025 11051709 6 at an interest rate of 5year the

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02(5) = $110,517.09. 6. At an interest rate of 5%/year, the present discounted value of the 3 payments is 8000[(1.05) –1 + (1.05) –2 + (1.05) –3 ] = $21,785.98. At an interest rate of 15%/year, the present discounted value of the 3 payments is 8000[(1.15) –1 + (1.15) –2 + (1.15) –3 ] = $18,265.80. As the interest rate increases, the discount factor attached to future net returns increases, and thus the present discounted value of the opportunity falls. 7. In 2010, Exponentia’s GDP/capita is 200,000/800 = 250 zlotniks/person. The growth rate of real per capita GDP (measured in 2010 dollars) is simply the rate of growth of nominal GDP (= 10%/year) minus the rate of inflation (= 5%/year) minus the rate of growth of population (= 2%/year), or 3%/year. Denoting per capita income by X , we have the following relation: X t = 500 = X 0 e g t = 250 e .03 t , from which M ATH M ODULE S olutions to Exercises 9
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it is possible to determine that t = [ ln (500/250)]/.03 = 23.105 years, so that Exponentia will have doubled its GDP per capita in about 2033. 8. There are several ways to calculate the average annual rate of increase in the price of Wedgios relative to other consumer goods. The simplest exact method is the fol- lowing. The relative price of Wedgios P W in 2010 was 12/80 = 0.15, and in 2013 it was 13/96 = .13541667. P W therefore declined over the 3-year period to 0.902778 of its 2010 level. Letting g Pw be the average annual relative price increase, and solving (1 + g Pw ) 3 = 0.902778, we have (1 + g Pw ) = (0.902778) 1/3 = 0.96648167, and so g Pw = 0.96648167 – 1 = –0.03351833, or an average rate of –3.35%/year. The relative price of Wedgios declined over the period by 3.35%/year, on average. [Note that in this example, we did not round off the intermediate results, but saved the rounding off until the final stage. This is a good practice to follow in calculating with growth rates, since when compounding occurs, minor initial discrepancies can be considerably magnified by the end of a problem. To the nearest tenth, for exam- ple, 1/3 = 0.3, but calculate and compare the values for (4/3) 50 and (1.3) 50 .] M9-2 MATH MODULE 9: SOLUTIONS TO EXERCISES
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