Simulating an AS-AD model for an open econony with flexible exchange rates and a mixture of backward-looking and forward-looking expectations
Following up on Exercise 6 in the previous chapter, this exercise seeks to deepen your understanding of the workings of an economy with flexible exchange rates by asking you to implement the following model on the computer:
Inflation expectations: $\quad \pi_i^t=\varphi \pi^f+(1-\varphi) \pi_{1-1}, \quad 0 \leq \varphi \leq 1$.
Monetary policy: $\quad i_t=r^f+\pi_{i+1}^i+h\left(\pi_t-\pi^f\right)$,
Uncovered interest rate parity: $\quad i_1=i^f+e_{i+1}^e-e_e$,
Exchange rate expectations: $\quad \epsilon_{1+1}^e-\epsilon_{\mathrm{t}}=-\theta\left(e_{\mathrm{t}}-\epsilon_{\mathrm{t}-1}\right)$,
Foreign real interest rate: $\quad r^i=i^i-\pi^i$,
SRAS: $\quad \pi_1=\pi_i^e+\gamma\left(y_1-y\right)+s_t$.
Real exchange rate: $\quad e_i^f=e_{t-1}^f+e_i-e_{i-1}+\pi^f-\pi_i$.
Equation (76) gives the average expected domestic inflation rate, assuming that a fraction $\varphi$ of the population has 'weakly rational' expectations which are anchored by the central bank's inflation target, $\pi^{\star}=\pi^f$, whereas the remaining fraction of the population has static expectations, expecting that this year's inflation rate will correspond to the inflation rate observed last year, $\pi_{1-1}$. In the special case where $\varphi=1$, we obtain the basic model analysed in the main text of this chapter.
1. Show by means of $(75)-(80)$ that the equation for the aggregate demand curve is:
$$
\begin{aligned}
& \pi_t=\pi^f+\left(\frac{\beta_1}{\hat{\beta}_1}\right) e_{i-1}^f-\left(\frac{1}{\hat{\beta}_1}\right)\left(y_t-\hat{y}-z_t\right), \\
& \hat{\beta}_1=\beta_1+\beta_2 h+\beta_1 \theta^{-1}(1-\varphi+h), \quad z_t=-\beta_2\left(t^f-\xi^f\right)+z_p,
\end{aligned}
$$
To facilitate implementation of the model on the computer, we now want to reduce it to two difference equations in the output gap and the inflation gap. Defining $\hat{y}_1=y_2-\hat{y}$ and $A_l=\pi_1-\pi^f$, and using $(76)-(80)$ to derive $e_{\mathrm{r}}-e_{h_1}=-\theta^{-1}(1-\varphi+h)$, the model consisting of (B1)-(83) may be summarized as:
AD: $\quad k_{\mathrm{t}}=\left(\frac{\beta_2}{\beta_1}\right) e_{t-1}^i-\left(\frac{1}{\beta_1}\right)\left(\hat{y}_{\mathrm{t}}-z_{\mathrm{t}}\right)$.
SRAS: $\quad \hat{\pi}_{\mathrm{t}}=(1-\psi) \hat{\pi}_{t-1}+\gamma \hat{y}_t+s_t$,
Real exchange rate: $\quad e_r^r=\epsilon_{\mathrm{T}-1}^{\mathrm{r}}-\left[1+\theta^{-1}(1-\phi+h)\right] t_l$.
