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 condition. d. Determine the firm's marginal cost function: [pic]. e. According to microeconomic theory, perfectly competitive firms will maximize profits by producing at the quantity where price equals marginal cost. Show that the theory holds in this example.
6. Consider a monopolist's linear demand function P = 12 - 2Q, where p is the price of the good and q is its quantity. The monopolist's total cost function is [pic]. Determine the level of output at which the monopolist should produce in order to maximize her profits. What is the optimal price that corresponds with this quantity?
7. A domestic auto producer is facing intense competition in the United States market from Asian auto imports. The executive officer decides that one way to counter this competition is by producing at that quantity at which total costs are minimized. The firm's cost structure can be illustrated by the function [pic]. a. Calculate the cost-minimizing quantity. b. Assume that the total revenue function is TR(Q) = 8Q - Q2. Show that the profit-maximizing quantity is different from the cost-minimizing quantity.
8. Assume that an individual's total labor compensation is a function of that person's education level, E, and years of experience, X, in a given profession. This compensation function (C) is C = -2E2 + 78E - 2X2 + 66X - 2EX. Find the stationary point that maximizes the compensation function to determine what combination of education and experience will give the highest level of this individual's labor compensation.
9. The sales (S) of a journal is a function of the number of pages devoted to important stuff about economics (E) and the number of pages devoted to everything else (U): [pic]. What should be the number of pages devoted to economics articles and the number of pages devoted to articles on other topics in order to maximize sales?
10. Consider a firm's Cobb-Douglas production function: [pic]. The firm is a price-taker in the input markets. Labor is paid an hourly wage of w = 12 and the price of capital is r = 6. The firm sells its output at a price of P = 4 per unit. Maximize the profit function [pic] to determine the optimum level of each input the firm should use.

B. Equality-Constrained Optimization
1. You own a farm that produces two types of wheat, type x and type z. You are under contract to deliver 12 tons of wheat, in any combination of your choosing, to the bread manufacturer operating in your farm district. Using the substitution method, find the combination of crops that minimize the cost of fulfilling this contract given your cost function C = 3x2 - 4xz + 9z2 - 8z + 36. Confirm the second-order condition for minimum.
2. In the theory of individual labor supply, it is assumed that an individual derives utility from both income, which is earned through work, and leisure. Income (I) is determined by the number of hours worked (L) multiplied by the hourly wage rate (w), so that I = wL. Assume that in each day, a total number of 24 hours is available for either work (L) or leisure (R) such that L + R = 24 and that the hourly wage is $4. The individual's labor-leisure utility function is U(I,R) = 4IR2. Using the substitution method, determine how the individual will balance labor and leisure in order to maximize utility.
3. For your dessert, you have decided to have some combination of cappuccino and pudding that minimizes the amount of calories you consume, while still providing you with a certain level of gustatory pleasure. Each ounce of cappuccino has 50 calories, and each ounce of pudding has 100 calories. The utility from consuming cappuccino (C) and pudding (P) is measured in "utils" and the "production of utils" from consuming cappuccino and pudding is represented by the function [pic]. You decide to consume 15 utils of dessert in a way that minimizes the number of calories. Set this up as a constrained minimization problem. Using the substitution method, find the optimal number of ounces of cappuccino and pudding. Use the second-order condition to show that this solution represents a minimum rather than a maximum number of calories.
4. You allocate $24 per week for the purchase of cookies and apples at your school's cafeteria. Your utility from eating cookies and apples is given by [pic]. Assume that cookies cost $1 each and apples cost $0.50. a. Set up the Lagrangian function associated with this problem. Solve for the optimal proportion of cookies to apples. Given your budget constraint, how many cookies and apples will you buy each week? b. Verify the second-order condition for maximum. c. Solve for the Lagrangian multiplier. What is interpretation of this parameter? What happens to this parameter if you allocate twice as much per week t the purchase of cookies and apples? How do you interpret this change?
5. The utility you derive from exercises (X) and watching movies (M) is described by the function [pic]. Currently you have 4 hours each day you can devote to either watching movies or exercising. Set up the Lagrangian function for finding the optimal amount of time spent at each activity. Solve for the optimal amounts of time exercising and watching movies. Confirm that the bordered hessian is negative-semi-definite. Also, provide an interpretation of the Lagrange multiplier in this problem.
6. Suppose it is late in the semester and you have two exams left. You must decide how to allocate your working time during the study period. After eating, sleeping, exercising and maintaining some human contact, you have 12 hours each day in which to study for your exams. You have figured out that your grade point average, GPA, from your two courses, Mathematical Methods and Literary Methods, takes the form [pic], where m is the number of hours per day spent studying for Mathematical Methods and L is the number of hours per day spent studying for Literary Methods. What is the optimal number of hours per day spent studying for each course? If you follow this strategy what will your GPA be?
7. The Cobb-Douglas production function takes the form [pic], where Q is the amount of output, K is the amount of capital input, and L is the amount of labor input. Suppose that a firm faces a linear cost-of-inputs function C = wL + rK, where C is the cost of inputs, w is the wage rate and r is the rental rate on capital. a. Set up a Lagrangian function reflecting the constrained optimization problem of obtaining the most output given a budget C to spend on inputs. Solve this for the optimal levels of capital and labor. b. Set up a Lagrangian function reflecting the constrained optimization problem of spending the least amount on inputs given that the level of output must equal to the amount Q. Solve this for the optimal levels of capital and labor.
8. Determine whether each function is homogeneous and, if so, of what degree. a. [pic] b. [pic]
9. Consider the production function [pic]. a. Determine whether the production function is homogeneous. If so, of what degree? b. Take the partial derivatives of the function and show that they are homogeneous of degree k-1 (given that the production function is homogeneous of degree k). c. Now, using the Euler's Theorem, show that [pic].
10. A proportional increase in all inputs in a production function increases the scale of production. If there are constant returns to scale, then output will increase equi-proportionally to the increase in all inputs. If there are increasing returns to scale, an increase in all inputs will lead to a more than proportionate increase in output. If there are decreasing returns to scale, then output will increase less than proportionately with an increase in all inputs. Consider the production function [pic]. a. Using Euler's Theorem, prove that this production function exhibits constant returns to scale when (+( = 1. (Hint: This production function is homogeneous of degree k = (+(.)
What condition on (+( is necessary for increasing returns to scale? For decreasing returns to scale? b. Consider the production function y = f(x1,x2) = x1 x2 defined over the domain x1 > 0 and x2 > 0. Also, consider the functions g(y) = ln(y) and j(y) = y2.
Is f(x1,x2) a homogeneous function? If so, of what degree?
Is g(y) a homothetic function? Is g(y) a homogeneous function in the arguments x1 and x2? If so, what is its degree?
Is j(y) a homothetic function? Is j(y) a homogeneous function in the arguments x1 and x2? If so, what is its degree?
Consider the following Cobb-Douglas production function, which is homogeneous of degree 1 in capital and labor: [pic]. The value of the output (Q) includes the payment made to the labor, i.e., the wages paid to the labor (wL), which is equal to [pic] in a competitive labor market. Also, the value of the output includes the payment made to the capital suppliers (rK), which is equal to [pic]. Show that the sum of the total factor payments (wL + rK) equals the value of the output, i.e., wL + rK = Q, such that wL + rK = ( Q + (1-() Q, where ( = 0.6.

