91Ó°ÊÓ

In the commodity market, we look at the balance between consumption and investment on one side and total output (or income, noted as Y) on the other side.
The consumption function, defined as C, depends on income and is written as: \[ C = aY + b \] Here, ‘a’ is the marginal propensity to consume (0 < a < 1) and ‘b’ is the autonomous consumption (b > 0).
The investment function, noted as I, depends on the interest rate, r, and is expressed as: \[ I = cr + d \] In this case, ‘c’ is less than zero (c < 0) indicating that investment decreases as interest rates rise, and ‘d’ represents autonomous investment (d > 0).
For the commodity market to be in equilibrium, total production must equal total consumption and investment. The equilibrium condition is: \[ Y = C + I \] Plugging in the consumption and investment functions, we get: \[ Y = aY + b + cr + d \] Rearranging to isolate Y and r gives us the equation: \[ (1 - a)Y - cr = b + d \]

The money market is where the equilibrium between money supply (\( M_S \)) and money demand (\( M_D \)) is determined. The money supply is assumed to be constant and represented as \( M_S^* \).
Money demand depends on income (Y) and the interest rate (r) and can be represented by the equation: \[ M_D = k_1Y + k_2r + k_3 \] Here, \( k_1 > 0 \) indicates that money demand rises with an increase in income. \( k_2 < 0 \) indicates that money demand decreases as the interest rate increases. \( k_3 > 0 \) represents the autonomous money demand.
For the money market to be in equilibrium, money supply must equal money demand: \[ M_S^* = k_1Y + k_2r + k_3 \] Rearranging to isolate Y and r, we get: \[ k_1Y + k_2r = M_S^* - k_3 \]

We use matrix algebra to solve systems of linear equations in economics. Combining the equilibrium conditions for both the commodity and money markets, we form a system of equations:
From the commodity market equilibrium: \[ (1 - a)Y - cr = b + d \] From the money market equilibrium: \[ k_1Y + k_2r = M_S^* - k_3 \] We can write this system of equations in matrix form as:
\[ \begin{bmatrix}1 - a & -c \ k_1 & k_2 \end{bmatrix} \begin{bmatrix} Y \ r \end{bmatrix} = \begin{bmatrix} b + d \ M_S^* - k_3 \end{bmatrix} \] To solve for Y and r, we first find the inverse of the coefficient matrix: \[ \begin{bmatrix}1 - a & -c \ k_1 & k_2 \end{bmatrix}^{-1} \] Multiply both sides of the matrix equation by this inverse to isolate the solution vector: \[ \begin{bmatrix} Y \ r \end{bmatrix} = \begin{bmatrix}1 - a & -c \ k_1 & k_2 \end{bmatrix}^{-1} \begin{bmatrix} b + d \ M_S^* - k_3 \end{bmatrix} \]

The multiplier effect measures how changes in one of the external factors, like money supply, affect other variables, such as income and interest rate. Here, focus on the impact of money supply changes (\( M_S^* \)) on the interest rate (r).
The interest rate's sensitivity to changes in the money supply is captured through the matrix solution. Specifically, notice how changes in \( M_S^* \) affect the interest rate in the equation:
\[ \begin{bmatrix} Y \ r \end{bmatrix} = \frac{1}{k_2 - ak_2 + ck_1} \begin{bmatrix} k_2 & c \ -k_1 & 1 - a \end{bmatrix} \begin{bmatrix} b + d \ M_S^* - k_3 \end{bmatrix} \] Analyzing this equation reveals that the term involving \( M_S^* \) shows a negative relationship with r due to \( k_2 < 0 \).
This means that as the money supply increases, the interest rate decreases. This showcases the multiplier effect where a change in one variable leads to a change in another.

Interest rates are critical for many economic decisions. They are determined by the interaction of money supply and demand. From the previous sections, we have the equation:
\[ k_1Y + k_2r = M_S^* - k_3 \] Rearranging to solve for r, we get:
\[ k_2r = M_S^* - k_3 - k_1Y \] \[ r = \frac{M_S^* - k_3 - k_1Y}{k_2} \] Because \( k_2 < 0 \), the equation indicates that r decreases as \( M_S^* \) increases since an increase in money supply lowers the interest rate.
The components \( k_1Y \) and \( k_3 \) further adjust the interest rate based on income and autonomous demand, respectively. This relationship plays a fundamental role in macroeconomic policy and financial markets as central banks manipulate money supply to influence interest rates and thus, economic activity.