_{t}}{c_{t+1}\left( 1+g_{t+1}^{c}\right) }\\&r_{t}k_{t}=Z\eta _{t}\left( k_{t}\right) ^{\alpha }-w_{t}\left( \overset{\_}{L}-L_{t}^{Y}\right) \\&a_{t}=r_{t}+q_{t}\left( 1-\delta \right) +\pi \left( q_{t}-1\right) \left( r_{t}+\rho _{t+1}a_{t+1}\xi \left( 1-\delta \right) \right) \\&q_{t}=\rho _{t+1}a_{t+1} \\&b_{t}=\left[ 1+\pi \left( q_{t}-1\right) \right] \rho _{t+1}\left( 1+g_{t+1}\right) b_{t+1} \\&g_{t+1}^{N}=t_{t} \\&t_{t}=Zk_{t}^{\alpha }\left( \overset{\_}{L}-L_{t}^{Y}\right) ^{1-\alpha }\\&(1+g_{t+1}^{N})k_{t+1}=\left( 1-\delta \right) k_{t}+\pi r_{t}k_{t}+\pi \rho _{t+1}\left[ a_{t+1}\left( \xi \left( 1-\delta \right) \right) k_{t}+(1+g_{t+1}^{N})b_{t+1}\right] \\&r_{t}=\alpha \left( Z\eta _{t}\right) ^{\frac{1}{\alpha }}\left( \frac{w_{t}\left( \overset{\_}{L}-L_{t}^{Y}\right) ^{\alpha }}{1-\alpha }\right) ^{\frac{\alpha -1}{\alpha }}\\&c_{t}+X_{t}+\pi r_{t}k_{t}+\pi \rho _{t+1}\left[ a_{t+1}\left( \xi \left( 1-\delta \right) \right) k_{t}+(1+g_{t+1}^{N})b_{t+1}\right] =AX_{t} ^{\sigma }\\&w_{t}=\left( 1-\sigma \right) A\left( X_{t}\right) ^{\sigma }(L_{t} ^{Y})^{-\sigma } \end{aligned}$$

1.7 Stochastic bubbles burst

We also study the case when bubbles burst stochastically. The setting is similar to stochastic bubbles in the baseline model. We only report the simulation result here in Fig. 10. Similar to the relationship between stochastic and unanticipated bursts in baseline model, the pattern of both these bursts are similar. The intuition is also similar to the one discussed before.

Fig. 10

Burst of stochastic bubbles in reallocation effects model