# Small Open Economy

Opening our model up to trade.

- Modified: Jan 1st, 2022

## ¶ Changes to The Model

There are two changes we make to switch from a closed economy model to a small open economy model. And the name of this kind of model hints at the changes we must make.

Small
The interest rate is exogenously equal to the world interest rate.
Open
There is now a term for net exports in output demand.

For the economy to be Open means that there is international trade in goods/assets. We represent this by adding a term for net exports to the equation for output demand.

$Y_{d}=C(r)+I(r)+G+NX$ $Y_{d}'=c' - (1-\delta)K' + G' + NX'$

In the model, Net exports are free floating, and can attain any value which allows output supply to equal output demand. This also means that domestic output demand, defined as $C+I+G$ isn’t necessarily equal to output supply in equilibrium. The difference is made up for by importing or exporting.

The economy is Small: in the sense that the country as a whole is a price-taker in the international market for goods and assets. Assume that when individuals make intertemporal decisions, they have access to a large international market. This market is so big that the domestic economy can’t affect the global interest rate $r_{w}$. Thus the real interest rate is now exogenous.

$r=r_{w}$

In other words, we are adding one constraint (fixed $r$) and simultaneously removing another (that $NX=0$).

The asset market now clears because of trade instead of price changes.

Note that the real wage $w$ is not exogenously fixed. The market clearing condition for labor doesn’t change.

### ¶ Full Definition of SOE Model from Chapter 16

Definition of the Small Open Economy 2-Period Model

Given the exogenous parameters $\lbrace K,h,h’,z,z’,G,G’, \textcolor{#f00}{r_w}\rbrace$, a competitive equilibrium is any set of endogenous prices $\lbrace w,w’\rbrace$ and allocations $\lbrace c,c’,l,l’,N_{s}=h-l,N_{s}^{‘}=h’-l’,N_{d},N_{d}^{\prime},I,K^{\prime},T,T^{\prime},\textcolor{#f00}{NX},\textcolor{#f00}{NX’}\rbrace$ that satisfy the following conditions:

• Representative Consumer, taking prices as given, solves:
$\max_{c,c',l,'l} \left[u(c,l)+\beta u(c',l')\right]$ \begin{aligned} \text{s.t. }\ \ \ \ & c\geq0, \ \ \ \ c'\geq0, \ \ \ \ 0\leq l \leq h, \ \ \ \ 0\leq l' \leq h' \\ & c+\frac{c'}{1+r}\leq\left[w(h-l)+\pi-T\right]+\frac{w'(h'-l')+\pi'-T'}{1+r}\\ \end{aligned}
• Representative Firm, taking prices as given, solves:
$\max_{c,c',l,'l} \left[u(c,l)+\beta u(c',l')\right]$ \begin{aligned} \text{s.t. }\ \ \ \ & N_d\geq0, \ \ \ \ N_d'\geq0, \ \ \ \ K'\geq 0\\ & K' = (1-\delta)K + I\\ & \pi = zF(K,N_d) - wN_d - I\\ & \pi' = z'F(K',N_d') - w'N_d' + (1-\delta)K' \end{aligned}
• Markets Clear:
\begin{aligned} N_d &= N_s = h-l\\ N_d' &= N_s' = h'-l'\\ zF(K,N_d) &= c + G + I + \textcolor{#f00}{NX}\\ z'F(K',N_d') &= c'+ G' - (1-\delta)K' + \textcolor{#f00}{NX'}\\ \end{aligned}
• Government Budget is balanced:
$G+\frac{G^{\prime}}{1+r}=T+\frac{T^{\prime}}{1+r}$

Changes from the closed economy version of the model are highlighted in red.

## ¶ How does this change the effects of shocks in the model?

The labor market works pretty much the same as before. Our agents are price-takers, which means they were already behaving as if the interest rate were exogenous.

But now shocks which would change the interest rate will instead just cause a change in $NX$, shifting the $Y_{d}$ curve to cancel out that change in interest rates.

• Increase in $z$:
• Like in closed economy (ch 11), this causes the $N_{d}$ and $Y_{s}$ curves to shift right.
• But, unlike in a closed economy, the $r$ is fixed, so the $Y_{d}$ must now shift to bring the economy back into equilibrium.
• So equilibrium $NX$ will increase. Much of the extra production will thus be shipped overseas.
• Increase in $z’$:
• The firm wants to invest more, which causes $Y_{d}$ to shift right.
• Then NX decreases to cancel out this shift and keep the economy in its original equilibrium for $Y,r,N,w$.
• Essentially, all that’s happened is that net exports have been turned into investment.
• Increase in $G$:
• In the closed economy, the direct effects are a shift rightwards in the $Y_d$ curve, as well as a shift rightwards in the $N_s$ curve because of higher taxes.
• The latter effect still happens. Higher $T$ causes $N_{s}$ and $Y_{s}$ curves to shift right.
• But the shift in the Y_{d} curve is counteracted by a decrease in $NX$.
• So output goes up, but not as much as in the closed economy.
• Increase in $r_w$:
• Increase in $r$ makes the $N_{s}$ curve shift right as we move up along the $Y_{s}$ curve.
• And the $Y_{d}$ curve shifts to bring this into equilibrium as this extra output is exported.
• Note that the firm is investing less and consumer is consuming less because of the higher interest rate. So some of the preexisting output is being exported as well.