A Contagious Malady? Open Economy Dimensions of Secular Stagnation

Abstract

Conditions of secular stagnation—low interest rates, below target inflation, and sluggish output growth—characterize much of the global economy. We consider an overlapping generations, open economy model of secular stagnation, and examine the effect of capital flows on the transmission of stagnation. In a world with a low natural rate of interest, greater capital integration transmits recessions across countries as opposed to lower interest rates. In a global secular stagnation, expansionary fiscal policy carries positive spillovers implying gains from coordination, and fiscal policy is self-financing. Expansionary monetary policy, by contrast, is beggar-thy-neighbor with output gains in one country coming at the expense of the other. Similarly, we find that competitiveness policies including structural labor market reforms or neomercantilist trade policies are also beggar-thy-neighbor in a global secular stagnation.

Appendices

Appendix A: Existence, Uniqueness, and Local Determinacy

Here we provide formal proofs for various propositions presented in the body of the text.

Proposition 2

If the international lending constraint is binding, $r^n > 0$, $r^{n *} < 0$, $\bar{\Pi } = \bar{\Pi }^{*} = 1,$ and $\gamma ^{*} > 0$, there exists a unique, locally determinate secular stagnation equilibrium in the creditor country with $i^{*} = 0$, $\Pi ^{*} < 1$, and $Y^{*} < Y_f^{*}$.

Proof

Under the assumptions of the proposition and the monetary policy rule, the zero lower bound is binding for the creditor country and the equilibrium real interest rate in steady state is given by $r^{*} = \frac{1}{\Pi ^{*}}$. Equilibrium inflation and output in steady state in the creditor country solve the following equations:

$$\begin{aligned} Y^{*}&= D^{*} + K^{*} + \psi ^{*} \Pi ^{*} \end{aligned}$$

(48)

$$\begin{aligned} Y^{*}&= \left( \frac{1-\frac{\gamma ^{*}}{\Pi ^{*}}}{1-\gamma ^{*}}\right) ^{\frac{\alpha }{1-\alpha }} Y^{*}_{f} \end{aligned},$$

(49)

where $\psi ^{*} = \frac{1+\beta }{\beta }\left( 1+g\right) \left( D^{*}+\frac{1+r^n}{1+\beta }K^{*}\right) > 0$. We may define the difference equation $\Delta \left( \Pi ^{*}\right)$ by taking the difference between (48) and (49) . This function is continuous in $\Pi ^{*}$ with $\Delta (\gamma ^{*}) > 0$ and $\Delta (1) < 0$ . Therefore, there exists a $\gamma ^{*}< \Pi ^{*} < 1$ such that $\Delta (\Pi ^{*}) = 0$. Since $\Pi ^{*} < 1$, it follows that $Y^{*} < Y_f^{*}$.

To establish uniqueness, we first assume that their exist multiple distinct values of $\Pi ^{*}$ at which $\Delta (\Pi ^{*}) = 0$. Graphically, in inflation-output space (output on the x-axis), the AS curve [Equation (49)] lies above the AD curve [Equation (49)] when inflation equals $\gamma ^{*}$ and the AS curve lies below the AD curve for inflation at unity. Thus, if multiple steady states exist, given that AS is a continuous function, there must exist at least three distinct points at which the AS and AD curve intersect.

At the first intersection point, the slope of AS curve crosses the AD line from above and, therefore, at the second intersection the AS curve crosses the AD curve from below. Since the AD curve is a line, the AS curve is locally convex in output in this region. Similarly, between the second and third intersection, the AS curve is locally concave in output. Thus, as output $Y^{*}$ increases, the AS curve must first have a positive second derivative followed by a negative second derivative.

We compute the second derivative of inflation with respect output of the AS curve and derive the following expression (we drop the $*$ for simplicity):

$$\begin{aligned} \frac{d^2 \Pi }{dY^2}&= G(Y) \left( \left( 1+\phi \right) \left( 1-\gamma \right) \left( \frac{Y}{Y^f}\right) ^{\phi } + \left( \phi - 1\right) \right) \end{aligned}$$

(50)

$$\begin{aligned} G\left( Y\right)&= \frac{\phi \gamma \left( 1-\gamma \right) \left( \frac{Y}{Y^f}\right) ^{\phi }}{Y^2\left( 1-\left( 1-\gamma \right) \left( \frac{Y}{Y^f}\right) ^{\phi }\right) }\end{aligned}$$

(51)

$$\begin{aligned} \phi&= \frac{1-\alpha }{\alpha } \end{aligned}.$$

(52)

As can be seen, over the region considered, the function $G\left( Y\right)$ is positive and, therefore, the convexity of the AS curve is determined by the second term. This term may be negative if $\phi < 1$, but this expression is increasing in Y between 0 and $Y_f$. Therefore, the second derivative cannot switch signs from positive to negative. Thus, we have derived a contradiction by assuming multiple steady states. Therefore, there must exist a unique intersection point.

