In a simple macroeconomic model, the equilibrium level of income is reached when the total planned spending in the economy equals the total income or output produced. This total planned spending is also known as Aggregate Demand (AD). In a basic model with only households and firms, Aggregate Demand consists of Consumption (C) and Investment (I).

The equilibrium condition is given by:

\(Y = C + I\)

where \(Y\) represents the level of income.

Understanding the Given Functions

We are provided with the following equations:

• Consumption Function: \(C = 40 + 0.8Y\)
• Investment: \(I = 10\)

The consumption function \(C = 40 + 0.8Y\) shows that consumption depends on two parts: autonomous consumption (40), which does not depend on income, and induced consumption (\(0.8Y\)), which depends on the level of income. The coefficient 0.8 is the Marginal Propensity to Consume (MPC), indicating that 80% of any extra income is spent on consumption.

The investment function \(I = 10\) indicates that investment is autonomous, meaning it does not depend on the level of income in this simplified model.

Calculating Equilibrium Income Level

To find the equilibrium level of income (\(Y\)), we substitute the given equations for \(C\) and \(I\) into the equilibrium condition \(Y = C + I\).

\(Y = (40 + 0.8Y) + 10\)

Now, we need to solve this equation for \(Y\).

First, combine the constant terms on the right side:

\(Y = 50 + 0.8Y\)

Next, collect all terms containing \(Y\) on one side of the equation. Subtract \(0.8Y\) from both sides:

\(Y - 0.8Y = 50\)

Simplify the left side:

\(0.2Y = 50\)

Finally, isolate \(Y\) by dividing both sides by 0.2:

\(Y = \frac{50}{0.2}\)

To simplify the division, we can multiply the numerator and denominator by 10:

\(Y = \frac{500}{2}\)

Performing the division:

\(Y = 250\)

Thus, the equilibrium level of income is 250.

We can verify this by substituting \(Y = 250\) back into the consumption function and the equilibrium condition:

• \(C = 40 + 0.8 \times 250 = 40 + 200 = 240\)
• \(I = 10\)
• \(C + I = 240 + 10 = 250\)
• Since \(Y = 250\) and \(C + I = 250\), the equilibrium condition \(Y = C + I\) is satisfied.

The calculated equilibrium level of income is 250.

Revision Table: Key Concepts

Additional Information: The Simple Keynesian Model

This problem is based on a simple Keynesian model of income determination. This model assumes that prices and wages are fixed and that output is determined by the level of aggregate demand. Key assumptions of this model include:

• Closed Economy: No international trade (no exports or imports).
• No Government: No government spending or taxes.
• Fixed Price Level: The analysis focuses on changes in real output or income.

In this context, the equilibrium income is the level of income at which firms are willing to produce exactly what is demanded by households (for consumption) and firms (for investment).

The multiplier effect is also an important concept in this model. The simple expenditure multiplier is given by \( \frac{1}{1 - MPC} \). In this case, MPC is 0.8, so the multiplier is \( \frac{1}{1 - 0.8} = \frac{1}{0.2} = 5 \). This means that any initial change in autonomous spending (like autonomous consumption or investment) will lead to a five-fold larger change in the equilibrium level of income.

For example, if investment increased from 10 to 20 (a change of 10), the new equilibrium income would be \(Y = 40 + 0.8Y + 20 \implies 0.2Y = 60 \implies Y = 300\). The change in income is \(300 - 250 = 50\), which is 5 times the initial change in investment ( \(5 \times 10 = 50\) ).