经济代写|EC1B1 Macroeconomics

Assume that population is equal to 1 . Consider the following IS-LM curve.
$$
\begin{gathered}
y=c(y-\tau)+\iota(\rho)+g \
\frac{m^s}{P}=k(\rho) y
\end{gathered}
$$
where price level $P$, tax revenue $\tau$, government expenditure $g$ and money supply $\mathrm{m}^s$ are given. Two equations determine GDP, $y$ and the interest rate $\rho$. Suppose that
$$
\begin{aligned}
c(y-\tau) & =c_0+c_1 \times(y-\tau) \
k(\rho) & =\frac{k}{\rho} \
\iota(\rho) & =-a \rho
\end{aligned}
$$
where $c_0, c_1, k$, and $a$ are parameters.

(a) Assume that the interest rate $\rho$ is constant. Derive GDP $y$, as a function of government expenditure and tax revenue. Show that $\frac{d y}{d g}>1$ and explain its reason.
Answer Equation (2) can be rewritten as
$$
\begin{aligned}
y & =c(y-\tau)+\iota(\rho)+g \
& =c_0+c_1 \cdot(y-\tau)+-a \rho+g .
\end{aligned}
$$
By solving this equation with respect to $y$, we have
$$
y=\frac{g+c_0-\tau c_1-a \rho}{1-c_1}
$$
GDP is a function of government expenditure and tax revenue. By differentiating this equation with respect to $g$,
$$
\frac{d y}{d g}=\frac{1}{1-c_1}>1
$$

Figure 1: GDP and interest rate in equilibrium.
Figure 2: Effect of an increase in tax revenue.
for all $c_1 \in(0,1)$. This implies that an increase in the government expenditure generates the multiplier effect. The increase in government expenditure rises not only the nation’s income directly but also the consumption indirectly.

(b) Derive the GDP, $y$, and the interest rate, $\rho$, as the function of $P, g, m^s, \tau$. Answer Given $P, g, m^s, \tau$, there are two equations (2) and (3) and two unknown variables, $y$ and $\rho$. From (2),
$$
y=-\frac{a}{1-c_1} \rho+\frac{g+c_0-\tau c_1}{1-c_1}
$$
and from (3), we have
$$
\rho=\frac{k P}{m^s} y
$$
Therefore, we can derive $y$ and $\rho$ as an intersection of these two lines (see Figure 1).
$$
\begin{aligned}
y\left(g, \tau, m^s, P\right) & =\frac{m^s\left(c_0-c_1 \tau+g\right)}{a k P+m^s\left(1-c_1\right)} \
\rho\left(g, \tau, m^s, P\right) & =\frac{P k\left(c_0-c_1 \tau+g\right)}{a k P+m^s\left(1-c_1\right)} .
\end{aligned}
$$

(c) What is the effect of an increase in tax revenue, $\tau$, on GDP and the interest rate? Explain economic logic behind your answer.

Answer We analyze the effect of an increase in tax revenue, $\tau$, on GDP and the interest rate. Note that IS curve depends on $\tau$ but LM curve does not depend on $\tau$. When tax revenue increases, IS curve shifts down (See Figure 2). Both GDP and the interest rate decrease. The economic logic for this result is as follows. First, IS curve implies that an increase in $\tau$ directly decreases demand and GDP (see (IS)):
$$
\tau \uparrow \Rightarrow y \downarrow
$$
Second, the decrease in GDP reduces the demand for money $\left(m^d\right)$. When the price level and money supply are constant, this actually decreases the interest rate in the money market (see (LM)):
$$
\tau \uparrow \Rightarrow y \downarrow \Rightarrow m^d \downarrow \Rightarrow \rho \downarrow
$$
Since investment $i=\iota(\rho)=-a \rho$, a decrease in $\rho$ leads to an increase in the investment $i$ :
$$
\tau \uparrow \Rightarrow y \downarrow \Rightarrow m^d \downarrow \Rightarrow \rho \downarrow \Rightarrow i \uparrow
$$

Figure 3: Effect of an increase in government expenditure.
Figure 4: Effect of an increase in money supply.
This indirectly increases $y$
$$
\tau \uparrow \Rightarrow y \downarrow \Rightarrow m^d \downarrow \Rightarrow \rho \downarrow \Rightarrow i \uparrow \Rightarrow y \uparrow
$$
In this model, the direct effect dominates the indirect effect. Therefore, the increase in $\tau$ decreases both GDP and the interest rate.

(d) What is the effect of an increase in government expenditure, $g$, on GDP and the interest rate? Explain economic logic behind your answer.

Answer Note that IS curve depends on $g$ but LM curve does not depend on $g$. When government expenditure, $g$, increases, IS curve shifts up (see Figure 4). Both GDP and the interest rate increase. The economic logic for this result is as follows. First, the IS curve implies that an increase in $g$ directly increases demand and GDP (see (IS)):
$$
g \uparrow \Rightarrow y \uparrow
$$
Second, the increase in GDP increases the demand for money. When the price level and money supply are constant, this actually increases the interest rate in the money market (see $(\mathrm{LM}))$ :
$$
g \uparrow \Rightarrow y \uparrow \Rightarrow m^d \uparrow \Rightarrow \rho \uparrow
$$
Since $i=\iota(\rho)=-a \rho$, an increase in $\rho$ leads to the decreases in the investment $i:$
$$
g \uparrow \Rightarrow y \uparrow \Rightarrow m^d \uparrow \Rightarrow \rho \uparrow \Rightarrow i \downarrow
$$
This indirectly decreases $y$
$$
g \uparrow \Rightarrow y \uparrow \Rightarrow m^d \uparrow \Rightarrow \rho \uparrow \Rightarrow i \downarrow \Rightarrow y \downarrow
$$
In this model, the direct effect dominates the indirect effect. Therefore, the increase in $g$ increases both GDP and the interest rate.