Chapter 11, Problem 2PA

 (a)

To determine

The graphical representation of the planned expenditure as a function of income.

Explanation of Solution

The Keynesian cross model is a basic model of income determination, which describes that there is one level of national income that occurs at the point where the planned expenditure (PE) equals the actual expenditure (AE). The planned expenditure is the sum total of consumption expenditure (C), investment expenditure (I), and the government expenditure (G).

$\text{PE = C + I + G}\text{. }$ (1)

Now substitute the respective values in Equation (1) to get the value of planned expenditure.

$\begin{array}{c}\text{PE = 120 +0}\text{.8}\left(\text{Y}-400\right)+\text{ 200 + 400}\\ =120+0.80\text{Y}-320+200+400\\ =0.80\text{Y}+400\end{array}$

Therefore, the planned expenditure as a function of income is $\text{PE}=0.80\text{Y}+400$.

Figure 1 illustrates the planned expenditure as a function of income.

In Figure 1, the vertical axis measures the planned expenditure (PE), and the horizontal axis measures the income and output levels where the interaction between the planned expenditure curve and the actual expenditure curve determines the equilibrium level of income and output.

Economics Concept Introduction

Planned expenditure: The planned expenditure is the amount households, firms, and the government would like to spend on goods and services.

Actual expenditure: The actual expenditure is the actual amount households, firms, and the government spend on goods and services.

 (b)

To determine

The equilibrium income.

Explanation of Solution

In Keynesian cross model, the level of equilibrium national income occurs at the point where the planned expenditure equals the actual expenditure.

$\begin{array}{c}\text{Actual Expenditure }=\text{ Planned Expenditure }\\ \text{Y}=\text{PE}\text{. }\end{array}$ (2)

Now, substitute the respective values in Equation (2) to solve the value of equilibrium level of income.

$\begin{array}{c}\text{Y}=\text{PE}\\ \text{Y}=0.80\text{Y}+400\\ \left(1-.80\right)\text{Y}=400\\ \text{Y}=\frac{400}{.2}\\ =2,000\end{array}$

Therefore, the equilibrium level of income is $2,000, which is shown in Figure 1.

 (c)

To determine

The impact of an increase in the government purchases on equilibrium GDP.

Explanation of Solution

As described in part (a), the initial planned expenditure is $\text{PE}=0.80\text{Y}+400$. If the government purchase increases to 420, the new planned expenditure function will be $\text{PE}=0.80\text{Y}+420$, which means the government purchase increases by 20$\left(420-400\right)$.

Now, substitute the respective values in Equation (2) to solve the new equilibrium income.

$\begin{array}{c}\text{Y}=\text{PE}\\ \text{Y}=0.80\text{Y}+400\\ \left(1-.80\right)\text{Y}=400\\ \text{Y}=\frac{400}{.2}\\ =2,000\end{array}$

Therefore, the new equilibrium level of income is $2,100, which indicates that an increase in government purchases of 20 increases income by 100. This is because of the multiplier effect of government purchase on national income.

This multiplier effect on income due to an increase in government purchase can be calculated using the following formula.

$\text{Government purchases mulitiplier}=\frac{1}{\left(1-\text{MPC}\right)}$ (3)

Substitute the respective values into Equation (3).

$\begin{array}{c}\text{Government purchases mulitiplier}=\frac{1}{\left(1-0.8\right)}\\ =5\end{array}$

Therefore, the value of government purchases multiplier is 5.

Economics Concept Introduction

Government purchases multiplier: The government purchases multiplier indicates the ratio of change in equilibrium income to the change in government spending.

 (d)

To determine

Explain that the level of government purchases is needed to achieve an income of 2,400.

Explanation of Solution

As described in part (b), the original equilibrium income is 2,000 to achieve an income level of 2,400; there is a need for an increase of 400 $\left(2,400-2,000\right)$ over the original level of income, where the value of government purchases multiplier is 5 (described in part (c)).

Now, the increase in government purchase to achieve an income level of 2,400 can be calculated as follows:

$\begin{array}{c}\text{Government purchase}=\frac{\text{Initial government purchase}}{\text{Multiplier}}\\ =\frac{400}{5}\\ =80\end{array}$

Therefore, the government purchases must increase by 80 to achieve a level of 480 for achieving an income of 2,400.

Now, substitute the respective values in Equation (2) to solve the new equilibrium income.

$\begin{array}{c}\text{Y}=\text{PE}\\ \text{Y}=0.80\text{Y}+480\\ \left(1-.80\right)\text{Y}=480\\ \text{Y}=\frac{480}{.2}\\ =2,400\end{array}$

Therefore, an increase of government purchases of 80 increases income by 400.

Economics Concept Introduction

Government purchases multiplier: The government purchases multiplier indicates the ratio of change in equilibrium income to the change in government spending.

 (e)

To determine

Explain the level of government taxes needed to achieve an income of 2,400.

Explanation of Solution

As described in part (b), the original equilibrium income is 2,000 to achieve an income level of 2,400; there is a need for an increase of 400 $\left(2,400-2,000\right)$ over the original level of income.

Now the value of tax multiplier can be calculated as follows:

$\begin{array}{c}\text{Tax mulitiplier}=\frac{-\text{MPC}}{\left(1-\text{MPC}\right)}\\ =\frac{-0.8}{\left(1-0.8\right)}\\ =-4\end{array}$

Therefore, the value of tax multiplier is $-4$ .

Now, the value of government taxes can be calculated as follows:

$\begin{array}{c}\text{Government purchase}=\frac{\text{Initial government purchase}}{\text{Multiplier}}\\ =\frac{400}{-4}\\ =-100\end{array}$

Therefore, the government taxes must decrease by 100 to achieve an income of 2,400, which means the government taxes must be 300.

Economics Concept Introduction

Tax multiplier: The tax multiplier indicates that the amount income changes in response to a $1 change in taxes.