Aggregate Demand and Fiscal Policy in Closed and Open Economies

Posted on Aug 20, 2024 in Economy

Distribution of Aggregate Demand in a Closed Economy (Long Term)

Exercise 1

Consider a closed economy operating at full employment, characterized by the following system of equations:

Based on this information:

1. Find the total saving function of the economy. Calculate the partial derivative of this function with respect to the interest rate and provide an intuition for the results.
2. Find the effect on the interest rate of increases in public spending. Express your result in terms of model parameters.
3. Explain intuitively the effects on the interest rate of a standalone increase in investment.

Guideline

Exercise 1

1)

The higher the interest rate, the cheaper future consumption (savings) becomes, so that current consumption is postponed. This is achieved today by increasing household savings (and thus total savings).

2)

The economy is in equilibrium when savings equals investment:

Fully differentiating the above expression:

We know that dY = 0 since it depends on the factor endowment (which is constant) and that taxes are also fixed (the problem does not state why these would vary). Given the above, the expression reduces to:

The effect on the interest rate derived from the policy of increasing spending is greater when the sensitivity of investment is higher and the sensitivity of consumer spending is lower.

3)

An increase in investment creates excess investment or a shortage of savings in the economy that is corrected with a higher interest rate. By raising the interest rate, investment falls and household saving rises, bringing the economy back into equilibrium. The adjustment occurs until these variables equalize.

In this case, the crowding-out effect is total and occurs in consumption.

Open Economy in the Short Term (Mundell-Fleming)

Consider a small open economy of the Mundell-Fleming type with a flexible exchange rate, described by the following set of equations:

Based on this information:

1. Find the interest rate, the level of output, and the exchange rate that create equilibrium in the economy.
2. Suppose the government decrees an increase in tax revenue of $10. Given this scenario, what is the multiplier of fiscal policy? Calculate how much the nominal exchange rate must vary to achieve the multiplier you mentioned in the previous step.
3. If the authority conducts monetary policy, calculate the multiplier of monetary policy and explain the change in the trade balance caused by this situation.
4. If the authority carries out a protectionist policy that involves reducing imports by $10, calculate the change in equilibrium income and the real exchange rate resulting from this policy.
5. Resolve issues 3 and 4, but now considering an economy with a fixed exchange rate.

Guideline

1)

Since it is an economy with perfect capital mobility, the domestic interest rate will coincide with the external rate.

Thus:

The LM function is determined as follows:

The newly found LM function depends only on known parameters (note that it is inelastic to the real exchange rate). With this, the equilibrium output of the economy is determined. Then:

The equilibrium real exchange rate clears the goods market:

2)

In a Mundell-Fleming open economy, expansionary fiscal policy does not increase income. What changes is the composition of aggregate demand, which now has more government spending at the expense of lower net exports.

Conceptually:

Step 1

Increased spending represents an excess of aggregate demand. If firms do not plan to reduce inventories, this leads to more production. The increase in production raises the demand for real balances, generating an excess demand for money and hence a rise in interest rates and a fall in bond prices.

Step 2

The rise in the local interest rate makes it temporarily higher than the foreign interest rate. This arbitrage opportunity leads to excess demand for local currency (since the instruments that pay the highest rates are denominated in local currency, not dollars), which is corrected by an appreciation of the local currency. The appreciation of the local currency leads to a fall in net exports, reducing aggregate demand, output, and interest rates to their initial levels (remember that i = i* always holds, and in this case, i* never changed).

The only result of the policy is a complete crowding-out effect: what goes up in spending falls in exports.

Mathematically, we are asked to determine the appreciation of the local currency that”reverse” the increase in aggregate demand.

We know from previous steps:

Conceptually, we said that dY = 0. If only the real exchange rate (appreciating) and spending vary, the income differential is determined as follows:

Then:

3)

In this case, monetary policy is fully effective. We only need the LM curve to find the new equilibrium:

The trade balance is:

As is the change in the value of the dollar:

We know that in equilibrium:

Then:

If we know that:

then:

Then:

Thus, the trade balance is modified by:

4)

When protectionist policies are implemented, the income effects on autonomous imports are zero. Since autonomous imports fall by 10, the resulting exchange rate change is identical to that obtained when spending increased by 10.

5)

CHECK THE PAPER

Part One: Exercises on the Real Sector of a Closed Economy

1. Final Goods Market and Analysis of Budget Multipliers

Consider an economy in which the real interest rate is given at a level””.

As usual, there is a consumption function, an investment function, and government spending, where the sum of these three demands generates aggregate expenditure.

For simplicity, assume that consumption depends only on disposable income, investment depends only on the real interest rate, and government spending is autonomous. Finally, consider that the role of taxation in the economy consists only of lump-sum taxes.

a) Show that if the government increases spending and taxes by the same amount, the equilibrium output level varies in the same direction as spending.

b) If now, the tax structure considers a positive tax rate (t), find an expression to show how much government spending and lump-sum taxes should vary simultaneously so that the budget remains constant under a policy of increasing government spending. (20 points)

Hint: If you have a function:

The total differential of the function is given by:

Standard Exercise 1

We know that in this case, equilibrium output is given by:

The fiscal budget is given by (for simplicity, we assume zero transfers):

For statement

Taking the differential of equilibrium output:

Since , we have:

=

The last equation confirms the statement’s proposition.

b)

In this case, the fiscal budget is defined as (again, we assume zero transfers):

Where Y* denotes equilibrium output.

