ECN 134 SOLUTION KEY 2 1 i The firm must invest today the amount x which solves the equation x 1 08 27 1 500 000 i e x 187 780 ii PV of the Smiths o er 115 000 PV of the Joneses o er iii a P V 1000 0 1 1 1 13 150000 112 700 You should choose the Smiths o er 10 000 b P V 1 500 1 1 0 1 c P V 1 2420 1 12 0 1 4 545 20 000 1 2 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 nut trees the price of trees will be bid up or the price of the final product will be bid down through increasing 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 the equilibrium 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 Thus if the trees bear fruit for 120 years the equilibrium price is 0 24 Because we value future payments less than current payments the higher the interest rate or the further into the future the greater the discount Ten times as productive does not mean ten times the value An imperfect analogy might be to diminishing marginal returns since each increase in the tree s productivity comes further o each increase must be worth less and less 2 3 i Use the annuity formula solving for the annual cash flow 1 1 x 0 14 1 1 14 10 10 000 1 Hence x 10 000 0 14 1 1 14 10 1917 1 ii Of the first payment of 1917 1400 are interest and the di erence of 517 goes towards repaying the debt which will thus have been reduced to 9483 Note that this is equal to the future value in year 1 of the remaining 9 annual payments note also that in the first year you have paid back less than 10 of your debt iii After the fifth payment you still need to pay back five installments whose value exactly equals what you owe since you will exactly repay your debt after 10 years the value of the remaining five payments is thus simply 1 1 0 14 1 1 145 1917 1 6581 7 The amount should be greater than one half of ten thousand since in the early years a large fraction 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 and solve for t 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 23 and 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 takes longer Since you pay o less in each period you need to pay o the debt over a longer period and ultimately pay more 3
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