StudySpace: Economics in Development, 6th Edition

Worked Example: Inflation and Real Returns on Financial Assets

In an inflationary environment, nominal interest rates overstate the real cost of borrowing and the real return on lending or holding financial assets. Suppose that the (nominal) interest rate on a one-year loan is 15 percent (so i = 0.15). A bank that lends $1,000 at this interest rate will be repaid an amount of $1,000 (1 + i) = $1,150 one year later. If the expected inflation rate over the next year is 10 percent (so p = 0.10), then the expected real value of this repayment is $1,150/(1 + p) = $1,045.45. The lender expects a real return—after adjusting for inflation—of $45.45 on the $1,000 loan. This represents a real interest rate of just over 4.54 percent. Algebraically, $1,045.45 = $1,000(1 + r), so r = 0.0454 = 4.54 percent.

Suppose, however, that the actual inflation rate is 25 percent (p = 0.25). Then the $1,150 repayment of principal plus interest has an inflation-adjusted value of $1,150/(1 + p) = $920. The lender earns a real return of –$80 on the $1,000 loan, so the real interest rate is –8 percent. Formally, $920 = $1,000(1 + r), so r = –0.08 = –8 percent. In this case, borrowing is an obvious bargain, but saving is a losing proposition. If savers foresee that inflation will exceed the nominal interest rate, then savings will be diverted to physical assets like jewelry and gold, which offer a hedge against inflation, or other financial assets held abroad (capital flight). Financial intermediation is repressed. This is an example of shallow financial policy.

In these numerical examples, the real interest rate is found by applying equation 13–4 from the textbook:

r = (1 + i)/(1 + p) – 1.

A simpler and more familiar relationship is given by equation 13–5 of the text: r = i – p. This is actually an approximation, not an exact formula. The approximation is satisfactory when p is fairly small, but less accurate for larger values. In the examples above, with 10 percent inflation the approximation gives a real interest rate of 15 – 10 = 5 percent, compared to the true value 4.54 percent. With 25 percent inflation the approximation gives a real interest rate of 15 – 25 = –10 percent, compared to the true value of –8 percent. If the inflation rate were 100 percent (as in Zambia in 1991), then the approximation gives a result that is double the true value of –42.5 percent—which is bad enough!

It is worse still if interest earnings are subject to income tax. In this case the interest reward accruing to savers is diminished by the tax obligation. Instead of i, one would insert i(1 – t) into the formulas. Suppose that the tax rate is 30 percent, so t = 0.30. With an interest rate of 15 percent, the after-tax nominal return is i(1 – t) = 0.105 = 10.5 percent. If the inflation rate is 25 percent, then the real interest rate is not –8 percent but

r = [1 + i(1 – t)]/(1 + p) – 1 = – 0.116 = –11.6%.

Money is a special case. Narrow-money balances are held in the form of cash and demand deposits, which earn zero interest in most countries. So i = 0 in the formula (and income taxes are irrelevant here). With 10 percent inflation, the real return on money balances is r = 1.00/(1 + p) – 1 = –0.09, or –9 percent. With 25 percent inflation, the real return is –20 percent. The decline in the purchasing power of money constitutes a genuine loss of command over real resources for money holders. The resources can be utilized by the government, which prints money to make its various payments. So this is an inflation tax. If the inflation tax is too onerous, due to runaway inflation, then the private sector will shun holding domestic money balances. Monetization, with all of its attendant efficiency advantages, suffers.

Exercises

1. It is your turn to calculate the real interest rate and the inflation tax.

a. Table 13–1 shows the nominal interest rates and inflation rates for five countries in mid-1995. The inflation rate is based on price data for the latest 12 months. Assume that savers used these inflation figures as the basis for formulating their expectations of the inflation rate for the near future.

Click here for Table 13-1

(i) Calculate the expected real interest rate in each of the five countries and put your answer in column 3.

(ii) For comparison, also calculate the approximation to the real interest rate, using the simple formula r = i – p; put your answer in column 4. Before going on, notice the pattern of differences between the figures in columns 3 and 4.