As established before, it must be the case that the AS curve has a lower slope than the AD curve at the point of intersection. The slope of the AS curve is

$$\begin{aligned} \frac{d\Pi ^{*}}{dY^{*}}&= \frac{1-\alpha }{\alpha }\frac{1}{\gamma ^{*}}\frac{\Pi ^{*}}{Y^{*}}\left( \Pi ^{*} - \gamma ^{*}\right) \end{aligned}.$$

(53)

If the slope of the AS curve is less than the slope of the AD curve at the intersection point, then it must be the case that

$$\begin{aligned} \frac{1-\alpha }{\alpha }\frac{\Pi ^{*}}{Y^{*}}\left( \frac{\Pi ^{*}}{\gamma ^{*}}-1\right)< \left( \psi ^{*}\right) ^{-1}\\ \frac{1-\alpha }{\alpha }\frac{\psi ^{*}\Pi ^{*}}{Y^{*}}\left( \frac{\Pi ^{*}}{\gamma ^{*}}-1\right)< 1\\ \frac{1-\alpha }{\alpha }\frac{Y^{*}-D^{*}-K^{*}}{Y^{*}}\left( \frac{\Pi ^{*}}{\gamma ^{*}}-1\right)&< 1\\ s_y\frac{\alpha }{1-\alpha } + 1> \frac{\Pi ^{*}}{\gamma ^{*}}\\ \frac{\gamma ^{*}}{\Pi ^{*}}\left( s_y\frac{\alpha }{1-\alpha } + 1\right)&> 1\end{aligned}.$$

Linearizing the equilibrium conditions around the secular stagnation steady state, we obtain the following linearized AD and AS equations:

$$\begin{aligned} 0&= E_{t} \pi _{t+1} - s_y y_{t}^{*} + d_{t}^{*} + s_d d_{t-1}^{*} \end{aligned}$$

(54)

$$\begin{aligned} y_{t}^{*}&= \gamma _w^{*} y_{t-1} + \gamma _w^{*} \frac{\alpha }{1-\alpha } \pi _{t}^{*} \end{aligned},$$

(55)

where $d_{t}^{*}$ is the collateral shocks and various coefficients are given in terms of their steady-state values.

$$\begin{aligned} \gamma _w^{*}&= \frac{\gamma ^{*}}{\bar{\Pi }^{*}}\\ s_y&= \frac{\bar{Y}^{*}}{\bar{Y}^{*} - \bar{D}^{*}-\bar{K}^{*}}\\ s_d&= \frac{\bar{D}}{\bar{Y}^{*} - \bar{D}^{*}-\bar{K}^{*}} \end{aligned}.$$

Substituting (55) into (54), we obtain a forward-looking difference equation in $y_{t}^{*}$. The local determinacy condition requires the coefficient on $E_{t} y_{t+1}^{*}$ to be less than one. This condition is the same as the slope condition. Therefore, the unique secular stagnation steady state is always locally determinate as required. $\square$

Proposition 3

If the international lending constraint at $K^{*}=0$, $\bar{\Pi }=1,$ and $\gamma > 0$, then, if $r^{n} > 0$, $i > 0$, $\frac{\partial r}{\partial IR}<0,$ and $\frac{\partial Y}{\partial IR}=0$. If $r^{n} < 0$, $i=0$, $\frac{\partial r}{\partial IR}>0,$ and $\frac{\partial Y}{\partial IR}<0$.

