Question

Show why a \(\$ 10\) billion reduction in government purchases of goods and services will have a larger effect on real GDP than a \(\$ 10\) billion reduction in government transfers by completing the accompanying table for an economy with a marginal propensity to consume \((M P C)\) of \(0.6 .\) The first and second rows of the table are filled in for you: on the left side of the table, in the first row, the \(\$ 10\) billion reduction in government purchases decreasesa. When government purchases decrease by \(\$ 10\) billion, what is the sum of the changes in real GDP after the 10 rounds? b. When the government reduces transfers by \(\$ 10\) billion, what is the sum of the changes in real GDP after the 10 rounds? c. Using the formula for the multiplier for changes in government purchases and for changes in transfers, calculate the total change in real GDP due to the \(\$ 10\) billion decrease in government purchases and the \(\$ 10\) billion reduction in transfers. What explains the difference? (Hint: The multiplier for government purchases of goods and services is \(1 /(1-M P C)\). But since each \(\$ 1\) change in government transfers only leads to an initial change in real GDP of \(M P C \times \$ 1\), the multiplier for government transfers is $M P C /(1-M P C) .)$

Short answer

Compare the differences and explain the results. Answer: The effect of a decrease in government purchases and a decrease in government transfers on real GDP depends on the multipliers for changes in those variables. The multiplier for changes in government purchases is greater than the multiplier for government transfers, which means the decrease in government purchases has a larger negative effect on real GDP compared to the decrease in government transfers.

Step-by-step solution

Find the change in real GDP for each round

First, we need to calculate the change in real GDP for each round following the decrease in government purchases and transfers. The change in real GDP for a given round can be calculated as follows: Change in real GDP = initial change * MPC^round Note that the initial change in government purchases is given as \(\$ 10\) billion, and the initial change in government transfers is given as \(-\$ 10\) billion.

Sum the changes in real GDP for the given 10 rounds

Next, for both scenarios, we will sum up the changes in real GDP after the 10 rounds to find the total change in real GDP at the end of the 10 rounds. Total change in real GDP = Sum of changes in real GDP for each round For government purchases: Sum of real GDP changes = \(-10 \times (1 + 0.6 + 0.6^2 + 0.6^3 + ... + 0.6^{10})\) For government transfers: Sum of real GDP changes = \(-10 \times 0.6 \times (1 + 0.6 + 0.6^2 + 0.6^3 + ... + 0.6^{10})\)

Calculate the multipliers for changes in government purchases and transfers

Now, we will calculate the multipliers for changes in government purchases and transfers using the provided formulas. Multiplier for changes in government purchases: \(1 /(1-MPC) = 1/(1-0.6)\) Multiplier for changes in government transfers: \(MPC /(1-MPC) = 0.6/(1-0.6)\)

Calculate the total change in real GDP using the multipliers

Next, we will use the multipliers to calculate the total change in real GDP due to the \(\$ 10\) billion decrease in government purchases and the \(\$ 10\) billion reduction in transfers. Total change in real GDP (government purchases) = initial change * multiplier = \(-10\) billion * multiplier for changes in government purchases Total change in real GDP (government transfers) = initial change * multiplier = \(-10\) billion * 0.6 * multiplier for changes in government transfers

Compare the differences and explain the results

Finally, we will compare the total changes in real GDP for both scenarios and explain the results. The multiplier for changes in government purchases is greater than the multiplier for government transfers, which means the decrease in government purchases has a larger effect on real GDP.