An economy has a marginal propensity to consume of \(0.5,\) and \(Y^{*},\) income- expenditure equilibrium GDP, equals \(\$ 500\) billion. Given an autonomous increase in planned investment of \(\$ 10\) billion, show the rounds of increased spending that take place by completing the accompanying table. The first and second rows are filled in for you. In the first row, the increase of planned investment spending of \(\$ 10\) billion raises real GDP and \(Y D\) by \(\$ 10\) billion, leading to an increase in consumer spending of \(\$ 5\) billion \((M P C \times\) change in disposable income) in row \(2,\) raising real GDP and \(Y D\) by a further \(\$ 5\) billion. a. What is the total change in real GDP after the 10 rounds? What is the value of the multiplier? What would you expect the total change in \(Y^{*}\) to be based on the multiplier formula? How do your answers to the first and third questions compare? b. Redo the table starting from round 2 , assuming the marginal propensity to consume is \(0.75 .\) What is the total change in real GDP after 10 rounds? What is the value of the multiplier? As the marginal propensity to consume increases, what happens to the value of the multiplier?

Step by step solution

01

Identify the Given Parameters

In this exercise, we are given the marginal propensity to consume (MPC) of 0.5, income-expenditure equilibrium GDP (\(Y^*\)) of \(\$500\) billion, and an autonomous increase in planned investment of \(\$10\) billion.

02

Find the Multiplier Formula

The multiplier formula is given by: \(k = \frac{1}{1-MPC}\)

03

Calculate the Value of the Multiplier

Using the MPC of 0.5, we can find the multiplier: \(k = \frac{1}{1-0.5} = 2\)

04

Calculate the Rounds of Increased Spending

Using the given investment increase of \(\$10\) billion and the multiplier of 2, we can calculate the increase in real GDP after 10 rounds of spending. Round 1: \(10 * (0.5)^0 = 10\) \\ Round 2: \(10 * (0.5)^1 = 5\) \\ Round 3: \(10 * (0.5)^{2} = 2.5\) \\ Round 4: \(10 * (0.5)^{3} = 1.25\) \\ Round 5: \(10 * (0.5)^{4} = 0.625\) \\ Round 6: \(10 * (0.5)^{5} = 0.3125\) \\ Round 7: \(10 * (0.5)^{6} = 0.15625\) \\ Round 8: \(10 * (0.5)^{7} = 0.078125\) \\ Round 9: \(10 * (0.5)^{8} = 0.0390625\) \\ Round 10: \(10 * (0.5)^{9} = 0.01953125\)

05

Calculate Total Change in Real GDP

To find the total change in real GDP after 10 rounds, we sum the amounts of the increased spending calculated in Step 4: \(10 + 5 + 2.5 + 1.25 + 0.625 + 0.3125 + 0.15625 + 0.078125 + 0.0390625 + 0.01953125 = 19.98046875\), approximately \(\$20\) billion. To check the agreement of the calculated values with the multiplier formula, we multiply the increase in investment (\(\$10\) billion) by the multiplier (\(2\)): \(10 * 2 = \)\$20$ billion. The answers to the first and third questions match, as the total change in real GDP is approximately \(\$20\) billion in both cases. PART B

06

Alter the Marginal Propensity to Consume

As per the question, we change the marginal propensity to consume to 0.75.

07

Calculate the New Multiplier

Using the new MPC of 0.75, we calculate the new multiplier: \(k = \frac{1}{1-0.75} = 4\).

08

Calculate the Rounds of Increased Spending

Using the new MPC of 0.75 and the new multiplier of 4, we can proceed to calculate the increase in real GDP after 10 rounds of spending, starting from round 2 (since round 1 remains unchanged). Round 1: (unchanged) \(10 * (0.75)^0 = 10\) \\ Round 2: \(10 * (0.75)^1 = 7.5\) \\ Round 3: \(10 * (0.75)^{2} = 5.625\) \\ Round 4: \(10 * (0.75)^{3} = 4.21875\) \\ Round 5: \(10 * (0.75)^{4} = 3.1640625\) \\ Round 6: \(10 * (0.75)^{5} = 2.373046875\) \\ Round 7: \(10 * (0.75)^{6} = 1.77978515625\) \\ Round 8: \(10 * (0.75)^{7} = 1.3348388671875\) \\ Round 9: \(10 * (0.75)^{8} = 1.001129150390625\) \\ Round 10: \(10 * (0.75)^{9} = 0.7508468627929688\)

09

Calculate the Total Change in Real GDP

Add the amounts from Step 8: \(10 + 7.5 + 5.625 + 4.21875 + 3.1640625 + 2.373046875 + 1.77978515625 + 1.3348388671875 + 1.001129150390625 + 0.7508468627929688 = \$37.747559967\) The total change in real GDP after 10 rounds is approximately \(37.75\) billion.

10

Analyze the Effects of Changing MPC

Comparing the results of part (a) and part (b), we can see that as the marginal propensity to consume increases, the value of the multiplier (\(k\)) also increases (from \(2\) to \(4\)). This increase in the multiplier leads to a greater total change in real GDP after 10 rounds of spending (\(\$20\) billion to \(37.75\) billion).

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