The Complexico mine is for lease. This mine

has been worked heavily and is approaching depletion. It is becoming increasingly

difficult to extract rich ore. In fact, if xis the amount of gold remaining in the mine at

the beginning of a year, the cost to extract z

(Note that as xdecreases, it becomes more difficult to obtain gold.) It is estimated

that the current amount of gold remaining in the mine is x0 =50,000 ounces. The

price of gold is $400/oz. We are contemplating the purchase of a 10-year lease of the

Complexico mine. The interest rate is 10%. How much is this lease worth?

Years K-values

0 213.81

1 211.45

2 208.17

3 203.58

4 197.13

5 187.96

6 174.79

7 155.47

8 126.28

9 80.00

Consider the Complexico mine and assume a 10% constant

interest rate; also assume the price of gold is constant at $400/oz.

(a) Find the value of the mine (not a 10-year lease) if the current deposit is x0. In particular,

how much is the mine worth initially when x0 =50,000 ounces? [Hint: Consider the

recursive equation for Kk as k.]

(b) For the 10-year lease considered in the text, how much gold remains in the mine at

the end of the lease; and how much is the mine worth at that time?

(c) If the mine were not leased, but instead operated optimally by an owner, what would

the mine be worth after 10 years?