Proof

Given the inflation target, if $r_{n} > 0$, then the domestic nominal interest rate is $i = r_{n} > 0$. Since the inflation target is attained, the wage norm does not bind and $Y = Y_{f}$. The expression for the real rate in steady state is given below:

$$\begin{aligned} 1+r =\frac{1+\beta }{\beta }\frac{D}{(Y_{f}-D)+\frac{1+\beta }{\beta }(IR-B^{g})} \end{aligned}.$$

For small changes in IR, $Y = Y_{f}$, therefore, $\frac{\partial Y}{\partial IR}=0$ and, from the previous equation $\frac{\partial r}{\partial IR}<0$ as required.

Given the inflation target, if $r_{n} < 0$, then the domestic nominal interest rate is $i = 0$. As shown in Proposition 2, a unique secular stagnation steady-state exists with $\Pi < \bar{\Pi }$ and $Y < Y_{f}$. Observe that $\frac{d\Pi }{dY} < \left( \frac{1+\beta }{\beta } D\right) ^{-1}$—the slope of the AS curve is less than the slope of the AD curve evaluated at the secular stagnation steady state.

In the stagnation steady state, output is given implicitly by the following equation where $\Pi \left( Y\right)$ is output inflation relationship from the AS curve:

$$\begin{aligned} \frac{1}{\Pi \left( Y\right) }&= \frac{1+\beta }{\beta }\frac{D}{(Y-D)+\frac{1+\beta }{\beta }(IR-B^{g})}\\ \Rightarrow \frac{1+\beta }{\beta } D \Pi \left( Y\right)&= Y - D + \frac{1+\beta }{\beta }\left( IR-B_g\right) \\ \Rightarrow \frac{\partial Y}{\partial IR}&= -\frac{1}{\frac{\beta }{1+\beta } -\frac{d\Pi }{dY} D} \end{aligned}.$$

In a secular stagnation steady state, the denominator is positive (since the slope of the AS curve is less than the slope of the AD curve at the intersection point), so the $\frac{\partial Y}{\partial IR}=0$. Since $\frac{d\Pi }{dY} > 0$, $\frac{\partial r}{\partial IR}>0$ as required. $\square$

Proposition 4

If $r^{W, Nat} < \bar{\Pi }^{-1}$, there exists a locally determinate secular stagnation equilibrium with $Y<Y_f$, $Y^{*}<Y_{f}^{*}$, $i=i^{*} = 0,$ and $\Pi < \bar{\Pi }$.

Proof

Under the assumptions of the proposition, monetary policy in both countries cannot track the world natural rate of interest and $i = i^{*} = 0$. Perfect capital market integration requires equalization of the domestic and foreign real interest rate, hence $\Pi = \Pi ^{*}$. Steady-state inflation, domestic output, and foreign output jointly satisfy the following equilibrium conditions:

$$\begin{aligned} \omega Y + \left( 1+\omega \right) Y^{*}&= D^{W} + \psi ^{W} \Pi \end{aligned}$$

(56)

$$\begin{aligned} Y&= \left( \frac{1-\frac{\gamma \bar{\Pi }}{\Pi }}{1-\gamma }\right) ^{\frac{\alpha }{1-\alpha }} Y_{f} \end{aligned}$$

(57)

$$\begin{aligned} Y^{*}&= \left( \frac{1-\frac{\gamma ^{*}\bar{\Pi }}{\Pi }}{1-\gamma ^{*}}\right) ^{\frac{\alpha }{1-\alpha }} Y^{*}_{f} \end{aligned},$$

(58)

where $D^{W} = \omega D + \left( 1-\omega \right) D^{*}$ and $\psi ^{W} = \frac{1+\beta }{\beta }\left( 1+g\right) D^{W}> 0$. We may define the difference equation $\Delta \left( \Pi \right)$ by taking the difference between (56) and the weighted sum of (57) and (58). Without loss of generality, assume that $\gamma < \gamma ^{*}$. We assume that output is bounded below by zero—that is, if $\Pi < \gamma ^{*}$, then $Y^{*} = 0$. Given this assumption, the function $\Delta \left( \Pi \right)$ is continuous (but not necessarily differentiable), with $\Delta \left( \gamma \right) > 0$ and $\Delta \left( \bar{\Pi }\right) < 0$. Therefore, there exists a global inflation rate $\Pi _{ss}$ with $\Pi _{ss} < \bar{\Pi }$ implying that, in steady state, $Y_{ss} < Y_{f}$ and $Y_{ss}^{*} < Y_{f}^{*}$.

