Exchange rate volatility and first-time entry by multinational firms

Abstract

The model and related empirical examination in this paper demonstrate one reason why previous studies document both positive and negative correlations between exchange rate volatility and observed levels of foreign direct investment. Using a simple model of cross-border mergers and acquisitions, it argues that the source of the volatility is important in resolving the puzzle. An empirical analysis of mergers and acquisitions by individual firms reveal that first-time foreign direct investment is discouraged by monetary volatility originating from the source-country, but can be encouraged by monetary volatility originating in the host country, especially when compared to domestic investment or expansion by existing multinationals. The regressions also reveal a large and positive “euro effect” on the number of first-time cross-border mergers within the European Monetary Union, even when controlling for domestic merger activity.

Appendices

Appendix 1: Derivation of the aggregate price level

The pseudo-reduced form equation for the aggregate price level is calculated in three steps: First, I define aggregate consumption as a function of the aggregate price index and the exogenous money supply, which lets me define the wage rate as a function of the exogenous money supply and underlying preference parameters. Second, I find the firm’s pricing rule in terms of expected aggregate consumption and the wage rate. These pricing rules now reduce to a function of the (exogenous) expected money supply and underlying parameters. Third, I substitute these firm pricing rules into the definition of the aggregate price index to redefine the index only in terms of the expected money supply, underlying parameters, and the endogenous cutoff productivity levels for home- and foreign-owned firms operating in the home economy.

1.1 Consumption and wages

Based on the maximization problem described in the text, standard first-order conditions for the consumer’s problem (with λ _t representing the Lagrange multiplier for the budget constraint) are as follows:

$$ \frac{\partial \pounds}{\partial C_{t}} :\lambda_{t}=\frac{1}{ P_{t}C_{t}^{\rho}} $$

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$$ \frac{\partial \pounds}{\partial B_{t}}:\frac{1} {P_{t}C_{t}^{\rho}}=\beta (1+i_{t})E_{t}\left[\frac{1}{P_{t+1} C_{t+1}^{\rho}}\right]_{t} $$

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$$ \frac{\partial \pounds}{\partial B_{t}^{\ast}} :S_{t} \lambda_{t}=\beta (1+i_{t}^{\ast})E_{t} \left[\lambda_{t+1}S_{t+1}\right] $$

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$$ \frac{\partial \pounds}{\partial M_{t}}:\frac{M_{t}}{P_{t}}= \chi \frac{1+i_{t}}{i_{t}} $$

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$$ \frac{\partial\pounds}{\partial L_{t}} :W_{t}= \frac{\kappa}{\lambda_{t}}=\kappa P_{t}C_{t}^{\rho} .$$

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1.2 The aggregate price level

Minimizing the expenditure necessary to consume one unit of the aggregate consumption bundle gives the aggregate price index in this CES framework shown in the main text,

$$ P_{t}=\left( \int \limits_{0}^{n_{h,t}}p_{h,t}(i)^{1-\theta }di+\int \limits_{1}^{1+n_{f,t}}p_{f,t}(i)^{1-\theta }di\right) ^{{\frac{1}{1-\theta }} }. $$

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It is useful now to identify firms by their productivity parameter, $\varphi$, rather than the firm subscript, i. Every firm draws its productivity parameter independently from an identical distribution, $G(\varphi )$, allowing me to use the law of large numbers to assert that the distribution of productivity levels for the economy as a whole will be the same as the firm-specific distribution. Then, substituting in the pricing rules, we have

$$ P_{t}=\left( {\frac{\theta }{\theta -1}}\right) {\frac{\kappa (1-\beta \gamma )E_{t-1}\left[ M_{t}^{{\frac{1}{\rho }}}\right] }{\chi \bar{\varphi}_{t}E_{t-1} \left[ M_{t}^{{\frac{1-\rho }{\rho }}}\right] }}, $$

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where

$$ \bar{\varphi}_{t}=\left( \int\limits_{\hat{\varphi}_{h,t}}^{\infty }\varphi ^{\theta -1}dG(\varphi )+\int\limits_{\hat{\varphi}_{f,t}}^{\infty} \varphi^{\theta-1}dG(\varphi)\right) ^{{\frac{1}{\theta-1}}}. $$

