1. Use Excel's Solver to find the solution to the following problems: a. In Sheet 1,maxQ1,Q2Q1(505Q1)+Q2(10010Q2)(90+20(Q1+Q2) b. In Sheet 2, maxK,L5K+8L s.t. 100=K0.5L0.5 (Hint: use positive initial values for K and L ) 2. A firm owns a private mine. In one time period, profit of the firm is =ln(1+) where qt is extraction level at time t. When the firm extracts from the mine, the resource stock, Rt, declines. The firm knows the initial resource stock is R0=1. The firm plans to extract for 10 time periods. Any remaining resource stock has no value to the firm, i.e. 10=0. The firm has a discount rate of 5% and wishes to maximize aggregate profit. a. Write the firm's problem and all constraints. b. Set up the problem in Sheet 3. Copy everything from Sheet 3 to Sheet 4 and use solver to find the optimal extraction path over time and the maximum profit of the firm. Graph extraction over time and stock over time in separate figures in Sheet 4. 3. Person A owns a lake of fish. The person wishes to maximize aggregate profit over 10 time periods. Any fish harvested at time t,Yt, is sold at a price p=10. Total cost is 2cYt2 where c=1. The discount rate is 5%. The fish grows according to the following function: F(Xt)=rXt(1Xt/K) where r=0.5,K=1 and the initial stock is X0=0.2. The person does not value any remaining stock, i.e. 10=0. One cannot have negative harvest, negative stock and one cannot harvest more than the available stock. a. Write the firm's problem and all constraints. b. Set up the problem in Sheet 5 . Use an initial guess of 0.05 for harvest at each time. Use solver to derive optimal harvest over time that maximizes aggregate profit in Sheet6. 1. Use Excel's Solver to find the solution to the following problems: a. In Sheet 1,maxQ1,Q2Q1(505Q1)+Q2(10010Q2)(90+20(Q1+Q2) b. In Sheet 2, maxK,L5K+8L s.t. 100=K0.5L0.5 (Hint: use positive initial values for K and L ) 2. A firm owns a private mine. In one time period, profit of the firm is =ln(1+) where qt is extraction level at time t. When the firm extracts from the mine, the resource stock, Rt, declines. The firm knows the initial resource stock is R0=1. The firm plans to extract for 10 time periods. Any remaining resource stock has no value to the firm, i.e. 10=0. The firm has a discount rate of 5% and wishes to maximize aggregate profit. a. Write the firm's problem and all constraints. b. Set up the problem in Sheet 3. Copy everything from Sheet 3 to Sheet 4 and use solver to find the optimal extraction path over time and the maximum profit of the firm. Graph extraction over time and stock over time in separate figures in Sheet 4. 3. Person A owns a lake of fish. The person wishes to maximize aggregate profit over 10 time periods. Any fish harvested at time t,Yt, is sold at a price p=10. Total cost is 2cYt2 where c=1. The discount rate is 5%. The fish grows according to the following function: F(Xt)=rXt(1Xt/K) where r=0.5,K=1 and the initial stock is X0=0.2. The person does not value any remaining stock, i.e. 10=0. One cannot have negative harvest, negative stock and one cannot harvest more than the available stock. a. Write the firm's problem and all constraints. b. Set up the problem in Sheet 5 . Use an initial guess of 0.05 for harvest at each time. Use solver to derive optimal harvest over time that maximizes aggregate profit in Sheet6