To establish that this steady state is locally determinate, we observe that, graphically, at $\Pi = \gamma$ the global AS curve [weighted sum of Equations (57) and (58)]. At $\Pi = \bar{\Pi }$, the global AD curve lies above the AS curve. Thus, there exists at least one equilibrium in which the AD curve is locally steeper than the AS curve. We first derive the condition for local determinacy. The log-linearized equilibrium conditions for a symmetric stagnation equilibrium are given below:

$$\begin{aligned} E_{t} \pi _{t+1}&= \bar{\omega } s_y y_{t} + \left( 1-\bar{\omega }\right) y_{t}^{*} + shocks \end{aligned}$$

(59)

$$\begin{aligned} y_{t}&= \gamma _{w} y_{t-1} + \gamma _{w} \phi \pi _{t} \end{aligned}$$

(60)

$$\begin{aligned} y_{t}^{*}&= \gamma _{w}^{*} y_{t-1}^{*} + \gamma _{w}^{*} \phi \pi _{t} \end{aligned},$$

(61)

where $\phi = \frac{\alpha }{1-\alpha }$ and the other coefficients are defined below:

$$\begin{aligned} \gamma _{w}&= \frac{\gamma \bar{\Pi }}{\Pi _{ss}}\\ \gamma _{w}^{*}&= \frac{\gamma \bar{\Pi }}{\Pi _{ss}} \\ s_y&= \frac{\omega Y_{ss}+\left( 1-\omega \right) Y_{ss}^{*}}{\omega (Y_{ss}-D)+\left( 1-\omega \right) (Y_{ss}^{*} - D^{*})}\\ \bar{\omega }&= \frac{\omega Y_{ss}}{\omega Y_{ss}+\left( 1-\omega \right) Y_{ss}^{*}} \end{aligned}.$$

This linearized system can be expressed as

$$\begin{aligned} A E_{t} x_{t+1}&= B x_{t} + shocks \\ E_{t} x_{t+1}&= A^{-1} B x_{t} + A^{-1} shocks \end{aligned},$$

where $x_{t} = [\pi _{t},y_{t-1},y_{t-1}^{*}]'$ and the A, B are square matrices with suitably defined coefficients. Local determinacy requires that the matrix $A^{-1}B$ has exactly one eigenvalue outside the unit circle.

Since the matrix B has a row of zeros, one eigenvalue of the system is zero. The characteristic polynomial that determines the remaining eigenvalues is

$$\begin{aligned} \lambda ^2 - \left( \phi s_y \left( \bar{\omega } \gamma _{w} - \left( 1-\bar{\omega }\right) \gamma _{w}^{*}\right) + \gamma _{w} + \gamma _{w}^{*}\right) \lambda + \gamma _{w} \gamma _{w}^{*} \left( 1+s_y \phi \right) = 0 \end{aligned}.$$

Since the characteristic polynomial is positive at $\lambda = 0$, the condition that ensures local determinacy is that the characteristic polynomial is negative at $\lambda = 1$. This condition requires

$$\begin{aligned} 1+ \gamma _{w} \gamma _{w}^{*} \left( 1+s_y \phi \right) < \phi s_y \left( \bar{\omega } \gamma _{w} - \left( 1-\bar{\omega }\right) \gamma _{w}^{*}\right) + \gamma _{w} + \gamma _{w}^{*} \end{aligned}.$$

(62)

It remains to show that this local determinacy condition is identical to the slope condition that must be satisfied in equilibrium. The slope of the global AS curve and global AD curve is given below:

$$\begin{aligned} \frac{dY^{W}_{AS}}{d\Pi }&= \phi \left( \omega \gamma _{w} \frac{Y_{ss}}{\Pi _{ss} - \gamma \bar{\Pi }} + \left( 1-\omega \right) \gamma ^{*} \frac{Y_{ss}^{*}}{\Pi _{ss}-\gamma ^{*}\bar{\Pi }}\right) \\ \frac{dY^{W}_{AD}}{d\Pi }&= \psi ^{W} \end{aligned}.$$