(iii) Finally, calculate the inflation-tax rate on money balances; put your answer in column 5. (Hint: If you earn a real return on money balances of –X percent, your money holdings are subject to a tax of X percent.)

b. Who benefits when the inflation rate exceeds the nominal interest rate on a financial asset: borrowers (issuers of financial assets) or lenders (holders of the financial assets)? Briefly explain.

c. Consider how taxes affect the interest rate analysis. In Zambia in mid1992, the controlled interest rate on bank deposits was 48 percent and the inflation rate was 112 percent.

(i) The real interest rate on bank deposits was percent.

(ii) Interest earned on bank deposits was subject to 10 percent income tax. Hence, the 48 percent interest rate produced a nominal after-tax yield for depositors of percent. (For help, see the Worked Example.)

(iii) The real after-tax interest rate on deposits was percent. (iv) How did this affect the supply of deposit funds to the banking system?

d. Continuing the example of Zambia, while the inflation rate was 112 percent the controlled interest rate on bank loans was 68 percent.

(i) The real interest rate on bank loans was percent.

(ii) Most borrowers getting bank loans were corporations facing a 35 percent tax on income. Since interest costs are deducted from the amount of income subject to tax, 35 percent of the interest cost is effectively offset by a smaller tax bill. Therefore, the 68 percent loan rate represented a net-of-tax nominal interest rate of percent.

(iii) The corresponding real after-tax interest rate on bank loans was percent. (iv) How did this affect the demand for loan funds from the banking system?

e. Think about your answers concerning the supply of deposit funds and the demand for loan funds in Zambia in 1992.

(i) How did the interest rate controls affect the overall balance between supply and demand for finance?

(ii) What does this imply about the way the banks allocate credit?

(iii) What does this imply about the market for informal finance in Zambia?

2. This exercise analyzes the efficiency gains from financial intermediation and the efficiency costs of shallow finance. Each family in Kapital saves $100 per year. Initially there are no banks or other financial intermediaries, so household savings are either invested in hens (productive capital) or jewelry (unproductive assets). The inflation rate is zero, and we will ignore tax considerations. Hens cost $1 each; the rate of return on marginal additions to the flock varies inversely with the number of birds. The rate of return on jewelry is zero.

FIGURE 13–1

a. The top panel of Figure 13–1 refers to family A. The bottom panel refers to family B. Line RA shows the relationship between the investment in hens and the rate of return earned on the marginal hen, for family A. Line RB shows the corresponding relationship for family B. The rate of return on jewelry is always zero.

(i) Members of family A are outstanding hen farmers. They invest all $100 of their savings and earn a rate of return on the marginal hen of percent (see point A0). Only a lack of funds prevents them from investing in even more hens.

(ii) Members of family B are terrible hen farmers. If they have more than hens, the marginal rate of return becomes negative (see point B0).

(iii) Family B invests only $ in productive assets (hens); the remaining $ of their savings go for jewelry.

b. In aggregate, the two families save $200. But the savings are not being allocated to investment as efficiently as possible. Explain why.

c. Banks now appear. In addition to hens and jewelry, families can hold their savings in the form of deposits that pay 10 percent interest. Also, loans are now available at an interest rate of 15 percent.

(i) Curve RB shows that family B earns a rate of return that exceeds the deposit rate (10 percent) when they invest up to $ of their savings in hens.

(ii) Now that interest-bearing bank deposits are available, how will Family B allocate its savings to earn the highest rate of return? Explain.

(iii) So banks receive deposit funds of $ from family B.

(iv) Examine curve RA. Why would family A decide not to hold any of its $100 in savings in bank deposits?

(v) Family A wants to obtain a bank loan at 15 percent interest to purchase hens, in addition to the hens that they can buy with their $100 of self-finance. Explain why.

(vi) How much do they want to borrow? (Hint: Borrowing is profitable as long as the marginal rate of return on hens exceeds 15 percent.)

d. In aggregate the two families still save $200. In what respect has the appearance of banks enhanced efficiency in the allocation of the savings?

e. Suppose nominal interest rates on loans and deposits remain unchanged, but the inflation rate rises to 100 percent per year.

