Answer all parts of the question

Graph a typical indifference curve for the following utility functions and determine whether they obey the assumption of diminishing MRS. Pick a utility such as 10. a. U(x, )) = 3x + y b. U(x, )) = xy c. U(x, J ) = Vx + d. U(x,J) = Vx2 - yz e. U(x, y) = x2/3 yl f. U(x, )) = logx + log y where log is the natural log (In) 2. Suppose a consumer's preferences for two goods can be represented by the Cobb- Douglas utility function U(x, )) = Ax J', where A, a, and Bare positive constants. a. What is MRS ,? b. Is MRS., diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve? c. On a graph with x on the horizontal axis and y on the vertical axis, draw a typical indifference curve. Indicate on your graph whether the indifference curve will intersect either or both axes. 3. Ch 3, Problem 3.6 For the following sets of goods draw two indifference curves, U, and U2, with U2 > Up- Draw each graph placing the amount of the first good on the horizontal axis. a. Hot dogs and chili (the consumer likes both and has a diminishing marginal rate of substitution of hot dogs for chili) b. Sugar and Sweet'N Low (the consumer likes both and will accept an ounce of Sweet'N Low or an ounce of sugar with equal satisfaction) . Peanut butter and jelly (the consumer likes exactly 2 ounces of peanut butter for every ounce of jelly) d. Nuts (which the consumer neither likes nor dislikes) and ice cream (which the consumer likes) Econ 3070 Prof. Barham 4. Repeat Problem 2 for the quasi-linear utility function U(x, y) = 2 x + y. Chapter 4 Problems: 5. Julie has preferences for food, F, and clothing, C, are derived by a utility function U(F,C)=FC. Food costs $1 a unit and clothing costs $2 a unit. Julie has $12 to spend on food and clothing. a. Give the equation for Julie's budget line. If food is on the vertical axis, what is the slope of the budget line? b. Graph Julie's budget line. Place food on the vertical axis and clothing on the horizontal axis. c. On the same graph, draw an indifference curves that is tangent to his budget line. d. Julie is a utility maximize, write the objective function. e. Write down the full optimization problem with the objective function and the constraint. d. Using calculus and algebra, find the basket of food and clothing that maximizes Julie's utility (ie. solve the maximization problem you wrote down in e) (Assume julie can purchase fractional amounts of both goods.) Do not solve this using the5. Julie has preferences for food, F, and clothing, C, are derived by a utility function U(F,C)=FC. Food costs $1 a unit and clothing costs $2 a unit. Julie has $12 to spend on food and clothing. a. Give the equation for Julie's budget line. If food is on the vertical axis, what is the slope of the budget line? b. Graph Julie's budget line. Place food on the vertical axis and clothing on the horizontal axis. c. On the same graph, draw an indifference curves that is tangent to his budget line. d. Julie is a utility maximize, write the objective function. e. Write down the full optimization problem with the objective function and the constraint. d. Using calculus and algebra, find the basket of food and clothing that maximizes Julie's utility (ie. solve the maximization problem you wrote down in e) (Assume Julie can purchase fractional amounts of both goods.) Do not solve this using the tangency condition taught in the book solve it like in class. 6. Each day Peter, who is in the third grade, eats lunch at school. He only likes liver (1) and onions (N), and these provide him a utility of U(L, N) = In(LN). Liver costs $4.00 per serving, onions cost $2.00 per serving, and Peter's mother gives him $8.00 to spend on lunch. a. Give the equation for Peter's budget line. If liver is on the vertical axis, what is the slope of the budget line? b. Graph Peter's budget line. Place the number of liver servings on the vertical axis and the number of onion servings on the horizontal axis. c. On the same graph, draw several of Peter's indifference curves, including one that is tangent to his budget line. 2 Econ 3070 Prof. Barham d. Peter is a utility maximize, write the objective function. e. Write down the full optimization problem with the objective function and the constraint. d. Using calculus and algebra, find the basket of liver and onions that maximizes Peter's utility. (Assume Peter can purchase fractional amounts of both goods.) Mark this basket on your graph. Do not solve this using the tangency condition taught in the book solve it like in class. 7. Ch 4, Problem 4.6 Jane likes hamburgers (H) and milkshakes (M). Her indifference curves are bowed in and toward the origin and do not intersect the axes. The price of a milkshake is $1 and the price of a hamburger is $3. She is spending all her income at the basket she is currently consuming, and her marginal rate of substitution of hamburgers for milkshakes is 2. Is she at an optimum? If so, show why. If not, should she buy fewer hamburgers and more milkshakes, or the reverse? 8. Ch 4, Problem 4.13 Toni likes to purchase round trips between the cities of Pulmonia and Castoria and other goods out of her income of $10,000. Fortunately, Pulmonian Airways provides air service and has a frequent-flyer program. A round trip between the two cities normally costs $500, but any customer who makes more than 10 trips a year gets to make additional trips during the year for only $200 per round trip. a. On a graph with round trips on the horizontal axis and "other goods" on the vertical axis, draw Toni's budget line. (Hint: This problem demonstrates that a budget line need not always be a straight line.)5. (30 points) A consumer has the utility function over goods r and y, u(x, y) = 12ry'. Let the price of good r be given by pr, let the price of good y be given by py, and let income be given by M. (a) What is the slope of the consumer's indifference curve at the consump- tion bundle. (1,1) ? (b) Derive the consumer's generalized demand function for goods r and y. (c) If we have px = 2, py = 2, and M = 24, compute the utility marimizing consumption bundle.Question: Changes in the prices of key commodities can have a significant impact on a company's bottom line. According to a September 27, 2007, article in the Wall Street Journal, "Now, with oil, gas and electricity prices soaring, companies are beginning to realize that saving energy can translate into dramatically lower cogs." Another Wall Street Journal article, dated September 9, 2007, states, Higher grain prices are taking an increasing financial toll." Energy Is an input into virtually all types of production; corn is an input into the production of beef, chicken, high-fructose corn syrup, and ethanol (the gasoline substitute fuel). a. Explain how the cost of energy can be both a fixed cost and a variable cost for a company. b. Suppose energy is a fixed cost and energy prices rise. What happens to the company's average total cost curve? What happens to its marginal cost curve? Illustrate your answer with a diagram. c. Explain why the cost of corn is a variable cost but not a fixed cost for an ethanol producer. d. When the cost of corn goes up, what happens to the average total cost curve of an ethanol producer? What happens to its marginal cost curve? Illustrate your answer with a diagram.d. When variable costs increase, so do average total costs and marginal costs. Both curves will shift upward. In panel {b} of the accompanying diagram, the movement of the average total cost curve is illustrated by the shiftfrorn its initial position, ATE, to its new position, ATE-'2. The movement of the marginal cost curve is illustrated by the shift from its initial position, M0,, to its new position, MG? Question: Marty's Frozen Yogurt is a small shop that sells cups of frozen yogurt in a university town. Many owns three frozen-yogurt machines. His other inputs are refrigerators, frozen-yogurt mix, cups, sprinkle toppings, and, of course, workers. He estimates that his daily production function when he varies the number of workers employed {and at the same time, of course, yogurt mix, cups, and so on] is as shown in the accompanying table. Quantity of labor {workers} Quantity of frozen yogurt [cups] a. What are the xed inputs and variable inputs in the production of cups of frozen yogurt? b. Draw the total product curve. Put the quantity of labor on the horizontal axis and the quantity of frozen yogurt on the vertical axis. c. What is the marginal product of the first worker? The second worker? The third worker? Why does marginal product decline as the number of workers increases