Our friendly cotton grower from the previous chapter still uses water and land for which production obeys a Cobb‐Douglas production function:Q = 0.203 L 0.12 W 0.88Where L = ha of land; W = water in...

Our friendly cotton grower from the previous chapter still uses water and land for which production obeys a Cobb‐Douglas production function:Q = 0.203 L 0.12 W 0.88Where L = ha of land; W = water in 1000 cubic meters applied; Q = tons of cotton produced. This time, we knowinput costs. The price of water is $100 per thousand cubic meters. The price of land is $200 per ha.1. Short run: Derive her short run total, average, and marginal cost functions if she produces cotton with landfixed for a 10 ha farm. Hint: with only one variable input, there is no optimization required to determine the optimized ratio of thetwo inputs. Each unit of the variable input produces one possible level of output as land is fixed at 10. Fromthat, you can find the relation between total input cost and total output. Then you should be able to calculatetotal, average, and marginal cost.2. Show that the absolute marginal product of both L and W depends only on the ratio of L to W. Why is this animportant fact to show? 3. Long Run: Next, suppose land is no longer fixed at 10 ha. Our grower can deliberating adjust acreage inproduction. Both land and water are now variable inputs, so she operates in the long run. She wishes tominimize the long run total cost of supplying 28 tons of cotton. How much land and how much water should she use to minimize the costs of supplying that amount ofcotton? What is her minimized total cost when she is successful? Big principle: Show that the ratio of themarginal product of land to the marginal product of water must equal the ratio of the price of land to the priceof water when supplying those 28 tons?4. Short run again: To change our story again, suppose total water supply, W, is limited to 150 units (1000 cubicmeters), but that 28 tons of cotton must still be produced. How much land should be used, now viewing landas a variable input? How much does this new water constraint add to our grower’s total costs of producingthose 28 tons compared to your long run optimized solution where both inputs are variable in 3 above? 5. What additional information does our grower need to discover her profit‐maximizing level of output?