Real Wage Flexibility, Economic Fluctuations, and Exchange Rate Regimes

Abstract

We study the desirability of real wage flexibility and its impact on welfare for each exchange rate regime, using a small open economy model with unemployment fluctuation. The main findings regarding the stabilizing role of real wage flexibilities can be summarized as two-fold. First, the more flexible real wage leads to smaller fluctuation in the output gap, reduces welfare losses regardless of exchange rate regimes. The increase in real wage flexibilities, however, may have destabilizing effects on domestic price and real wage inflation and may have a negative effect on welfare for both exchange rate regimes. Overall, contrary to conventional wisdom, increase in real wage flexibilities may be welfare-reducing. Second, in the presence of real wage rigidities smoothing the exchange rate fluctuations may increase the welfare loss.

Appendix

In this appendix we derive a second-order approximation to the utility of the representative household around the efficient steady state. As has been discussed in the main text, we restrict our study to the special case of σ = η = 1. Frequent use is made of the following fact:

$$ \frac{X_{t}-X}{X}=x_{t}+\frac{1}{2}{x^{2}_{t}} \, , $$

where x _t is the log deviation from steady state for the variable X _t . The second-order Taylor approximation of the household i ^′ s period t utility, U _t (i), around a steady state and intergrating across households yields

$$\begin{array}{@{}rcl@{}} {{\int}_{0}^{1}}(U_{t}(i)-U)di &\,\,\simeq\,\,& U_{C}C\left[c_{t}+\left( \frac{1}{2}+\frac{C}{2}\frac{U_{CC}}{U_{C}}\right){c_{t}^{2}}\right]\\&+& U_{N}N\left[n_{t}+\left( \frac{1}{2}+\frac{N}{2}\frac{U_{NN}}{U_{N}}\right){N_{t}^{2}}\right]+t.i.p. \, , \end{array} $$

where t.i.p. stands for terms independent of policy.

Using the fact$\ \frac {C}{2}\frac {U_{CC}}{U_{C}}=-\frac {1}{2}$ and$\ \frac {1}{2}+\frac {N}{2}\frac {U_{NN}}{U_{N}}=\frac {1+\varphi }{2}$ and the market clearing condition$\ c_{t}=(1-\delta )y_{t}+\delta y^{*}_{t}$, we have

$$\begin{array}{@{}rcl@{}} {{\int}_{0}^{1}}(U_{t}(i)\,-\,U)di \!\simeq \!U_{C}C(1-\delta)y_{t}\,+\, U_{N}N\left[\!\!{\int}^{1}_{0}n_{t}(i)di\,+\,\frac{1+\varphi}{2}{{\int}^{1}_{0}} {n^{2}_{t}}(i)di\!\right]\,+\,t.i.p. , \end{array} $$

Define aggregate employment as$\ N_{t} = {{\int }^{1}_{0}} N_{t}(i)di$, or, in terms of log deviations from the steady state and up to a second-order approximation,

$$n_{t} + \frac{1}{2}{n_{t}^{2}} \,\simeq\, {{\int}^{1}_{0}}\tilde{n}_{t}(i)di+\frac{1}{2}{{\int}^{1}_{0}}\tilde{n}_{t}(i)^{2}di \, . $$

Note also that

$$\begin{array}{@{}rcl@{}} {{\int}_{0}^{1}}n_{t}(i)^{2}di & =& {{\int}_{0}^{1}}\left( n_{t}(i)-n_{t}+n_{t}\right)^{2}di\\ & =& \tilde{n}_{t}^{2} - 2n_{t}\epsilon_{w}{{\int}^{1}_{0}}\left( w_{t}(i)-w_{t}\right)di + {\epsilon_{w}^{2}}{{\int}^{1}_{0}}\left( w_{t}(i)-w_{t}\right)^{2}di\\ & =& {n_{t}^{2}}+{\epsilon_{w}^{2}}var_{i}\left\{w_{t}(i)\right\} \, , \end{array} $$

where we have used the labor demand function$\ n_{t}(i)-n_{t}=-\epsilon _{w}\left (w_{t}(i)-w_{t}\right )$, and the fact that$\ {{\int }^{1}_{0}}\left (w_{t}(i)-w_{t}\right )di=0$ and that ${{\int }^{1}_{0}} \left (w_{t}(i)-w_{t}\right )^{2}di = var_{i}\left \{w_{t}(i)\right \}$ is of second order.

