Question

(More difficult) An economy has the following consumption function: $$ C=200+0.8 D I . $$ The government budget is balanced, with government purchases and taxes both fixed at \(\$ 1,000\). Net exports are \$100. Investment is \$600. Find equilibrium GDP. What is the multiplier for this economy? If \(G\) rises by \(\$ 100,\) what happens to \(Y ?\) What happens to \(Y\) if both \(G\) and \(T\) rise by \(\$ 100\) at the same time?

Short answer

Answer: The equilibrium GDP for this economy is $5500. Question: What is the multiplier for this economy? Answer: The multiplier for this economy is 5. Question: What will be the new equilibrium output if government purchases (G) increase by $100? Answer: The new equilibrium output, when G increases by $100, will be $6000. Question: What will be the new equilibrium output if both government purchases (G) and taxes (T) increase by $100? Answer: The new equilibrium output, when both G and T increase by $100, will be $5600.

Step-by-step solution

Understand the Aggregate Demand (AD) function

Aggregate demand in an economy is the sum of consumption (\(C\)), investment (\(I\)), government purchases (\(G\)), and net exports (\(NX\)). In this case, we are given the consumption function \(C = 200 + 0.8DI\), investment \(I = 600\), government purchases \(G = 1000\), and net exports are \(NX = 100\). We will use this information to find the equilibrium GDP.

Calculate the AD function

Using the information provided, the AD function can be written as: $$AD = C + I + G + NX = (200 + 0.8DI) + 600 + 1000 + 100$$

Equilibrium GDP

Equilibrium GDP (\(Y\)) is achieved when aggregate demand equals the income (\(Y\)), thus: $$Y = AD$$ or $$Y = (200 + 0.8DI) + 600 + 1000 + 100$$ Since we are given that the government budget is balanced with taxes equal to \(T\), we can assume \(DI = Y - T\). So, $$Y = (200 + 0.8(Y - 1000)) + 600 + 1000 + 100$$

Solve for equilibrium GDP

Now let's solve for \(Y\): $$Y = 200 + 0.8(Y - 1000) + 600+ 1000 + 100$$ $$Y = 200 + 0.8Y - 800 + 600 + 1000 + 100$$ $$0.2Y = 1100$$ $$Y = 5500$$ So, the equilibrium GDP is \(\$5500\) in this economy.

Calculate the multiplier

The multiplier in this economy is calculated by taking the inverse of \((1 - MPC)\), where \(MPC\) is the marginal propensity to consume. In this case, \(MPC = 0.8\). Therefore, the multiplier can be calculated as follows: $$Multiplier = \frac{1}{1-0.8} = \frac{1}{0.2} = 5$$ So, the multiplier for this economy is 5.

Analyze the effect of an increase in G on equilibrium output

If \(G\) rises by \(\$100\), we need to find the new equilibrium level of output. Using the multiplier effect, we know that the increase in output will be the multiplier times the change in the initial spending, which in this case is the increase in government purchases \(G\). $$Change\,in\,Output = Multiplier \times Change\,in\,G = 5 \times 100 = 500$$ So, when \(G\) rises by \(\$100\), the equilibrium output (\(Y\)) will increase by \(\$500\) to reach a new equilibrium output level of \(\$6000\).

Analyze the effect of an increase in both G and T on equilibrium output

If both \(G\) and \(T\) rise by \(\$100\) at the same time, the overall change in GDP will depend on both the direct initial spending effect and the offsetting tax effect. Since taxes reduce disposable income by \(100\), the change in consumption will be: $$Change\,in\,C = MPC \times Change\,in\,T = 0.8 \times 100 = 80$$ Therefore, the net increase in aggregate demand will be: $$Change\,in\,AD = Change\,in\,G - Change\,in\,C = 100 – 80 = 20$$ Using the multiplier effect once again: $$Change\,in\,Output = Multiplier \times Change\,in\,AD = 5 \times 20 = 100$$ So, when both \(G\) and \(T\) rise by \(\$100\), the equilibrium output (\(Y\)) will increase by \(\$100\) to reach a new equilibrium output level of \(\$5600\).

Key concepts

Consumption Function

The consumption function describes how households plan their consumption based on their income levels. In a simple way, it's a formula to estimate total consumption spending in the economy. It consists of two main parts: autonomous consumption and induced consumption.

• Autonomous consumption refers to the minimum level of consumption that would still occur even if there was no income, represented by a constant (200 in our example).
• Induced consumption depends on disposable income (DI), which is income after taxes. It increases linearly with income, dictated by the marginal propensity to consume (MPC), here 0.8.

The consumption function formula here is given as \[ C = 200 + 0.8DI \]This means that for every extra dollar of disposable income, consumption increases by 80 cents. It emphasizes the role of disposable income in determining how much households are likely to spend, affecting the overall economy's health.

Government Budget

A government budget in an economic context refers to the balance of its revenue (primarily through taxes) and its spending (like government purchases). In this problem, the government has a balanced budget, meaning its revenue equals its expenditures, both set at $1000.

Key impacts of a government budget:

• Balanced Budget: No deficit or surplus, stabilizing economic conditions without further borrowing or unnecessary tax cuts.
• Taxes: They reduce the disposable income available for consumption and can impact economic growth depending on the tax level and structure.
• Government Spending: Directly influences aggregate demand since it’s a component alongside consumption and investment.

Adjustments in government spending or taxes can significantly influence the economic equilibrium, impacting GDP and employment.

Multiplier Effect

The multiplier effect describes how initial changes in spending can lead to larger changes in overall economic output. The basic idea is that one person's spending becomes another's income, causing a chain reaction of spending. In this exercise, the multiplier is determined using the formula \[ \text{Multiplier} = \frac{1}{1 - MPC} = \frac{1}{0.2} = 5 \]

With an MPC of 0.8, the economy has a multiplier of 5, meaning an initial increase in government spending of \(100 leads to a total GDP increase of \)500.

• This effect magnifies any initial increase in spending across the economy.
• Conversely, reductions in spending can similarly amplify negative impacts on GDP.

Essentially, the multiplier effect plays a crucial role in fiscal policy and economic planning by predicting the full impact of changes in spending.

Aggregate Demand

Aggregate demand (AD) represents the total demand for all goods and services in an economy at a given price level and in a given time period. It is a crucial concept in determining the overall economic health and GDP level. The AD in this scenario is calculated as:\[ AD = C + I + G + NX \]Where:

• \(C\): Total consumption as determined by the consumption function.
• \(I\): Investment, fixed at \(600.
• \(G\): Government purchases, set at \)1000.
• \(NX\): Net exports at $100.

The equilibrium GDP is found when the aggregate demand equals the total economic output \( Y \). It forms the basis to solve for GDP where changes in consumer behavior or policy, like increases in government spending or taxation, shift the whole demand curve, affecting the macroeconomic stability.