We are tasked with finding the equilibrium level of income and consumption in a Keynesian framework, where the consumption function, investment, government spending, and taxes are provided. We will solve this using matrices.

1. Key Variables & Functions

• Consumption function (C): $C = 0.6Y + 10$
• Planned Investment (I): $I = 12$
• Government Spending (G): $G = 20$
• Tax function (T): $T = 0.2Y + 25$
• Another Consumption function (C2): $C = 0.9Y_d + 70$, where $Y_d = Y - T$ is disposable income.

2. Equilibrium Condition

The equilibrium condition is when planned aggregate spending (sum of consumption, investment, and government spending) equals the total income, $Y$. This can be expressed as:

$Y = C + I + G$

3. Consumption Function Substitution

First, let's write the equation for consumption based on $Y_d = Y - T$:

$C = 0.9(Y - T) + 70 = 0.9(Y - (0.2Y + 25)) + 70$

Now simplify the consumption function:

$C = 0.9(Y - 0.2Y - 25) + 70$ $C = 0.9(0.8Y - 25) + 70$ $C = 0.72Y - 22.5 + 70$ $C = 0.72Y + 47.5$

4. Aggregate Demand Function

Now, substitute the new consumption function and other values into the aggregate demand equation:

$Y = C + I + G$ $Y = (0.72Y + 47.5) + 12 + 20$

Simplifying this:

$Y = 0.72Y + 79.5$

5. Solving for Equilibrium Income

Rearrange the equation to isolate $Y$:

$Y - 0.72Y = 79.5$ $0.28Y = 79.5$ $Y = \frac{79.5}{0.28}$ $Y = 284.64$

So, the equilibrium income is approximately 284.64.

6. Find Equilibrium Consumption

Substitute $Y = 284.64$ into the consumption function:

$C = 0.72Y + 47.5$ $C = 0.72(284.64) + 47.5$ $C = 204.74 + 47.5$ $C = 252.24$

So, the equilibrium consumption is approximately 252.24.

7. Solution in Matrix Form

We can represent this system in matrix form as follows:

Let the vector of unknowns be:

Y \\ C \end{bmatrix} = \begin{bmatrix} 1 & 0.72 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} Y \\ C \end{bmatrix} + \begin{bmatrix} 79.5 \\ 47.5 \end{bmatrix}$$ This system can be solved using standard methods in matrix algebra, as outlined in the steps above. ### Conclusion - The equilibrium income ($$Y$$) is approximately **284.64**. - The equilibrium consumption ($$C$$) is approximately **252.24**. Would you like to explore the matrix formulation further or have any specific questions? --- ### Related Questions: 1. How does the consumption function affect the equilibrium level of income? 2. How would changes in government spending (G) influence the equilibrium income? 3. What would happen if the tax function were changed to $$T = 0.3Y + 20$$? 4. How do changes in the marginal propensity to consume (MPC) affect the equilibrium? 5. Can this model be expanded to include exports and imports? ### Tip: When working with matrices in economic models, always ensure that the system is correctly specified (i.e., that the correct functions for consumption, taxes, and investment are used). This allows you to derive more accurate results for the equilibrium.