91Ó°ÊÓ

(1) For the commodity market $$ \begin{array}{ll} Y=C+I \\ C=a Y+b \quad(00) \\ I=c r+d \quad(c<0, d>0) \end{array} $$ where \(r\) is the interest rate. Show that, when the commodity market is in equilibrium, $$ (1-a) Y-c r=b+d $$ (2) For the money market $$ \begin{array}{lll} \text { (money supply) } & M_{\mathrm{s}}=M_{\mathrm{S}}^{*} & \left(M_{\mathrm{S}}^{*}>0\right) \\ \text { (total demand for money) } & M_{\mathrm{D}}=k_{1} Y+k_{2} r+k_{3} & \left(k_{1}>0, k_{2}<0, k_{3}>0\right) \\ \text { (equilibrium) } & M_{\mathrm{D}}=M_{\mathrm{S}} & \end{array} $$ Show that when the money market is in equilibrium $$ k_{1} Y+k_{2} r=M_{\mathrm{S}}^{*}-k_{3} $$ (3) (a) By solving the simultaneous equations derived in parts (1) and (2) show that when the commodity and money markets are both in equilibrium. $$ Y=\frac{k_{2}(b+d)+c\left(M_{\mathrm{S}}^{*}-k_{3}\right)}{(1-a) k_{2}+c k_{1}} $$ (b) Write down the money supply multiplier, \(\partial Y / \partial M_{s}^{*}\) and deduce that an increase in \(M_{s}^{*}\) causes an increase in \(Y\).

Step by step solution

01

- Commodity Market Equilibrium Setup

Write the commodity market equilibrium condition: \[ Y = C + I \] Substitute the given functions for consumption \(C = aY + b\) and investment \(I = cr + d\), resulting in: \[ Y = (aY + b) + (cr + d) \]

02

- Simplify the Equation

Combine like terms: \[ Y = aY + b + cr + d \] Rearrange to isolate terms involving \(Y\): \[ Y - aY = b + d + cr \] This simplifies to: \[ (1 - a)Y = b + d + cr \]

03

- Final Commodity Market Equation

Rearrange to show the equilibrium condition: \[ (1 - a)Y - cr = b + d \] Alternatively, \[ (1-a) Y - c r = b+d \]

04

- Money Market Equilibrium Setup

Write the money market equilibrium condition: \[ M_D = M_S \] Given the total demand for money \(M_D = k_1Y + k_2r + k_3\) and money supply \(M_S = M_S^*\), we get: \[ k_1Y + k_2r + k_3 = M_S^* \]

05

- Simplify the Money Market Equation

Rearrange to isolate terms involving \(Y\) and \(r\): \[ k_1Y + k_2r = M_S^* - k_3 \]

06

- Solve Simultaneous Equations

Substitute the commodity market equilibrium equation into the money market equation: \[ (1-a)Y - cr = b + d \] \[ k_1Y + k_2r = M_S^* - k_3 \] To solve for \(Y\) and \(r\), we first isolate \(r\) from the second equation: \[ r = \frac{M_S^* - k_3 - k_1Y}{k_2} \]

07

- Substitute \(r\)

Substitute \(r = \frac{M_S^* - k_3 - k_1Y}{k_2}\) into the first equation: \[ (1 - a)Y - c\left(\frac{M_S^* - k_3 - k_1Y}{k_2}\right) = b + d \] Simplify and solve for \(Y\): \[ (1 - a)Y - \frac{c(M_S^* - k_3 - k_1Y)}{k_2} = b + d \] Multiply through by \(k_2\): \[ k_2(1 - a)Y - c(M_S^*) + ck_3 + c(k_1Y) = k_2(b + d) \] Combine like terms to isolate \(Y\): \[ (k_2(1 - a) + ck_1)Y = k_2(b + d) + c(M_S^*) - ck_3 \] Simplify: \[ Y = \frac{k_2(b + d) + c(M_S^* - k_3)}{k_2(1 - a) + ck_1} \]

08

- Determine Money Supply Multiplier

The money supply multiplier \(\frac{\partial Y}{\partial M_s^*}\) is found by differentiating \(Y\) with respect to \(M_s^*\): \[ \frac{\partial Y}{\partial M_S^*} = \frac{c}{k_2(1 - a) + ck_1} \]

09

- Conclude Impact of Increase in Money Supply

Since \(c < 0\), the denominator \(k_2(1 - a) + ck_1\) is negative, meaning the derivative \(\frac{\partial Y}{\partial M_S^*} > 0\). Thus, an increase in \(M_S^*\) results in an increase in \(Y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Commodity Market

The commodity market is one of the key pillars of macroeconomic equilibrium. It involves the total production of an economy (denoted as Y) being equal to the sum of consumption (C) and investment (I). In this context, consumption is described as a function of income (Y):
\( C = aY + b \)
where \( 0 < a < 1 \) reflects the consumption propensity and \( b > 0 \) represents autonomous consumption, which occurs regardless of income. Investment (I) is represented as a function of the interest rate (r):
\( I = cr + d \)
where \( c < 0 \) indicates that higher interest rates generally reduce investment, and \( d > 0 \) represents autonomous investment. By substituting these into the equilibrium condition Y = C + I, we arrive at an equation reflecting equilibrium in the commodity market:
\( (1 - a)Y - cr = b + d \).

Money Market

The money market addresses the demand and supply for money in the economy. It is in equilibrium when the money supply (\(M_s\)) equals the total demand for money (\(M_D\)). Money demand is modeled as:
\( M_D = k_1Y + k_2r + k_3 \)
Here, \( k_1 > 0 \) indicates that higher income levels increase money demand, \( k_2 < 0 \) shows that higher interest rates reduce money demand, and \( k_3 > 0 \) reflects autonomous money demand. The money supply is given as \( M_s = M_S^* \), which is exogenously determined. In equilibrium,
\( M_D = M_S \), thus:
\( k_1Y + k_2r = M_S^* - k_3 \).

Equilibrium Analysis

Equilibrium is achieved when both the commodity and money markets reach their respective equilibrium conditions simultaneously. For the commodity market, the equilibrium equation is:
\( (1 - a) Y - cr = b + d \)
For the money market, it is:
\( k_1 Y + k_2 r = M_S^* - k_3 \)
These equations can be solved simultaneously to determine the values of Y (aggregate output) and r (interest rate) that satisfy both markets.

Simultaneous Equations

Solving simultaneous equations involves using both the commodity and money market equilibrium conditions to find the consistent values of Y and r. First, you isolate the interest rate (r) from one of the equations, often the money market:
\( r = \frac{M_S^* - k_3 - k_1Y}{k_2} \)
Then, substitute this expression for r into the commodity market equation:
\( (1 - a)Y - c\left(\frac{M_S^* - k_3 - k_1Y}{k_2}\right) = b + d \)
By simplifying this, we derive an expression for Y that addresses the interaction between output, interest rates, and exogenous factors such as money supply.

Money Supply Multiplier

The money supply multiplier reflects how changes in the money supply affect aggregate output. By differentiating aggregate output (Y) with respect to money supply (\(M_S^*\)), we get:
\( \frac{\partial Y}{\partial M_S^*} = \frac{c}{k_2 (1 - a) + ck_1} \)
Since c is negative, and by ensuring the right conditions, the denominator is negative, making the overall derivative positive. This implies that an increase in the money supply (\(M_S^*\)) leads to an increase in aggregate output (Y), highlighting the multiplier effect of money supply changes on economic activity.