Show that in a simple Keynesian model, equal expansion in tax and government expenditure does not always lead to balanced budget theorem.

The Keynesian Model and the Multiplier

The simple Keynesian model focuses on the relationship between aggregate demand (AD) and national income (Y). AD is composed of Consumption (C), Investment (I), Government Spending (G), and Net Exports (X-M). The multiplier effect explains how an initial change in spending can lead to a larger change in national income. The expenditure multiplier (k) is calculated as 1/(1-MPC), where MPC is the marginal propensity to consume.

Government Expenditure Multiplier

An increase in government expenditure (G) directly increases AD by the amount of the increase. This initial increase in spending leads to a further increase in income, which in turn leads to further increases in consumption, and so on. The total increase in income is equal to the government expenditure multiplier multiplied by the initial increase in G. For example, if MPC is 0.8, the multiplier is 1/(1-0.8) = 5. A $100 increase in G will lead to a $500 increase in Y.

Tax Multiplier

A change in taxes (T) affects AD indirectly through its impact on disposable income (Yd = Y - T). A decrease in taxes increases disposable income, leading to an increase in consumption. However, the tax multiplier is smaller in absolute value than the government expenditure multiplier. The tax multiplier is calculated as -MPC/(1-MPC). Using the same MPC of 0.8, the tax multiplier is -0.8/(1-0.8) = -4. A $100 decrease in T will lead to a $400 increase in Y.

The Balanced Budget Multiplier

The balanced budget multiplier is the sum of the government expenditure multiplier and the tax multiplier. In the simple Keynesian model, it is always equal to 1: k_G + k_T = 1/(1-MPC) + (-MPC)/(1-MPC) = 1. This suggests that an equal increase in G and a decrease in T will lead to an equal increase in Y. However, this holds true under specific assumptions.

Why the Theorem Doesn't Always Hold

The balanced budget multiplier of 1 is based on the assumption of a *simple* Keynesian model. Several factors can cause it to deviate from 1:

• Crowding Out Effect: Increased government spending might lead to higher interest rates, crowding out private investment. This reduces the overall impact on AD.
• Changes in Expectations: If individuals anticipate future tax increases to offset the current decrease, they may reduce their current consumption, diminishing the multiplier effect.
• Supply-Side Constraints: If the economy is operating near full capacity, increased government spending may lead to inflation rather than an increase in real output.
• Non-Linearities in MPC: The MPC isn't constant across all income levels. As income rises, the MPC may fall, reducing the multiplier effect.

Furthermore, the impact of taxes depends on *which* taxes are increased. Increasing taxes on investment income will have a different effect than increasing taxes on consumption. Similarly, the type of government expenditure matters – investment in infrastructure has a different multiplier effect than transfer payments.

Illustrative Example

Consider an economy with an MPC of 0.75. If the government increases spending by $100 billion and simultaneously increases taxes by $100 billion:

Component	Impact
Government Expenditure Multiplier	1/(1-0.75) = 4
Tax Multiplier	-0.75/(1-0.75) = -3
Net Impact on Income	(4 * $100 billion) + (-3 * $100 billion) = $100 billion

In this case, the balanced budget multiplier holds. However, if crowding out occurs, or if expectations lead to reduced consumption, the actual increase in income could be less than $100 billion.