A steeper slope for the AD curve relative to the AS curve implies

$$\begin{aligned} \frac{dY^{W}_{AS}}{d\Pi }> \frac{dY^{W}_{AD}}{d\Pi } \\ \phi \left( \omega \gamma _{w} \frac{Y_{ss}}{\Pi _{ss} - \gamma \bar{\Pi }} + \left( 1-\omega \right) \gamma ^{*}_{w} \frac{Y_{ss}^{*}}{\Pi _{ss}-\gamma ^{*}\bar{\Pi }}\right)> \psi ^{W} \\ \phi \left( \bar{\omega } \frac{\gamma _{w}}{1 - \gamma _{w}} + \left( 1-\bar{\omega }\right) \frac{\gamma ^{*}_{w}}{1-\gamma ^{*}_{w}}\right)> \psi ^{W} \frac{\Pi _{ss}}{Y^{W}} \\ \phi s_y \left( \bar{\omega } \gamma _{w} \left( 1-\gamma _{w}^{*}\right) + \left( 1-\bar{\omega }\right) \gamma _{w}^{*} \left( 1-\gamma _{w}\right) \right)> \left( 1-\gamma _{w}\right) \left( 1-\gamma _{w}^{*}\right) \\ \phi s_y \left( \bar{\omega } \gamma _{w} + \left( 1-\bar{\omega }\right) \gamma _{w}^{*} - \gamma _{w} \gamma _{w}^{*} \right)> 1 - \gamma _{w} - \gamma _{w}^{*} + \gamma _{w} \gamma _{w}^{*} \\ \phi s_y \left( \bar{\omega } \gamma _{w} - \left( 1-\bar{\omega }\right) \gamma _{w}^{*}\right) + \gamma _{w} + \gamma _{w}^{*}&> 1+ \gamma _{w} \gamma _{w}^{*} \left( 1+s_y \phi \right) \end{aligned},$$

where the last inequality is identical to the determinacy condition derived in Equation (62). $\square$

Proposition 5

If $r^{W,Nat} < \bar{\Pi }^{-1}$, $D^{W} > \left( 1-\omega \right) Y_{f}^{*}$, $\gamma > 0$, there exists a unique, locally determinate asymmetric secular stagnation with $r=r^{*}$, $Y < Y_{f}$, $Y^{*} = Y^{*}_{f}$, $i=0$, and $\Pi < \bar{\Pi }$.

Proof

Under the assumptions of the proposition and the monetary policy rule, the zero lower bound is binding for the home country and not binding for the foreign country. Nevertheless, real interest rates are equalized across both countries: $\frac{1}{\Pi } = r = r^{*} = \frac{i^{*}}{\bar{\Pi ^{*}}}$ where $i^{*} > 0$. Equilibrium inflation and output in the home country solve the following equations:

$$\begin{aligned} Y&= \frac{1}{\omega }\left( D^{W} - \left( 1-\omega \right) Y^{*}_{f} + \psi ^{W}\Pi \right) \end{aligned}$$

(63)

$$\begin{aligned} Y&= \left( \frac{1-\frac{\gamma \bar{\Pi }}{\Pi }}{1-\gamma }\right) ^{\frac{\alpha }{1-\alpha }} Y_{f} \end{aligned},$$

(64)

where $D^{W} = \omega D + \left( 1-\omega \right) D^{*}$ and $\psi ^{W} = \frac{1+\beta }{\beta }\left( 1+g\right) D^{W}> 0$. We may define the difference equation $\Delta \left( \Pi \right)$ by taking the difference between (63) and (64) . This function is continuous in $\Pi$ with $\Delta (\gamma ) > 0$ (since $D^W > \left( 1-\omega \right) Y_f$) and $\Delta (\bar{\Pi }) < 0$ since $r^{W,Nat} < \bar{\Pi }^{-1}$. Therefore, there exists a $\gamma< \Pi < \bar{\Pi }$ such that $\Delta (\Pi ) = 0$. Since $\Pi < \bar{\Pi }$, it follows that $Y < Y_f$.

Uniqueness of an asymmetric stagnation equilibrium under perfect integration is established identically as in Proposition 2. Graphically, the global AD curve (Equation (63) form a line in domestic inflation-output space. The domestic AS curve [Equation (64)] is identical to Equation (49) and cannot cross the AD curve more than once given that the second derivative cannot switch signs from positive to negative.