The differential of the budget is:

, where we have used the fact that dt = 0 (the statement asks for a change in lump-sum taxes and spending, not the tax rate).

Replacing the equilibrium income differential in the budget differential expression, we obtain:

Since the aim is to keep the budget constant, we set the differential to zero:

Developing the previous expression:

The last expression defines the relationship between the change in lump-sum taxes and spending that keeps the budget constant.

Economy of Short-Term Market of Goods and Multipliers

The government of a closed economy has the major objective of reaching full employment. It has two alternatives:

1. The first is to”induce transfers” which means implementing a transfer function of the form:
2. The second is to”not induce transfers” which means providing benefits to the private sector regardless of their income.

Based on this information and considering that the interest rate is fixed (i.e., we do not include the monetary sector analysis), answer:

a) If the government wants to implement an expansionary fiscal policy (increasing spending), explain which of the two alternatives can more quickly reach full employment. Provide both an economic and a mathematical proof. (Consider the interest rate as exogenous, assume the tax structure is based only on income, and assume that investment is sensitive to the interest rate.)

b) Graph and find the equation of the IS curve for each type of transfer structure, conceptually explaining the differences between them.

Pattern

To answer this question, we first find the equilibrium income with and without induced transfers and then calculate the spending multipliers for each alternative. The alternative associated with a higher multiplier is preferred by the government (this is obvious because the greater the leverage, the more effective the policy of increasing spending will be in quickly reaching full employment).

The alternative with the greater spending multiplier is the one without induced transfers. Consider the following analysis:

With induced transfers:

The transfer rate reduces the impact of changes in income on disposable income. Disposable income does not increase by the full amount of the increase in income but by a factor of (1-tr), where tr is the transfer rate. In this case, as income increases, total transfers decrease, absorbing some of the increase in disposable income (and therefore consumption and aggregate demand, which increases by a fraction b(1-tr)).

Without induced transfers:

Disposable income increases by the full amount of the increase in income (since there is no transfer rate). When income increases, transfers remain constant, so there is no dampening effect. (Therefore, aggregate demand rises by a fraction”” (the marginal propensity to consume) of the increase in disposable income.)

Since the government wants to achieve full employment by increasing its spending (and thus income), the policy will be more effective in achieving full employment in the case without induced transfers, where there is no dampening effect.

Mathematically:

The spending multipliers for each case are:

This confirms the economic intuition.

b)

First, we derive the respective IS curves:

Notation:

• Variables with (*) refer to the case without the transfer rate.
• Variables with (**) refer to the case with the transfer rate.

The slopes of the IS curves are:

Since:

It follows that:

Graphically:

The difference between the IS curves lies in their slopes. Note that for a given interest rate, the IS curve that implies a higher equilibrium income in the goods market is always IS*, i.e., the one without induced transfers.

Brief Conceptual

a) Explain the speculative and transactions motives for holding money according to Keynes.

b) Through a rigorous analysis, explain why the money supply (M₁) has a direct relationship with the market interest rate. How does your answer change if the banking system has 100 percent reserves?

Guideline 3

a) Speculative Motive

This motive reflects the inverse relationship between interest rates and the demand for money. Individuals hold a portfolio of assets consisting of money (the most liquid asset) and bonds. The price of a bond is the present discounted value of the future cash flows it will yield, which conceptually corresponds to the amount you should invest in an alternative asset to replicate the cash flow of the bond. Given this, the bond price has an inverse relationship with interest rates.

Thus, an increase (decrease) in the interest rate will decrease (increase) the price of a bond, leading to an increase (decrease) in the demand for bonds. As wealth is constant, individuals will buy more bonds by selling money. Thus, there is an inverse relationship between the interest rate and the demand for money.

Transactions Motive

This motive reflects the direct relationship between the demand for money and the level of income (GDP) in a country. An increase in GDP implies an increase in the quantity of goods and services available for sale, which will generate an increase in transactions. Since money is the universal medium of exchange, an increase in transactions also implies an increase in the demand for money, generating the direct relationship mentioned initially.

b)

Let’s analyze the case of a rise in market interest rates.

A rise in market interest rates increases the bank spread, which incentivizes banks to lend out more of their funds. Banks can do this by reducing their excess reserves (reserves held above the required level). This decrease in excess reserves reduces total reserves, which increases the money multiplier and the amount of money created through lending, thus increasing the money supply.

This explains the direct relationship mentioned.

If the banking system has 100 percent reserves, banks cannot reduce their excess reserves because they are required to hold all deposits as reserves. In this case, a rise in the market interest rate would not affect the money supply because banks would not be able to increase lending. Therefore, the money supply would be independent of the market interest rate.