Asset Market Volatility and New Keynesian Macroeconomics: A Game-Theoretic Approach

Abstract

This study develops a game-theoretic approach to asset market bubbles. In our model, portfolio investments consist of risk-free and risky assets. Risky assets attract more investors who may adopt a hawk strategy, because they promise a higher return than risk-free assets. However, the economy falls into a full-blown crisis state when the portion of investment devoted to risky assets exceeds a threshold. Furthermore, we incorporate the periodic bubbles in asset markets into a New Keynesian baseline model. Our simulation results show that high volatility in asset markets increases the variability of inflation and output dynamics, while uncertainty about the dynamic economy can be amplified and hence may be seen as shocks for an inefficient distribution of wealth.

Appendixes

1.1 Pure and Mixed Strategy Nash Equilibria

The pure strategy Nash equilibria are the same as the HD game: ($\text{ Risk-free }_{1}$, $\text{ Risky }_{2}$), ($\text{ Risky }_{1}$, $\text{ Risk-free }_{2}$). The best strategy is to choose the opposite strategy to that of the opponent. Similarly, the mixed strategy can be considered: $S_j=(1-\hat{\beta }_j,\ \hat{\beta }_j)$. In this case, $\hat{\beta }$ suggests the probability of choosing a risky strategy. Then, the payoff of agent j can be expressed as follows¹³:

$$\begin{aligned} \pi _j(S_j)= & {} S_j A_j S_{-j}^T,\quad S_j=(1-\hat{\beta }_j,\ \hat{\beta }_j),\nonumber \\ A_1= & {} \begin{pmatrix} r^{S}&{}r^{S}\\ r^{R}&{}-1 \end{pmatrix},\quad A_2=\begin{pmatrix} r^{S}&{}r^{R}\\ r^{S}&{}-1 \end{pmatrix},\quad 1>r^{R}>r^{S}>0. \end{aligned}$$

(24)

Hence, we can derive Eq. (25) from (24).

$$\begin{aligned} \pi _j(\hat{\beta }_j)=\hat{\beta }_j(-r^{S}+(1-\hat{\beta }_{-j})r^{R}-\hat{\beta }_{-j})+r^{S}. \end{aligned}$$

(25)

Figure 8 shows three Nash equilibria of two person HD game: $\{(\hat{\beta }_1^*,\ \hat{\beta }_2^*)\}=\{(1-(1+r^{S})/(1+r^{R}),\ 1-(1+r^{S})/(1+r^{R})),\ (1,0),\ (0,1)\}$.

Fig. 8

Mixed strategy Nash equilibria in a two-person game. NoteDashed line indicates the best response of player 1, while solid line is used to show the best response of player 2

1.2 Replicator Dynamics in the Crisis State

In this section, we show the derivation of wealth dynamics in the crisis state. For $\bar{\beta }_t$ is increasing in time t, there exists $\Lambda _1$ that satisfies the crisis condition: $\beta _{\Lambda _1}>\bar{R}$. Then, we obtain the total wealth as follows:

$$\begin{aligned} W_{\Lambda _1+1}(j)=(1-j)(1+r^S)w_0 (1+r^S+j(r^R-r^S))^{\Lambda _1}. \end{aligned}$$

(26)

Let ${\hat{t}}$ be the time index after crisis:

$$\begin{aligned} t=\Lambda _1+1+\hat{t},\quad \text {or}\quad {\hat{t}}:=t-\Lambda _1-1. \end{aligned}$$

Until the second crisis occurs, we arrive at:

$$\begin{aligned} W_{{\hat{t}}}(j)= (1+r^S+j(r^R-r^S))^{\hat{t}} W_{\Lambda _1+1}(j). \end{aligned}$$

From Eq. (26), we obtain the following equation for the period $t=\Lambda _1+1+{\hat{t}}$:

$$\begin{aligned} \bar{W}_t= & {} \int _0^1 W_{{\hat{t}}}(j) dj = \int _0^1 (1-j)(1+r^S)w_0 (1+r^S+j(r^R-r^S))^{\Lambda _1+{\hat{t}}}dj\\= & {} (1+r^S)w_0 \int _0^1 (1-j)(1+r^S+j(r^R-r^S))^{t-1}dj. \end{aligned}$$

Similarly, if the crisis state occurs $\Gamma$ times, $\bar{W}_t$ and $\bar{W}^R_t$ can be calculated as follows:

$$\begin{aligned} \bar{W}_t&= (1+r^S)w_0 \int _0^1 (1-j)^{\Gamma }(1+r^S+j(r^R-r^S))^{t-\Gamma }dj;\\ \bar{W}^R_t&= (1+r^S)w_0 \int _0^1 j(1-j)^{\Gamma }(1+r^S+j(r^R-r^S))^{t-\Gamma }dj. \end{aligned}$$

Now we derive the crisis period $t^*$ as a limiting case. From Eq. (17), we observe the first crisis occurs when:

$$\begin{aligned} \bar{\beta }_t = \frac{\Pi _t - r^S}{r^R-r^S} > \bar{R}. \end{aligned}$$