Answers to Practice Problem Set 3: Optimization

A. Unconstrained Optimization
1. Answers: a. (x = 4-1 = 3. When x = 1, y = 2, and when x = 4, y = 32. Therefore, (y = 32-2 = 30. [pic]=10. b. u = 1, v = 4, ( = 0.3; ( f(u) + (1-() f(v) = 23, whereas f[( u + (1-() v] = 19.22; therefore, the function is strictly convex. c. f'(x) = 4x; therefore, f '(u) = 4. f(v) = 32, whereas f(u) + f '(u) (v - u) = 14; therefore, strictly convex.
2. Answers: a. [pic] b. strictly convex c. strictly convex
3. strictly concave
4. Extreme values of Y are 13.72 (minimum) and 3.96 (maximum).
5. Answers: a. ((Q) = - Q3 + 4.5Q2 - 6Q + 7 b. Q = 2 c. ("(Q) = - 3 at Q = 2; therefore, maximum profit d. MC(Q) = 3Q2 - 9Q + 18 e. In the above expression for MC, substitute the profit-maximizing value of output (Q=2) and calculate MC. At Q=2, MC = 12, which is, indeed, equal to the market price.
6. Q = 5 and P = 2 for maximum profit
7. Answers: a. Cost-minimizing output is 14.47. b. Profit-maximizing output is 12, which is different from the cost-minimizing output.
8. E = 15 and X = 9
9. E = 70 and U = 30
10. K = 4, L = 2

B. Equality-Constrained Optimization
1. x = 8 and z = 4
2. L = 8 and R = 16
3. [pic]. C = P = 7.21
4. Answers: a. [pic]; A = 16, C = 16 b. [pic], i.e., negative definite; therefore, maximum c. ( = 0.25; this is the marginal utility of income. If the weekly budget is $48, instead of $24, then the maximum utility combination would be A = C = 32 and ( = 0.18 (you will have to solve the problem again with the new income figure). With higher income, marginal utility of income decreases.
5. X = 1.56 and M = 2.43. Determinant of bordered hessian = 0.27. The Lagrange multiplier is the marginal utility of time.
6. M = 4 and L = 8. Determinant of bordered hessian = 0.017. GPA = 4
7. Answers: a. [pic] b. [pic]
8. Answers: a. [pic]; therefore, homogeneous of degree 1 b. [pic]; therefore, not homogeneous
9. Answers: a. homogeneous of degree 7/12 (i.e., k=1/12) b. [pic]; f1(x1,x2) is homogeneous of degree k-1, i.e., -5/12; similarly, you can take the partial derivative of the production function with respect to x2 and show that f2(x1,x2) is homogeneous of degree -5/12. c. According to the Euler's Theorem, [pic]. Since both f1(.) and f2(.) are homogeneous of degree k-1, you can substitute them with [pic] and [pic], respectively, in the right hand side of the above equation. Then, multiply both sides of the equation by sk-1 to arrive at what you are supposed to show in this question.
10. Answers: We know that the production function is homogeneous of degree ((+(). Therefore, according to the Euler's Theorem, [pic]. When ((+() = 1, [pic]. If in the left hand side of this equation, we double both K and L, i.e., use 2K and 2L, you can show that the right hand side adds up to 2q, i.e., output also doubles; in other words, constant returns to scale.
If ((+() is not equal to 1, then[pic]. Doubling of K and L in the right hand side adds up to 2((+()q. If ((+() > 1, the output more than doubles, or the increasing returns to scale. If ((+() < 1, the output is less than double, or the decreasing returns to scale.
Answers:
homogeneous of degree 2 homothetic; not homogeneous in x1 and x2 homothetic; homogeneous of degree 4 in x1 and x2
Take the partial derivatives and substitute them on the right hand side of wL + rK = [pic] + [pic] and show that the right hand side adds up to 0.6 Q + 0.4 Q, i.e., ( Q + (1-() Q.