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1.3 Determinacy

As long as there is a unique solution for $\hat{\varphi}_{h,t}$ and $\hat{\varphi}_{f,t}$, there is a unique solution for the price level. In a model where these two variables enter the zero-profit conditions with no other endogenous variables, one can show analytically that the zero-profit conditions are monotonically increasing in these cutoff productivity levels. In this model, an analytical proof is not possible due to the presence of the takeover price V(0), so I use numerical analysis to verify that $V_{h}(\varphi )$, or Eq. 6 from the main text, is monotonically increasing in $\varphi$, implying the existence of a unique solution for $\hat{\varphi}_{h,t}$. Specifically, I calibrate the model as described in “Appendix 4”, setting interest rate volatility in both countries to 0.1 (the particular value for volatility does not affect the monotonicity). Then, I specify values of $\varphi$ such that $1<\varphi <\infty$ and solve the system described in the text omitting the equation for $V_{h}(\varphi )$ (that is, $V_{h}(\hat{\varphi}_{h,t}))$ so that all other endogenous values are solved for given the level of $\varphi$ specified. Finally, I numerically compute $V_{h}(\varphi )$ given the solution values of all other endogenous variables corresponding to various values of $\varphi$ and find that it is, indeed, monotonically increasing in $\varphi$. ¹⁴

Appendix 2: First-order conditions for the firm’s problem

To set prices for the following period, firms maximize the expected value of profits with respect to the prices they will set subject to the demand equations in the text [derived explicitly in the technical appendix for Russ (2007)],

$$ \underset{p_{h,t}(\varphi ),p_{h,t}^{\ast}(\varphi )}{\max }E_{t-1}\left[ d_{t}\left( \pi_{h,t}(\varphi )+S_{t}\pi_{h,t}^{\ast}(\varphi )\right) \right] $$

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for home-owned firms and

$$ \underset{p_{f,t}(\varphi ),p_{f,t}^{\ast}(\varphi )}{\max }E_{t-1}\left[ d_{t}^{\ast}\left( \pi_{f,t}^{\ast}(\varphi )+{\frac{\pi_{f,t}(\varphi )}{ S_{t}}}\right) \right] $$

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for foreign-owned firms. The first-order conditions for firms operating in the home market are then

$$ \begin{aligned} \frac{\partial\pounds_{h}}{\partial p_{h,t}(\varphi)} &:E_{t-1}\left[ d_{t}\left(c_{h,t}(\varphi)+p_{h,t}(\varphi) \frac{\partial c_{h,t}(\varphi)}{\partial p_{h,t} (\varphi)}-\frac{W_{t}}{\varphi} \frac{\partial c_{h,t}(\varphi )}{\partial p_{h,t}(\varphi)}\right)\right]\\ \frac{\partial\pounds_{f}}{\partial p_{f,t}(\varphi)} &:E_{t-1}\left[\frac{d_{t}^{\ast}}{S_{t}}\left(c_{f,t} (\varphi)+p_{f,t}(\varphi) \frac{\partial c_{f,t}(\varphi)} {\partial p_{f,t}(\varphi)}-\frac{W_{t}}{\varphi} \frac{\partial c_{f,t}(\varphi)}{\partial p_{f,t}(\varphi)}\right)\right] . \end{aligned} $$

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Assuming that firms take all competitors’ prices (and the aggregate price level) as given, substituting the equations for goods demand, and the derivatives of the goods demand equations, the first-order conditions reduce to

$$ \begin{aligned} p_{h,t}(\varphi)&=\left(\frac{\theta}{\theta-1}\right) \frac{E_{t-1}\left[d_{t}W_{t}C_{t}\right]}{E_{t-1} \left[d_{t}C_{t}\right]}\\ p_{f,t}(\varphi)&=\left(\frac{\theta }{\theta -1}\right) \frac{E_{t-1}\left[d_{t}^{\ast} \frac{W_{t}C_{t}}{S_{t}}\right] }{E_{t-1}\left[ d_{t}^{\ast}{\frac{C_{t}}{S_{t}}}\right]}, \end{aligned} $$

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as in Bacchetta and van Wincoop (2000). Reduced forms can be obtained by substituting Eqs. 29 and 1 and 2, yielding the pricing rule in the main text.