The next step is to derive a relationship between aggregate employment and output:

$$\begin{array}{@{}rcl@{}} N_{t} & =&{{\int}^{1}_{0}}{{\int}^{1}_{0}}N_{t}(z,i)didz ={{\int}^{1}_{0}}N_{t}(z){{\int}^{1}_{0}}\frac{N_{t}(z,i)}{N_{t}(z)}didz \\ & =&{\Delta}_{w,t}{{\int}^{1}_{0}}N_{t}(z)dz = {\Delta}_{w,t} \left( \frac{Y_{t}}{A_{t}}\right)^{\frac{1}{1-\alpha}} {{\int}^{1}_{0}}\left( \frac{Y_{t}(z)}{Y_{t}}\right)^{\frac{1}{1-\alpha}}dz \\ & =&{\Delta}_{w,t} {\Delta}_{p_{H},t} {{\int}^{1}_{0}}\left( \frac{Y_{t}(z)}{Y_{t}}\right)^{\frac{1}{1-\alpha}}dz \, , \end{array} $$

where$\ {\Delta }_{w,t} = {{\int }^{1}_{0}}\left (\frac {w_{t}(i)}{w_{t}}\right )^{-\epsilon _{w}}di$ and$\ {\Delta }_{p_{H},t} = {{\int }^{1}_{0}}\left (\frac {p_{H,t}(z)}{P_{H,t}}\right )^{-\epsilon _{p}}dz$.

Thus, the following second-order approximation of the relation between (log) aggregate output and (log) aggregate employment holds:

$$ n_{t}=\frac{1}{1-\alpha}(\tilde{y}_{t}-a_{t})+d_{w,t}+d_{p_{H},t} , $$

where$\ d_{w,t} = {\log {\int }^{1}_{0}}\left (\frac {w_{t}(i)}{w_{t}}\right )^{-\epsilon _{w}}di$ and$\ d_{p_{H},t} = {\log {\int }^{1}_{0}}\left (\frac {p_{H,t}(z)}{P_{H,t}}\right )^{-\epsilon _{p}}dz$.

Lemma 1

$d_{p_{H},t}=\frac {\epsilon _{p}(1-\alpha +\alpha \epsilon _{p})}{2(1-\alpha )^{2}}var_{z}\left \{p_{H,t}(z)\right \} .$

Proof

See Galí and Monacelli (2005). □

Lemma 2

$d_{w,t}=\frac {\epsilon _{w}}{2}var_{i}\left \{w_{t}(i)\right \} .$

Proof

See Erceg et al. (2000). □

Now, one-period aggregate welfare can be written as

$$\begin{array}{@{}rcl@{}} {{\int}^{1}_{0}}\frac{U_{t}(i)-U}{U_{c}C}di=-\frac{1-\delta}{2} &&\left[\left( \frac{1+\varphi}{1-\alpha}\right)\tilde{y}^{2}_{t} +\frac{\epsilon_{p}(1-\alpha+\alpha\epsilon_{p})}{(1-\alpha)}var_{z}\left\{p_{H,t}(z)\right\} \right.\\&&+ \epsilon_{w}(1-\alpha)\left[1+\varphi\epsilon_{w}\right]var_{i}\left\{w_{t}(i)\right\}\Bigg] + t.i.p. \, , \end{array} $$

where t.i.p. stands for terms independent of policy.

Lemma 3

$$\begin{array}{@{}rcl@{}} \sum\limits_{t=0}^{\infty} \beta^{t} var_{z}\left\{p_{H,t}(z)\right\} &=& \frac{\theta_{p_{H}}}{(1-\beta\theta_{p_{H}})(1-\theta_{p_{H}})} \sum\limits_{t=0}^{\infty}\beta^{t}\pi_{H,t}^{2} \, ,\\ \sum\limits_{t=0}^{\infty} \beta^{t} var_{i}\left\{w_{t}(i)\right\} &=& \frac{\theta_{w}}{(1-\beta\theta_{w})(1-\theta_{w})} \sum\limits_{t=0}^{\infty}\beta^{t}\pi_{w,t}^{2} \, . \end{array} $$

Proof

See Woodford (2005, Chapter 6). □

Collecting the previous results, we can write the second-order approximation to the small open economy’s aggregate welfare function as follows:

$$ \textbf{\Large{W}} \,=\,-\frac{1\,-\,\delta}{2} E_{0} \sum\limits^{\infty}_{t=0} \beta^{t} \left\{\left( \frac{1\,+\,\varphi}{1\,-\,\alpha}\right)\tilde{y}^{2}_{t}\,+\,\frac{\epsilon_{p}}{\lambda_{p_{H}}}\left( \pi_{H,t}\right)^{2}\!\,+\, \frac{\epsilon_{w}(1\!-\alpha)}{\lambda_{w}}\left( \pi^{R}_{w,t}\right)^{2}\right\} \!+ t.i.p. \, , $$

where$\ \lambda _{p_{H}}=\frac {(1-\theta _{p_{H}})(1-\beta \theta _{p_{H}})}{\theta _{p_{H}}(1-\alpha +\alpha \epsilon _{p})}(1-\alpha )$ and $\lambda _{w}=\frac {(1-\theta _{w})(1-\beta \theta _{w})}{\theta _{w}(1+\epsilon _{w}\varphi )}$.