(i) How will this inflation affect the willingness of family B to hold savings in the form of bank deposits?

(ii) Would it be profitable for someone to seek a bank loan to use for purchasing jewelry, which is completely unproductive? Explain.

(iii) In general, how does inflation affect the efficiency of savings allocations? Be sure to take into account the productivity of the various investments financed by bank loans, as well as the volume of intermediation.

3. This exercise explores the relationship between money supply growth and inflation.

a.

(i) In Harganya the 1996 level of GDP is Y = $1,000. The money supply is M = $300, so the ratio of money to GDP is M/Y = .

(ii) Real GDP is expected to grow by 6 percent during the following year. On the basis of past experience, the income elasticity of the demand for money is E = 1.5. So if real GDP grows by g_Y = 6 percent, the money supply can expand by g_M = percent without causing inflationary pressure. (Hint: g_Y ×E. Do you see why?)

(iii) As GDP grows 6 percent, from $1,000 to $1,060, an increase in the money supply to $ would maintain price stability.

(iv) The money supply can grow more rapidly than GDP without stimulating inflation because E (the income elasticity of demand for money) exceeds unity. Why is the value of E in low-income countries likely to exceed unity?

b. Suppose that the government of Harganya is content to hold the inflation rate to 10 percent.

(i) Simplifying a bit, assume that prices rise by 1 percent for every 1 percent that g_M exceeds the noninflationary value found above. Then 10 percent inflation is compatible with g_M = percent.

(ii) So the policy makers can permit M to increase from $300 to $ without the inflation rate exceeding 10 percent.

(iii) If the money supply were to grow by g_M = 25 percent, then inflation would spurt to a rate of percent.

4. The example from Exercise 3 continues here, with a focus on the relationships between government deficits, the growth of the money supply, and the availability of private credit. Expansion of the money supply is equal to the net inflow of international reserves (since this creates new deposits in local banks) plus the expansion of domestic credit to either the government or the private sector. Repeating equation 13–9 from the textbook,

ΔM = ΔDC + ΔIR.

a. Recall the 1996 conditions in Harganya: Y = $1,000, M = $300, and E = 1.5.

(i) In Exercise 3 you calculated that a 10 percent inflation rate would result if the money supply increased to M = $ . Call this the money supply target.

(ii) The government expects that the change in international reserves will be zero. In this case the money supply target can be achieved by permitting domestic credit to expand by ΔDC = $ .

(iii) This amount of domestic credit creation may consist of the government borrowing $50 and the private-sector borrowing $ .

(iv) Or it may consist of the government borrowing $20 and the private-sector borrowing $ .

(v) Or any other combination of government plus private-sector borrowing from the banking system adding up to $ .

b. Government revenues in Harganya are expected to fall short of expenditures by an amount equal to 5 percent of GDP during 1996.

(i) In absolute terms this means that the budget deficit is projected to be BD = $ .

(ii) This deficit is financed by borrowing from the central bank, that is, by domestic credit creation of an amount equal to BD. Given the credit creation needed to finance the government deficit, the money supply target can be achieved if credit expansion to the private sector is limited to $ .

(iii) Outstanding bank credit to the private sector previously totaled $250. Now it can increase to no more than $ .

(iv) In percentage terms, credit to the private sector can expand by only percent.

(v) With an inflation rate of 10 percent, this means that real bank credit to the private sector will by percent.

(vi) Is this consistent with the projected 6 percent rate of growth of real GDP? Explain briefly.

c.

(i) To avoid a real decline, credit to the private sector must expand in nominal value by no less than percent, to $ .

(ii) If this increase in nominal credit to the private sector were allowed, then the money supply target can be achieved only by holding the government budget deficit to no more than $ , or percent of GDP.

d. The government astonishes everyone by cutting the deficit enough to achieve the warranted value of ΔDC without squeezing the real volume of credit that is available to the private sector. Unhappily, an unexpected inflow of international reserves occurs, with the result that ΔIR = $24. (Recall that the government anticipated ΔIR = $0.) How will this unexpected event affect ΔM? How will it affect the rate of inflation? Give specific numerical answers.

