Excel Solver version of mine management problem:Excel Solver version of mine management problem: In this problem, we will explore an Excel Solver version of the mine management problem. Initially, the spreadsheet is set up so that the current marginal user cost (MUC) is the one that maximizes the present discounted value of the stream of net social benefits from the mine, under all of the conditions defined in the yellow box. Notice that there are two graphs over to the right side of the spreadsheet. Begin by exploring the set-up in the spreadsheet. a.) We assume a linear demand curve (marginal benefits to society) and that the firm is competitive (a price-taker). For the initial set-up of the spreadsheet, give the formula for price as a function of quantity. Identify the “choke price” where demand goes to zero for any price at or above this level. b.) Do marginal extraction costs depend upon the quantity of the mineral that the company extracts in the current period? Do they depend on how much of the mineral has already been removed from the mine? What is the formula for marginal extraction costs as a function of quantity extracted? c.) What is the current discount rate? How does the discount factor depend on the discount rate? Give the formula. d.) We are assuming that this firm is a competitive price taker, so it merely adjusts its quantity produced (extracted) so that price equals the current-value (undiscounted) overall marginal costs (MEC+MUC) associated with the current period’s extraction. This means that the firm determines extraction by setting the height of the demand curve equal to the overall marginal costs associated with this period’s extraction: In terms of the initial setup, write down the equimarginal condition for profit-maximizing output determination, and solve it for the formula that gives the firm’s profit-maximizing extraction level as a function of the sum of its undiscounted MEC and MUC. Can you find this formula anywhere in the spreadsheet? cost constant e.) How might your answer to part d.) be different if the firm was a monopolist (i.e. not a competitive price taker)? f.) Where in the spreadsheet do we find the present discounted value of the full stream of net benefits from the operation of the mine over this 40-year period? g.) Open Solver and inspect the settings. Which cell in the spreadsheet contains the thing we are trying to maximize (our objective)? Which cell contains the thing we are going to have Solver adjust until our objective is in fact maximized? h.) Note that we should constrain extraction to be non-negative (we aren’t going to be putting the mineral back in the ground). Likewise, if the resource stock goes to zero, it will stay there, and it cannot go negative. How are these constraints implemented in the spreadsheet? i.) What is the appropriate MUC under the default conditions? Now put the model through its paces. After each change, run Solver to re-optimize the initial period MUC before studying the consequences of the change you have made for the dynamically efficient extraction path. Remember that the MUC is a measure of what would be the appropriate royalty charge when a private firm is exploiting the resource, on a year-to-year lease, on public lands. j.) What happens to the efficient MUC at period 0 if new discoveries boost the size of the resource stock to 300? What happens to the “life of the mine”? (Be sure you have re-solved the optimization after you change the settings.) k.) Now put everything back. What happens to the efficient MUC at period 0 (the appropriate royalty rate in the public lands case), and the life of the mine if environmental groups succeed in persuading the government to limit the amount of the mineral that can be extracted to only 50 units, making the effective resource stock only 50 units. l.) Now put everything back again (and continue to do this at the beginning of all future questions as well). What would happen to the efficient MUC/royalty charge at period 0 and if the discount rate was decreased to 0.01? (Be sure to re-optimize.) m.) Fear of takeover by a foreign multinational that will export the mineral to another country may mean that future net benefits must be discounted much more than normal. What if uncertainty about the future drives the discount rate up to 0.50? What happens to the efficient MUC in period 0, and what happens to the life of the mine? n.) What happens if demand grows so that the vertical intercept of the demand curve is at 60, rather than 30 (while the slope stays the same)? What would be the efficient MUC (royalty)? What happens to the life of the mine? o.) What happens if marginal extraction costs are cut in half (say, due to an improvement in extraction technology? What would be the efficient MUC (royalty) and what would be the life of the mine?
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