It must be the case that the AS curve has a lower slope than the AD curve at the point of intersection. The slope of the AS curve is identical to Equation (53). If the slope of the AS curve is less than the slope of the AD curve at the intersection point, then it must be the case that

$$\begin{aligned} \frac{1-\alpha }{\alpha }\frac{\Pi }{Y}\left( \frac{\Pi }{\gamma \bar{\Pi }}-1\right)< \left( \frac{\psi ^{W}}{\omega }\right) ^{-1}\\ \frac{1-\alpha }{\alpha }\frac{\psi ^{W}\Pi }{\omega Y}\left( \frac{\Pi }{\gamma \bar{\Pi }}-1\right)< 1\\ \frac{1-\alpha }{\alpha }\frac{\omega Y-D^{W}-\left( 1-\omega \right) Y_{f}^{*}}{\omega Y}\left( \frac{\Pi }{\gamma \bar{\Pi }}-1\right)& < 1\\ s_y\frac{\alpha }{1-\alpha } + 1> \frac{\Pi }{\gamma \bar{\Pi }}\\ \frac{\gamma \bar{\Pi }}{\Pi }\left( s_y\frac{\alpha }{1-\alpha } + 1\right)& > 1 \end{aligned}.$$

The linearization of the global AD curve [Equation (63)] and the domestic AS curve [Equation (64)] around the asymmetric stagnation steady state imply identical expressions to the linearized equilibrium conditions in Proposition 2 where the coefficients are given by

$$\begin{aligned} \gamma _{w}&= \frac{\gamma \bar{\Pi }}{\Pi _{ss}}\\ s_y&= \frac{\omega Y_{ss}}{\omega Y_{ss} - D^{W} - \left( 1-\omega \right) Y_f^{*}} \end{aligned},$$

where $\Pi _{ss}$ and $Y_{ss}$ are the solution to steady-state equilibrium conditions (63) and (64). Substituting the linearized AS curve into the linearized AD curve as in Proposition 2 provides a forward-looking difference equation in $y_{t}$. Local determinacy requires the coefficient on $E_{t} y_{t+1}$ to be less than unity. This condition is identical to slope condition derived above implying that the asymmetric stagnation equilibrium is always locally determinate, as required. $\square$

Appendix B: Fiscal Policy Coordination

Non-Cooperative Game

The government in each country unilaterally chooses its own level of government spending G to minimize the deviations of output, inflation, and level of government spending from their own respective target levels. We assume both countries have identical aggregate supply curves—same full-employment level of output, labor share, and degree of wage rigidity. The policy objective and constraints are given below:

$$\begin{aligned} \min _{G, Y, Y^{*}, \Pi }&\left( Y - Y_{f}\right) ^2 + \left( \Pi - 1\right) ^2 + \left( G - G_{target}\right) ^2 \\ \text{ s.t. }\,\,\omega Y + \left( 1-\omega \right) Y^{*}&= D^{W} + \omega G + \left( 1-\omega \right) G^{*} + \psi ^{W} \Pi \\ Y&= \left( \frac{1-\frac{\gamma }{\Pi }}{1-\gamma }\right) ^{\frac{\alpha }{1-\alpha }} Y_{f} \\ Y^{*}&= \left( \frac{1-\frac{\gamma }{\Pi }}{1-\gamma }\right) ^{\frac{\alpha }{1-\alpha }} Y_{f} \end{aligned},$$

where $D^{W} = \omega D + \left( 1-\omega \right) D^{*}$ and $\psi ^{W} = \frac{1+\beta }{\beta }\left( 1+g\right) D^{W}$.

By substituting the domestic and foreign aggregate supply curves into the objective function and global aggregate demand curve, we obtain the following Lagrangian:

$$\begin{aligned} \mathcal {L}&= \frac{1}{2} \left( Y_{AS}\left( \Pi \right) - Y_{f}\right) ^2 + \frac{1}{2} \left( \Pi - 1\right) ^2 + \frac{1}{2} \left( G - G_{target}\right) ^2 \\&+ \lambda \left( \omega Y_{AS}\left( \Pi \right) + \left( 1-\omega \right) Y_{AS}^{*}\left( \Pi \right) - D^{W} - \omega G - \left( 1-\omega \right) G^{*} - \psi ^{W} \Pi \right) \end{aligned}.$$

