(Table: Exchange Rates and Prices) Suppose a computer costs $500 in the United States. With the price of the computer given in the local currency, the South African rand is _______.
A) undervalued by 12.0%
B) overvalued by 3.0%
C) undervalued by 12.5%
D) undervalued by 1.25%

Question

(Table: Exchange Rates and Prices) Suppose a computer costs $500 in the United States. With the price of the computer given in the local currency, the South African rand is _______. 
A) undervalued by 12.0% 
B) overvalued by 3.0% 
C) undervalued by 12.5% 
D) undervalued by 1.25%

Quizplus · Accepted Answer

To find the answer, we need to use the formula of relative PPP (purchasing power parity):

(local price of computer) / (exchange rate) = (price of computer in the US)

Let X be the local price of the computer in South Africa, and let E be the exchange rate of rands to dollars. We know that X = 500 for the same computer in the US.

So we have:

X / E = 500

Solving for E, we get:

E = X / 500

Now suppose that the local price of the computer in South Africa is ZAR Y. Then we have:

X = Y / 100

Substituting into the previous equation, we get:

E = (Y / 100) / 500 = Y / 50000

So the exchange rate is simply the local price of the computer in rands divided by 50000.

To determine whether the rand is undervalued or overvalued, we compare the actual exchange rate to the rate implied by PPP. The implied exchange rate is simply the ratio of the prices of the computer in South Africa and the US, but with the prices expressed in the same currency (dollars):

Implied exchange rate = (local price of computer) / (US price of computer) = X / 500

So the rand is undervalued if the actual exchange rate is lower than the implied exchange rate, and overvalued if the actual exchange rate is higher than the implied exchange rate.

To see this, suppose the actual exchange rate is ZAR R per dollar. Then one dollar can buy R rand. So the implied local price of the computer in South Africa, expressed in dollars, is:

500 / R

If the actual local price of the computer in South Africa, also expressed in dollars, is higher than this, then the rand is overvalued, since it takes more rand than implied by PPP to buy the same dollar-denominated good. Conversely, if the actual local price of the computer in South Africa, expressed in dollars, is lower than this, then the rand is undervalued, since it takes fewer rand than implied by PPP to buy the same dollar-denominated good.

Substituting the above formulas for X and E, we get:

Implied exchange rate = X / 500 = (Y / 100) / 500 = Y / 50000

Actual exchange rate = ZAR R per dollar = 1 / E = 50000 / Y

So the rand is undervalued if:

(50000 / Y) < (Y / 50000)

Multiplying both sides by 50000, we get:

25,000,000 < Y^2

Since Y is the local price of the computer in rands, it must be positive. Therefore, we can take the square root of both sides:

5,000 < Y

In other words, if the local price of the computer in South Africa is less than ZAR 5,000, then the rand is undervalued relative to the dollar, since it takes fewer rands than implied by PPP to buy the same computer. If the local price is more than ZAR 5,000, then the rand is overvalued.

In this problem, we are given that the local price of the computer in South Africa is ZAR 6,000. Since this is more than ZAR 5,000, the rand is overvalued. To find the percentage of overvaluation, we calculate the difference between the actual exchange rate and the implied exchange rate as a percentage of the latter:

(6000 / 50000) - (500 / 500) / (500 / 500) = 0.12 = 12%

So the correct answer is C: undervalued by 12.5%.

(Table: Exchange Rates and Prices) Suppose a Computer Costs $500