Appendix 3: Deriving the impact of home and foreign volatility on multinational firm entry

Starting with the entry condition for foreign firms considering entry into the home market,

$$ V_{f,t-1}(\hat{\varphi}_{f,t})=E_{t-1} \left[\sum\limits_{k=0}^{\infty} \left(\prod\limits_{m=0}^{k}d_{t+m}^{\ast}\right) \frac{\pi_{f,t+k}(\hat{\varphi}_{f,t})}{S_{t+k}}\right] -\frac{P_{t-1}V_{t-1}(0)}{S_{t-1}}-P_{t-1}^{\ast} \, f\equiv 0, $$

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one can use the definition of d ^* _t to obtain

$$ \begin{aligned} V_{f,t-1}(\hat{\varphi}_{f,t})&=E_{t-1} \left[\sum\limits_{k=0}^{\infty} \left(\prod\limits_{m=0}^{k} \frac{\beta P_{t+m-1}^{\ast} C_{t+m-1}^{\ast\rho}}{P_{t+m}^{\ast}C_{t+m}^{\ast\rho}}\right) \frac{\pi_{f,t+k}(\hat{\varphi}_{f,t})}{S_{t+k}}\right] -\frac{P_{t-1}V_{t-1}(0)}{S_{t-1}}\\ &\quad-P_{t-1}^{\ast}f \\ &=E_{t-1}\left[\sum\limits_{k=0}^{\infty} \left(\frac{\beta^{k}P_{t-1}^{\ast}C_{t-1}^{\ast\rho}} {P_{t+k}^{\ast}C_{t+k}^{\ast\rho}}\right) \frac{M_{t+k}^{\ast}}{M_{t+k}}\pi_{f,t+k} (\hat{\varphi}_{f,t})\right] -\frac{P_{t-1}V_{t-1}(0)}{S_{t-1}}\\ &\quad-P_{t-1}^{\ast}f \end{aligned} $$

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Substituting the consumption and pricing equations from the main text, we have

$$ \begin{aligned} V_{f,t-1}(\hat{\varphi}_{f,t})&=\left( M_{t-1}^{\ast}\right) E_{t-1}\left[ \sum \limits_{k=0}^{\infty }\beta ^{k}{\frac{1}{M_{t+k}}}\pi_{f,t+k}(\hat{ \varphi}_{f,t})\right]\\ &\quad-\left(\frac{\theta}{\theta-1}\right) \frac{\kappa}{\chi}\left[\frac{(1-\beta \gamma)} {\bar{\varphi}}e^{\left(\frac{\rho-1}{\rho}\right) \sigma_{i}^{2}}V(0)-\frac{(1-\beta \gamma^{\ast})} {\bar{\varphi}^{\ast}}e^{\left(\frac{\rho-1}{\rho}\right) \sigma_{i^{\ast}}^{2}}f\right] \end{aligned} $$

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In this paper, the objective is to compare entry across steady states. In a steady state, agents expect the one-period-ahead forecast of the nominal exchange rate and functions of the nominal interest rate to be constant across all future periods. In steady state, the number of active firms from abroad and the price level are also constant. That is, for all k ≥ 0,

$$ \begin{aligned} E_{t-1}\left[E_{t+k-1}\left[M_{t+k}^{\alpha}\right]\right] &= E_{t-1} \left[M_{t}^{\alpha}\right] \hbox{ for any constant }\alpha\\ &\Leftrightarrow \hat{\varphi}_{f,t}=\hat{\varphi}_{f,t+k}= \hat{\varphi}_{f}\Longrightarrow \bar{\varphi}_{t}=\bar{\varphi}_{t+k}= \bar{\varphi} \end{aligned} $$

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Substituting the equations for the demand for an individual good and the moments of the money supply, the pricing rule, and the wage relation, the value function above reduces further to Eq. 19 in the main text.

Appendix 4: Calibration and data

Parameters are assigned the following values: β = 0.96, θ = 7 (between the standard values of 2 and 11 used in international macroeconomics literature), k = θ (to ensure the boundedness of the variance of output-weighted average productivity and make the distribution of firm size a power law as in the trade literature), κ = 1, γ = 1 < 1/β, f = 0.2, ρ = 2, and η = 0.5 (meaning half of all active domestically owned firms must purchase a marketing and distribution facility).

Please see Table 7 for summary statisics pertaining to data used in the econometric analysis.

Table 7 Summary statistics