ECN 134SOLUTION KEY #21. i) The firm must invest today the amount x which solves the equationx × (1.08)27=1, 500, 000,i.e. x = $187, 780 .ii) PV of the Smiths’ offer: $ 115,000.PV of the Joneses offer:11.13150000 = $112, 700. You should choose the Smiths’ offer.iii) a) PV =10000.1= 10, 000 .b) PV =11.15000.1= 4, 545 .c) PV =11.1224200.1= 20, 000 .12. r=0%:Add the stream of payments: -150 + 8×(-10) + 12×(50 × 0.6) =130 .Perfect competition implies free entry and exit. Therefore, as long as positive profits exist in nuttrees, the price of trees will be bid up (or the price of the final product will be bid down throughincreasing supply). This drives the net present value of the nut tree to zero.-150 + 8×(-10) + 12×(50×p) = 0 has solution p = 23/60 =0.383 .r=4%:Use the annuity formula for each of the three streams:-150 + [-10/0.04×(1-1/1.048)] + 1/1.048×[50×0.6/0.04×(1 - 1/1.0412)] = -11.6 .The equilibrium price has to be higher since the NPV is currently less than zero. To find theequilibrium price, set NPV equal zero and solve for p:-150 + [-10/0.04×(1 - 1/1.048)] + 1/1.048× [50×p/0.04×(1-1/1.0412)] = 0.Then the equilibrium price is:0.63 .-150 + [-10/0.04×(1 - 1/1.048)] + 1/1.048× [50×p/0.04×(1-1/1.04120)] = 0.Th us, if the trees bear fruit for 120 years, the equilibrium price is:0.24 .Because w e value future payments less than current payments - the higher the in terest rate or thefurther into the future, the greater the discount.Ten times as productive does not mean ten times the ”value”. An imperfect analogy might be todiminishing marginal returns: since each increase in the tree’s productivity comes further off,eachincrease must be worth less and less.23. i) Use the annuity formula, solving for the annual cash flow:x·10.14(1-11.1410) =10,000 .Hence x=10,000×(0.14/(1-11.1410))= 1917.1 .ii) Of the first payment of $ 1917, $ 1400 are interest, and the difference of $ 517 goes towardsrepaying the debt, which will thus have been reduced to$ 9483 . Note that this is equal to thefuture value in year 1 of the remaining 9 annual payments; note also that in the first year, you havepaid back less than 10% of your debt.iii) After the fifthpayment,youstillneedtopaybackfive installments whose value exactly equalswhat you owe (since you will exactly repay your debt after 10 years); the value of the remaining fivepayments is thus simply10.14(1-11.145)×1917.1=6581.7 .The amount should be greater than one-half of ten-thousand since in the early years, a largefraction of the annual payment covers interest in addition to the repayment of the principal.iv) Use the annuity formula: 10000 = 1600/0.14 × (1-1/1.14t)andsolvefort:14/16 = 1 - 1/1.14t→1/1.14t= 1/8 and 1.14t=8.Use natural log: t = log(8)/log(1.14) yielding t=15.87 .Alternately, use the double approximation given in class for r=14%:(1+r)t= 2 but our right hand side is eight, so (1+r)t=23and thus (1+r)t/3=2.Then apply the rule: t=(0.7/0.14) × 3 = 15 years (approximately).v) Thus 15.87 × 1600 = 25392. But 10× 1917.135 = 19171.35. So case four tak es longer. Sinceyou pay off less in each period, you need to pay off the debt over a longer period and ultimately
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