The first-order conditions are given below:

$$\begin{aligned} 0&= \frac{dY_{AS}}{d\Pi }\left( Y_{AS}\left( \Pi \right) - Y_{f}\right) + \Pi - 1 + \lambda \left( \frac{dY_{AS}}{d\Pi } - \psi ^{W} \right) \\ 0&= G - G_{target} - \lambda \omega \end{aligned},$$

where $\lambda$ is the Lagrange multiplier on the global aggregate demand curve. Given that the domestic economy is in secular stagnation: $Y < Y_{f}$ and $\Pi < 1$ and $\frac{dY_{AS}}{d\Pi }> \psi ^{W} > 0$—and the slope of the AS curve exceeds the slope of the AD curve, the multiplier $\lambda > 0$ and the fiscal authority in secular stagnation always chooses a level of government spending that exceeds the target. The level of government spending above target is increasing in $\omega$.

Cooperative Game

We now consider the optimal level of government spending when both countries jointly maximize their welfare. The loss function of the global planner is the weighted sum of each country’s loss function. Given that the aggregate supply curve are identical and if we assume that the target level of government expenditures is the same, we obtain the following loss function subject to a global aggregate demand and global aggregate supply constraints:

$$\begin{aligned} \min _{G, Y, \Pi }&\left( Y - Y_{f}\right) ^2 + \left( \Pi - 1\right) ^2 + \left( G - G_{target}\right) ^2 \\ \text{ s.t. }\,\,Y&= D^{W} + G + \psi ^{W} \Pi \\ Y&= \left( \frac{1-\frac{\gamma }{\Pi }}{1-\gamma }\right) ^{\frac{\alpha }{1-\alpha }} Y_{f} \end{aligned},$$

where $D^{W} = \omega D + \left( 1-\omega \right) D^{*}$ and $\psi ^{W} = \frac{1+\beta }{\beta }\left( 1+g\right) D^{W}$. In the cooperative setup, Y is global output (instead of output of the domestic country only) and G is global government spending. Relative to the non-cooperative setup, the only difference is that the planner chooses G and $G^{*}$ simultaneously.

The first-order conditions for the optimal level of global government spending are given below:

$$\begin{aligned} 0&= \frac{dY_{AS}}{d\Pi }\left( Y_{AS}\left( \Pi \right) - Y_{f}\right) + \Pi - 1 + \lambda \left( \frac{dY_{AS}}{d\Pi } - \psi ^{W} \right) \\ 0&= G - G_{target} - \lambda \end{aligned},$$

where the multiplier is the same as in the case of the non-cooperative game. The only difference is that $\omega$ no longer appears in the second optimality condition.

Proposition 7

Consider two countries in symmetric secular stagnation with identical aggregate supply parameters, loss functions, and target levels of government spending. Then coordinated optimal government spending exceeds uncoordinated government spending. Coordination losses are maximized when $\omega = \frac{1}{2}$.

Proof

Global government spending under coordination and absent coordination are given below along with the Lagrange multiplier, $\lambda$:

$$\begin{aligned} G_{coop}&= G_{target} + \lambda \\ G_{non-coop}&= G_{target} + \lambda \left( \omega ^{2} + \left( 1-\omega \right) ^{2}\right) \\ \lambda&= \frac{\frac{dY_{AS}}{d\Pi }\left( Y_{f}-Y_{AS}\left( \Pi \right) \right) + 1- \Pi }{\frac{dY_{AS}}{d\Pi } - \psi ^{W}} \end{aligned},$$

where $G_{coop}$ is global government spending under cooperation and $G_{non-coop}$ is global government spending absent coordination. Since $\omega \le 1$, $G_{non-coop} \le G_{coop}$. The term $\omega ^{2} + \left( 1-\omega \right) ^{2}$ is minimized at $\omega = \frac{1}{2}$ implying that losses from coordination are maximized when the two countries have the same size. $\square$

Corollary

Consider N countries in a symmetric secular stagnation with identical aggregate supply parameter, loss function, and target levels of government spending. Global government spending absent coordination goes to zero as $N \rightarrow \infty$.

Proof

Since countries are identical, $\omega = \frac{1}{N}$. The optimality condition for government spending for each country is given by the first-order condition of the non-cooperative game. Therefore, global government spending absent coordination is given by the expression below:

$$\begin{aligned} G_{non-coop}&= G_{target} + \lambda N \omega ^{2}\\&= G_{target} + \lambda \frac{1}{N} \end{aligned},$$

where the second term goes to zero as $N \rightarrow \infty$. $\square$