From (84) and (86) it follows that:
while (85) implies that:
$$
\hat{\pi}_{\mathrm{t}}-\hat{\pi}_{l-1}=-\varphi \hat{t}_{t-1}+\gamma \hat{y}_{\mathrm{t}}+s_{\mathrm{r}}
$$
It also follows from (85) that:
$$
\hat{y}_t=\gamma^{-1} \hat{\mu}_i-\gamma^{-1}(1-p) \hat{t}_{t-1}-\gamma^{-1} s_t \text {. }
$$
2. Equate (87) and (88) to find an expression for $\hat{f}_{\mathrm{t}}$ and substitute the resulting expression into (85) to show that the model may be reduced to the following second-order difference equation in the output gap:
$$
\begin{aligned}
& \hat{y}_{t+2}-a_1 \hat{y}_{t+3}+a_0 \hat{y}_{\mathrm{t}}=\beta\left(z_{t+2}-z_{t+1}\right)-\beta(1-\varphi)\left(z_{t+1}-z_t\right]-\beta_1 \beta s_{t+2}+\beta h \beta_2 s_{t+1}, \\
& \beta=\frac{1}{1+\gamma \beta_1}, \quad a_1=\beta\left[2-\varphi+\gamma h \hat{\beta}_2\right], \quad a_0=\beta(1-\varphi) .
\end{aligned}
$$
Furthermore, insert (89) into (87) and use the definition of $\hat{\beta}_1$ to show that the model may alternatively be condensed to the following difference equation in the inflation gap:
$$
\hat{\pi}_{t+2}-a_1 \hat{\pi}_{t+1}+a_1 \hat{t}_{\mathrm{t}}=\gamma \beta\left(z_{i+2}-z_{1+1}\right)+\beta\left(s_{1+2}-s_{i+1}\right) .
$$
Verify that (90) collapses to Eq. (28) in the main text in the special case where $\varphi=1$. (Hint: remember to use the definition of $\hat{\beta}_1$,)
You are now asked to simulate the model (86), (90) and (91) on the computer to study the dynamics of output, inflation and the real exchange rate under flexible exchange rates. We proceed in a manner identical to the procedure followed in Exercise 6 of the previous chapter, but in case you haven't solved that exercise, we repeat all the steps here.
Using the definitions given in the appendix to Chapter 23, and assuming that trade is initially balanced and that the marginal and the average propensities to import are identical, one can demonstrate that:
$$
\beta_1=\frac{m\left(\eta_x+\eta_M-1\right)-\eta_D(1-\tau)}{1-D_r+m},
$$
where $m=M_y / P$ is the initial ratio of imports to GDP, $\eta_X$ and $\eta_M$ are the prico elasticities of export and import demand, $\tau=(\tilde{Y}-D) / \hat{Y}$ is the net tax burden on the private sector, $\left.\eta_D=-\left(\partial D / \partial E^{\prime}\right)\left(E^{\prime} / D\right)\right]$ is the elasticity of private domestic demand with respect to the real exchange rate (reflecting the income effect of a change in the terms of trade), and $D_Y$ is the private marginal propensity to spend.
Under the assumptions mentioned above, one can also show from the definition of $\beta_2$ in the appendix to Chapter 23 that:
$$
\beta_2=\frac{\eta(1-\tau)(1-m)}{1-D_\gamma+m}, \quad \eta=-\frac{D_r}{(1-\tau) Y},
$$
where $\eta$ is the change in private demand induced by a one percentage point change in the real interest rate, measured relative to private disposable income (we introduced this parameter in Chapter 18). Given these specifications, your flrst sheet should allow you to choose the parameters $\gamma, \varphi, h, \theta, m, \eta_X, \eta_3, \tau, \eta, \eta_D$ and $D_Y$ to calculate the auxiliary variables $\beta_2, \beta_2, \beta_1$ and $\beta$ (from which your spreadsheet can calculate the coefficients $a_1$ and $a_6$ in (90)).
