Question

An economy has a fixed price level, no imports, and no income taxes. \(M P C\) is 0.80 , and real GDP is \(\$ 150\) billion. Businesses increase investment by \(\$ 5\) billion. An economy has a fixed price level, no imports, and no income taxes. An increase in autonomous expenditure of \(\$ 2\) trillion increases equilibrium expenditure by \(\$ 8\) trillion. Calculate the multiplier and explain what happens to the multiplier if an income tax is introduced.

Short answer

The multiplier is 4. If an income tax is introduced, the multiplier decreases.

Step-by-step solution

Understand the Given Information

This problem involves a fixed price level economy with no imports and no income taxes, where the marginal propensity to consume (MPC) is 0.80. Real GDP is \(150 billion, and businesses increase investment by \)5 billion. Additionally, an increase in autonomous expenditure of \(2 trillion leads to an increase in equilibrium expenditure by \)8 trillion.

Define the Multiplier

The multiplier in an economy without taxes or imports can be calculated using the formula: \[ \text{multiplier} = \frac{1}{1 - MPC} }.\]

Calculate the Multiplier

Using the given MPC value of 0.80: \[ \text{multiplier} = \frac{1}{1 - 0.80} = \frac{1}{0.20} = 5 }\]

Confirm the Multiplier with Autonomous Expenditure Data

Using the given information that a \(2 trillion increase in autonomous expenditure leads to an \)8 trillion increase in equilibrium expenditure, the multiplier can be confirmed: \[ \text{multiplier} = \frac{\text{change in equilibrium expenditure}}{\text{change in autonomous expenditure}} = \frac{8}{2} = 4 }\]

Analyze the Difference

There seems to be a discrepancy between the two modes of calculating the multiplier. The correct contextual multiplier is confirmed through the given expenditure data, which is 4.

Introduce Income Tax

If an income tax is introduced, the marginal propensity to consume (MPC) effectively decreases because consumers have less disposable income to spend. This leads to a lower multiplier. Mathematically, if 't' is the tax rate, the formula for the multiplier adjusts to: \[ \text{multiplier} = \frac{1}{1 - MPC \times (1 - t)} }\]

Key concepts

Marginal Propensity to Consume (MPC)

The Marginal Propensity to Consume (MPC) is a crucial concept in understanding how economies react to changes in income. It signifies the portion of additional income that consumers are likely to spend rather than save. For example, an MPC of 0.80 means that for every extra dollar earned, 80 cents is spent and 20 cents is saved.

The MPC plays a key role in determining the multiplier effect, which shows how changes in spending can lead to changes in overall economic output. The higher the MPC, the larger the multiplier and the more pronounced the effects on the economy. This is because a higher MPC means more of every dollar is spent, thus circulating more money through the economy.

To see this in action, consider the formula for the multiplier in an economy with no taxes or imports: \[\text{multiplier} = \frac{1}{1 - MPC}\]. With an MPC of 0.80, the multiplier is 5, indicating that an initial increase in spending can magnify its impact on GDP by five times.

Autonomous Expenditure

Autonomous expenditure refers to the spending that does not depend on the current level of income or economic activity. This includes investments by businesses, government spending, and exports. In our example, an increase of autonomous expenditure by \(2 trillion led to an increase in equilibrium expenditure by \)8 trillion.

This shows the powerful impact of autonomous expenditures on the overall economy through the multiplier. When businesses invest more or the government increases spending, it sets off a chain reaction of increased spending by consumers and businesses.

To verify the multiplier using the given data: \[\text{multiplier} = \frac{\text{change in equilibrium expenditure}}{\text{change in autonomous expenditure}} = \frac{8}{2} = 4\].

This real-life example helps students see the direct connection between changes in autonomous expenditure and the resulting changes in equilibrium expenditure.

Income Tax Impact on Economy

Introducing income taxes impacts the economy by reducing consumers' disposable income. Lower disposable income means consumers will spend less, which decreases the Marginal Propensity to Consume (MPC).

A lower MPC in turn reduces the multiplier effect because each dollar of increased income results in less spending. This means that the economy will not experience as strong of a boost from increased spending or investment.

Mathematically, if 't' is the tax rate, the new formula for the multiplier is: \[\text{multiplier} = \frac{1}{1 - MPC \times (1 - t)}\].

As taxes increase, the term MPC \times (1-t) becomes smaller, leading to a smaller multiplier. For instance, if the MPC is still 0.80 but the tax rate is 25%, the modified multiplier is: \[\text{multiplier} = \frac{1}{1 - 0.80 \times (1 - 0.25)} = \frac{1}{1 - 0.80 \times 0.75} = \frac{1}{1 - 0.60} = 2.5\].

Equilibrium Expenditure

Equilibrium expenditure represents the point where total spending equals total output, meaning the economy is in a state of balance. Any change in autonomous expenditure, investment, or consumption affects equilibrium expenditure.

In the given example, an increase of \(2 trillion in autonomous expenditure resulted in an \)8 trillion increase in equilibrium expenditure, highlighting the multiplier effect in action.

The equilibrium is impacted not just by spending levels but also by factors like taxes and imports. When taxes are introduced or increased, they lower disposable income and reduce overall spending, which shifts the equilibrium to a lower level. Similarly, if there were imports, money spent on foreign goods wouldn't circulate within the domestic economy, also affecting equilibrium expenditure.

Understanding equilibrium expenditure helps students visualize how different economic policies or changes in autonomous expenditure can drastically alter the entire economic landscape.