5. The textbook says that interest rates have little effect on saving-consumption decisions in developing countries, although there is a clear link between real interest rates and the extent to which savings are held in the form of liquid financial assets. Let’s look at some empirical evidence.

Click here for Table13-2

FIGURE 13-2

a. Chapter 10 showed that GDS = Sg + Sp, where GDS is gross domestic savings, Sg is government savings, and Sp is private savings. Table 13–2 provides data on GDS and Sg as percentages of GDP, along with data on real interest rates (r), for 14 developing countries. This sample has been selected to cover a wide range of per capita incomes.

(i) From the data on GDS/Y and Sg/Y in columns 4 and 5, calculate Sp as a percentage of GDP and fill in the blanks in column 6.

(ii) In Figure 13–2, plot the 14 points representing the values for r and Sd for each country.

(iii) Does the graph reveal a significant relationship between real interest rates and savings rates for this sample of countries? Explain briefly.

(iv) You probably noticed that the private savings rate in Jordan is negative 23 percent of GDP and gross domestic savings also is negative. What does this mean? How is it possible?

b. Table 13–3 shows data on real interest rates (r) and growth of the ratio of broad money to GNP g(M2/Y) in Indonesia during the first decade following Indonesia’s successful stabilization program. The textbook explains that the ratio M2/Y often is used as an indicator of financial deepening.

Click here for Table 13-3

FIGURE 13–3

(i) In Figure 13–3, plot the ten points representing the values of r and g(M2/Y ).

(ii) Does the graph reveal a significant relationship between the real interest rate and financial deepening in Indonesia during this period? Explain briefly.

 

6. This exercise shows how changes in international reserves lead to changes in the money supply under a fixed exchange rate system. Consider Sombrero, a country where the peso is the domestic currency. For simplicity, assume that the dollar is the only foreign exchange unit. The government is committed to maintaining a fixed exchange rate of 20 pesos per dollar. Think of this as the price of the dollar in terms of the Sombrero peso.

a. Sombrero’s international transactions generate a supply of dollars (for example, from exports) that exceeds the demand for dollars (for example, for imports) by $100 million.

(i) With supply in excess of demand, free market forces would cause the price of the dollar to .

(ii) The government can prevent this change in the exchange rate by eliminating the excess supply of dollars. This can be accomplished by purchasing $ million in the foreign exchange market, to add to Sombrero’s stock of international reserves.

(iii) The government pays for the dollars with .

(iv) As a result of the government’s action to maintain the fixed exchange rate, the supply of domestic money in circulation increases by million pesos.

b. Suppose instead that international transactions create a demand for dollars that exceeds the supply of dollars by $50 million.

(i) With demand in excess of supply, free-market forces would cause the price of the dollar to .

(ii) The government can prevent this change in the exchange rate by $50 million in the foreign exchange market.

(iii) As a result of the government’s action to maintain the fixed exchange rate, Sombrero’s stock of international reserves will by $50 million. (iv) And the supply of domestic money in circulation will by million pesos.

c. Finally, suppose that the foreign exchange market is in equilibrium at the exchange rate of 50 pesos per dollar. But domestic inflation is zero, while the foreign inflation rate averages 15 percent.

(i) As prices in foreign markets rise, dollar earnings from Sombrero’s exports will .

(ii) Because foreign goods become increasingly expensive, the people of Sombrero will their purchases of imports.

(iii) Consequently, in Sombrero’s foreign exchange market the supply of dollars increasingly will the demand for dollars.

(iv) To prevent market forces from causing a change in the exchange rate, the government of Sombrero must dollars in the foreign exchange market.

(v) This government action will cause Sombrero’s stock of international reserves to .

(vi) And it will cause the domestic money supply to .

(vii) This change in the domestic money supply will tend to cause Sombrero’s inflation rate to .

 

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