Further, from Eq. (15),

$$\begin{aligned} \Pi _t = \frac{\frac{(1+r^R)^{t+2}-(1+r_S)^{t+2}}{t+2}}{\frac{(1+r^R)^{t+1}-(1+r^S)^{t+1}}{t+1}}-1. \end{aligned}$$

As $\Pi _t$ is increasing with regard to t in the normal state, $t^*$ can be calculated from:

$$\begin{aligned} \frac{\frac{(1+r^R)^{t^*+2}-(1+r_S)^{t^*+2}}{t^*+2}}{\frac{(1+r^R)^{t^*+1}-(1+r^S)^{t^*+1}}{t^*+1}} = (r^R-r^S)\bar{R} + r^S + 1. \end{aligned}$$

If $r^R$ is sufficiently large, $(\frac{1+r^S}{1+r^R})^{t^*}$ converges to 0. This can result in:

From this, we obtain the simulation results in Figs. 5 and 7, the parameter values for $r^S$, $r^R$, $\bar{R}$, the simulated period of $t^{**}$, and the theoretical period of $t^*$ given in Table 2.

1.3 Portfolio Strategy and Nash Equilibrium with Many Players

This section explains the simulation process for asset market dynamics. After initialization that all parameters are set to predefined initial values, all agents choose the strategy for their portfolios ($\beta _{j,t}$). Then, the artificial economy aggregates the amount of risky asset ($\bar{\beta }_t$), which determines the macro state (normal, crisis). Afterwards, the economy determines individual returns for the investment and the wealth will be distributed accordingly. Figure 9 shows that the process will be repeated until the final period.

Fig. 9

Flow chart of the simulation process

More precisely, we consider the best response of jth player ($\beta _j^*$) as the solution of maximization problem:

$$\begin{aligned} \arg \ \max _{\beta _j}&\ \Pi _j,\\ \Pi _j (\beta _j)&:= (1-\beta _j)r^SW_j+r^R\beta _jW_j\Psi \left( \bar{\beta }_t\le \bar{R}\right) -\beta _jW_j\Psi \left( \bar{\beta }_t>\bar{R}\right) , \end{aligned}$$

where $\Psi (\bullet )$ is the indicator function. Its function value is one if the input is true and zero otherwise. zero otherwise. In other words, $\Psi$ depends on $\bar{\beta }_{t}$ and $\bar{R}; \Psi (\bar{\beta }_{t} \le \bar{R})$, i.e., $\Psi =1$ when current state is normal, and zero otherwise.

$$\begin{aligned} \bar{\beta }_t := \frac{\sum \beta _j W_j}{\sum W_j}. \end{aligned}$$

Now we rewrite $\Pi _j$ with regard to $\beta _j$:

$$\begin{aligned} \Pi _j(\beta _j) = r^S W_j + \beta _j W_j \left( \Psi (1+r^R)-(1+r^S)\right) . \end{aligned}$$

The following three cases are based on $\Psi$:

• (i) $\Psi = 1$ regardless of $\beta _j$. In this case, $\beta _j^*=1$ because $\frac{\partial \Pi }{\partial \beta _j}>0$ (corner solution).

• (ii) $\Psi = 0$ regardless of $\beta _j$. In this case, $\beta _j^*=0$ because $\frac{\partial \Pi }{\partial \beta _j}<0$ (corner solution).

• (iii) $\Psi$ can be either 1 or 0 depending on $\beta _j$. This means that the sign of $\bar{R} - \bar{\beta }_t$ can be positive or negative by the value of $\beta _j$ (interior solution). In this case,

  $$\begin{aligned} \bar{R} - \bar{\beta }_t(\beta _j) = \bar{R} - \frac{\beta _jW_j + \sum _{k\ne j}\beta _k W_k}{W_j + \sum _{k\ne j} W_k} \end{aligned}$$

, where $\frac{\partial \bar{R} - \bar{\beta }_t}{\partial \beta _j}$ is always negative and $\bar{R} - \bar{\beta }_t$ is strictly decreasing with regard to $\beta _j$. Therefore $\beta _j^*=\bar{R} W_j + \sum _{k\ne j}(\bar{R} - \beta _k)W_k$, because $\Pi _j$ is increasing if $\beta _j <\beta _j^*$ and decreasing if $\beta _j > \beta _j^*$ (note that $\Psi =1$ when $\bar{R} - \bar{\beta }_t > 0$ by the definition of $\Psi (\bullet )$).

In fact, $\beta _j^*$ is the value for which $\bar{\beta }_t$ is equal to $\bar{R}$. In particular, if $0<\beta _j^*<1\quad \forall j$,—i.e., all $\beta _j^*$ are internal solution—the solution vector $\varvec{\beta }^*= (\beta _1^*,\ldots ,\beta _N^*)$ is the surface of the hyperplane, we immediately obtain:

$$\begin{aligned} \bar{R} = \sum _j\left( \frac{W_j}{\sum _k W_k}\beta _j^*\right) . \end{aligned}$$