Aggregate Demand & Open Economy Macroeconomics: A Comprehensive Analysis

Posted on Aug 29, 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:

• Find the total saving function of the economy. Calculate the partial derivative of this function relative to the interest rate and provide an intuition for the results.
• Find the effect on the interest rate of increases in public spending. Express your result in terms of model parameters.
• Explain intuitively the effects on the interest rate of a hike occurring in stand-alone 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 balance of the economy occurs 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. Given the above, the above expression reduces to:

The higher the sensitivities of investment and the lower the consumer spending, the greater the effect on the interest rate derived from the policy of increasing spending.

3)

An increase in investment creates over-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, rebalancing the economy. The adjustment occurs until these variables equalize again.

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:

• Find the interest rate, the level of output, and the exchange rate that create balance in the economy.
• Suppose the government decrees an increase in tax expense of $10. Given this scenario, what is the multiplier of fiscal policy, and how much must the nominal exchange rate vary to achieve the multiplier you mentioned in the previous form?
• If the authority conducts monetary policy, calculate the multiplier of monetary policy and explain the change in the trade balance caused by this situation.
• 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.
• 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, it is resolved with the equilibrium condition of the economy. Then:

The real exchange rate equilibrium is determined by market clearing of goods:

2)

In a Mundell-Fleming open economy, expansionary fiscal policy is ineffective, so income does not change. 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 creates an excess of aggregate demand. To avoid unplanned destocking of inventory, production increases. This increase in output 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 rate. This arbitrage opportunity leads to excess demand for local currency, which is corrected with a depreciation of the domestic currency. This depreciation reduces net exports, lowering aggregate demand, output, and interest rates back to their initial levels (remember that i = i* always holds, and i* never changed).

The only result of the policy is a complete crowding-out effect: What increases in spending is offset by a fall in exports.

Mathematically, we are asked for the depreciation of the domestic currency that brings aggregate demand back to its original level.

We know from previous steps:

Conceptually, we said that dY = 0. If only the real exchange rate (downward) 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:

The change in the value of the dollar is:

We know that in the IS curve:

Then:

If we know that:

Then:

Then:

Thus, the trade balance is modified as follows:

4)

With protectionist policies that reduce autonomous imports, the income effects are zero. Since autonomous imports fall by 10, the result for the exchange rate is identical to that obtained when spending rose by 10.

5)

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Part One: Exercises – 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 “r”.

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

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 in the same amount, the equilibrium output level varies as well as spending.

b) If now, the tax structure considers income taxes with a positive tax rate (t). Find an expression to show how much government spending and autonomous taxes should vary simultaneously so that the budget remains constant under a policy of increasing 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 expression confirms the statement.

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 autonomous 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 autonomous taxes and spending required to keep 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:

• The first is to “induce transfers”, meaning to generate a transfer function of the form
• The second is “not to induce transfers”, meaning to deliver 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 implements an expansionary fiscal policy (increasing spending), explain which of the two alternatives can more quickly reach full employment. Justify your answer with economic reasoning and mathematical proof. (Consider the interest rate as exogenous, assuming the tax structure is based only on income taxes, 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 Transfer Rate: 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 thus consumption and aggregate demand, which increases by a fraction b(1-tr)).

Without Transfer Rate: Disposable income increases by the full amount of the increase in income. When income increases, transfers remain constant, providing no cushioning effect. (Therefore, aggregate demand increases by a fraction “b” (marginal propensity to consume) of the increase in disposable income).

Since the government increases spending (and thus income) to achieve full employment, the policy will be more effective 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:

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 r, 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 Questions

a) Explain the speculative and transactional grounds for Keynes’ demand for money.

b) Through 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 Demand for Money

This refers to 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 discounted value of its future cash flows, which conceptually corresponds to the amount you should invest in an alternative asset to replicate the bond’s cash flows. 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 and a decrease (increase) in the demand for money, as wealth remains constant. Thus, there is an inverse relationship between the interest rate and the demand for money.

Transactional Demand for Money

This refers to the direct relationship between the demand for money and the income level of a country (GDP). An increase in GDP implies an increase in the stock of products available for sale, which will generate an increase in transactions. Since money is the universal medium of exchange, an increase in transactions also involves an increase in the demand for money, generating the direct relationship mentioned initially.

b)

Let’s consider an example of a rise in market interest rates.

A rise in market interest rates increases the bank spread, incentivizing banks to lend more funds, which is only possible by reducing voluntary reserves (not required reserves). This decrease in voluntary reserves reduces total reserves, increasing the money multiplier and thus the money supply.

This explains the direct relationship mentioned.

Variables (*) refer to the case without a transfer rate, while (**) implies a transfer rate.