Practice Problem Economics: Optimization

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Practice Problem Set 3 for Econ 4808: Optimization

A. Unconstrained Optimization
(Note: In each optimization problem, check the second-order condition.)
1. Consider the function: y = 2 x2. a. Determine the average rate of change of the function in the closed interval [1,4] of the argument. b. Determine whether the function is concave or convex, using geometric test and a specific value of ( = 0.3. c. Do the concavity/convexity test using the derivative conditions.
2. Consider the function: y = 2x. a. Determine the average rate of change of the function in the closed interval [1,4] of the argument. b. Determine whether the function is concave or convex, using geometric test and a specific value of ( = 0.3. c. Do the concavity/convexity test using the derivative conditions.
3. Use the derivative condition to test whether the function y = 8 + 10x - x2 is concave or convex over the domain [0,7].
4. In a cross-country study of the relationship between income per-capita (Y) and pollution (S), Grossman and Krueger estimate a cubic relationship as S = 0.083Y3 - 2.2Y2 + 13.5Y + X, where X represents other factors not linked to income. For this exercise, consider X as a fixed quantity for a country. Identify and characterize the extreme values of this function.
5. In the model of perfect competition, all firms are price-takers since they treat price (P) as a market-determined constant. Assume that P = 12. A firm's total revenue (TR) function is TR(Q) = P ( Q, where Q is the quantity of output of the firm. The total cost (TC) function of the firm is TC(Q) = Q3 - 4.5Q2 + 18Q - 7. a. Determine the firm's profit (() function. (Hint: ( = TR - TC) b. What quantity of output (Q) the firm should produce in order to maximize profits? c. Confirm that this quantity represents maximum profits for the firm by using the second-order…...

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