You should construct a deterministic as well as a stochastic version of the model. In the deterministic version you just feed an exogenous time sequence of the shock variables $z_1$ and $s_{\mathrm{t}}$ into the model. In the stochastic version of the model, the shock variables are assumed to be given by the autoregressive processes:
$$
\begin{array}{llll}
z_{\mathrm{t}+1}=\delta z_{\mathrm{T}}+x_{1+1}, & 0 \leq \delta<1, & x_{\mathrm{r}} \sim N\left(0, \sigma_x^2\right), & x_{\mathrm{t}} \text { i.i.d } \\
s_{t+1}=\omega s_{\mathrm{t}}+c_{\mathrm{t}+1}, & 0 \leq \omega<1, & c_{\mathrm{r}} \sim N\left(0, \sigma_e^2\right), & c_{\mathrm{t}} \text { i.i.d. }
\end{array}
$$
so your first sheet should also allow you to choose the autocorrelation coefficients $\delta$ and $\omega$ and the standard deviations $\sigma_\pi$ and $\sigma_\sigma$. We suggest that you simulate the model over 100 periods, assuming that the economy starts out in long-run equilibrium in the initial period 0 (so that all of the variables $y, f, e^{\prime}, x, c, z$ and $s$ are equal to zero in period 0 and period -1). From the web page for this book http://highered.mcgraw-hill.com/sites/0077104250/ student_view0/index.html you can download an Excel spreadsheat with two different 100period samples taken from the standardized normal distribution. Choose the first sample to represent the stochastic shock variable $x_{\text {, }}$ and the second sample to represent the shock variable $c_{\mathrm{r}}$ To calibrate the magnitude of the shocks $\mathrm{x}_{\mathrm{f}}$ and $c_u$ you must multiply the samples from the standardized normal distribution by the respective standard deviations $\sigma_s$ and $\sigma_c$
Apart from listing the parameters of the model, your first sheet should also list the standard deviations of output and inflation as well as the coefficient of correlation between output and inflation and the coefficients of autocorrelation for these variables (going back four periods) emerging from your simulations of the stochastic model version. It will also be useful to include diagrams illustrating the simulated values of the output gap, the inflation gap and the real exchange rate.
For the simulation of the deterministic version of the model, we propose that you use the parameter values
$$
\begin{aligned}
& \varphi=0.5, \quad \gamma=0.2, \quad m=0.3, \quad \eta_X=\eta_M=1.5, \quad \theta=2, \\
& h=0.5, \quad \tau=0.2, \quad \eta=0.5, \quad \eta_0=0.3, \quad D_{\mathrm{Y}}=0.8 . \\
&
\end{aligned}
$$
3. Use the deterministic model version to simulate the dynamic effects of a temporary negative demand shock, where $z_1=-1$ in period $t=1$ and $z_1=0$ for all $t \geq 2$. Ilustrate the evolution of the output and inflation gaps in diagrams and comment on your results. Investigate how the economy's adjustment to the shock depends on the conduct of monetary policy by varying the value of the parameter $h$. Try to explain your findings.
4. Vary (one by one) the parameters $\varphi$ and $\theta$ to explore how the economy's dynamic reaction to the temporary demand shock depends on the way expectations ars formed. Try to explain your results.
5. Now use the deterministic model to simulate the dynamic effects of a temporary negative supply shock, where $s_1=+1$ in period $t=1$ and $s_1=0$ for all $t \geq 2$. Illustrate the evolution of the output and inflation gaps in diagrams and comment on your results. How do the results depend on the monetary policy parameter $h$ ? Does monetary policy face a dilemma? Explain.
6. Simulate the stochastic version of the model, setting $\delta=\omega=0.5$, and $\sigma_x^2=\sigma_{\mathrm{e}}^2=1$, while maintaining the other parameter values given above. From this point of departure, try to adjust the values of $\varphi, \delta, \omega, \sigma_x^2$ and $\sigma_{\varepsilon}^2$ so as to achieve a better match between the modelsimulated standard deviations and coefficients of correlation and autocorrelation for output and inflation and the corresponding data for the UK economy presented in Tables 13.2-13.4. For your information, the standard deviation of the cyclical component in the UK inflation rate was 0.0086 , according to the quarterly data underlying Tables 13.3 and 13.4 . (When trying to reproduce the observed correlation between output and inflation, you should only focus on the contemporaneous coefficient of correlation. Moreover, note that our simplified model cannot be expected to reproduce the data with great accuracy.) Assuming that your parameter values are plausible, what are the implications of your analysis for the relative importance of demand shocks and supply shocks as drivers of the UK business cycle?