Consumer Behavior and Utility Maximization: Indifference Curves and Budget Constraints, Summaries of Law

Download Consumer Behavior and Utility Maximization: Indifference Curves and Budget Constraints and more Summaries Law in PDF only on Docsity! -2- Short Questions 1. Consider an individual whose preferences for goods X and Y are given by the following set of indifference curves. a. Is the utility function of this person Cobb-Douglas, Leontieff, or Linear Utility? It is a Linear Utility function. b. Suppose the budget constraint of this individual is steeper than the indifference curves. Will this person consume only good X, only good Y, or some of both goods? He will consume only good Y. -3- 2. One budget constraint (the thick straight line) and a few indifference curves (thin curved lines) are plotted in the figure below. Both goods, X and Y are desirable to the consumer, i.e., utility is higher for indifference curves that are further from the origin. a. In the above figure, clearly indicate and label the utility maximizing choice of consumption levels of X and Y. b. Which of the following expressions are (or is) true at the utility maximizing choice of consumption? Circle the ones that are known to be correct on the basis of the above figure. This is because the optimum is a corner solution in which he only consumes X , i.e., X gives more utility per dollar than Y. This is because the slope of the indifference curve is equal to the MRS=ratio of marginal utilities. A relatively steep indifference curve (steeper than a slope of -1) means that it takes more than one unit of Y to make up for the loss of one unit of X, and thus good X gives the consumer more satisfaction that good Y (at the margin). This is because the slope of the budget constraint is equal to the ratio of prices. In particular, a relatively flat budget constraint (i.e., one with slope flatter than -1) implies that the good in the Y axis is more expensive that the good in the X axis. -6- b. Suppose that this consumer wants to spend 100 dollars on wine, and that French wine costs 10 dollars per bottle while California wine costs 5 dollars per bottle. Draw this consumer’s budget constraint for consumption of wine with the quantity of California on the vertical axis and French wine on the horizontal axis. Label all points and slopes. The consumer’s budget constraint is given by Solving for California wine, in order to facilitate the drawing of the budget constraint, we get: The plot of the budget constraint is given by the figure below: -7- c. How much California and French wine will this consumer buy ? Show your answer diagrammatically using the budget constraint and an indifference curve. The consumer will maximize his utility subject to his budget constraint. Graphically, this means getting on the furthest out indifference curve that touches his budget set. The solution can been seen graphically in the figure below, where the indifference curves are in thin lines and the budget constraint in thicker line. This consumer will consume 20 bottles of California wine and no French wine. -8- 2. A consumer’s utility from consumption during his working years and consumption during his retirement years is given by: where and indicate his consumption in thousands of dollars during the work years and retirement years, respectively. The consumer earns a total of 600 thousand dollars during his work years. He can put any amount of this money in a bank that earns NO interest, and withdraw this money for consumption during his retirement years. a. If he saves 100 in the bank during his working years, what will CW and CR be ? What if he saves 200 in the bank ? In general, what will his consumption be if he saves S dollars ? If he saves 100 during his working years his consumption during his working years will be 600-100=500 thousand. His consumption in his retirement years will be 100 thousand. If he saves 200 thousand his consumption during his working years will be 600-200=400 thousand. His consumption during his retirement years will be 200 thousand. In general if he saves S thousand during his working years CW will be 600-S and CR will be S. b. What will his utility be if he saves S dollars during his working years ? Plugging the consumption in terms of the amount of savings into his utility function yields: -11- 3. A consumer’s utility from the consumption of French and California wine is given by: a. Plot the indifference curve of this consumer that corresponds to, say, utility level 5 with quantity of California wine on the vertical axis and French wine on the horizontal axis. Label all points and slopes. The indifference curve for utility level equal to 5 is given by the formula: If we would like to plot this indifference curve with California wine on the vertical axis we should solve the above equation for C. The graph of this indifference curve is shown below: -12- b. Suppose that this consumer wants to spend 100 dollars on wine, and that French wine costs 10 dollars per bottle while California wine costs 5 dollars per bottle. Draw this consumer’s budget constraint for consumption of wine with the quantity of California on the vertical axis and French wine on the horizontal axis. Label all points and slopes. The consumer’s budget constraint is given by Solving for California wine, in order to facilitate the drawing of the budget constraint, we get: The plot of the budget constraint is given by the figure below: -13- c. How much California and French wine will this consumer buy ? Show your answer diagrammatically using the budget constraint and an indifference curve. The consumer will maximize his utility subject to his budget constraint. Graphically, this means getting on the furthest out indifference curve that touches his budget set. The solution can been seen graphically in the figure below, where the indifference curves are in thin lines and the budget constraint in thicker line. This consumer will consume 20 bottles of California wine and no French wine. d. How high must the price of California wine go before the consumer is indifferent between California and French wines ? For the consumer to be indifferent between California and French wine the psychic tradeoff between the two, which is one for one, must be equal to their monetary tradeoff. This means that the price of California wine must be equal to the price French wine (10 dollars per bottle) for the consumer to be indifferent between them. -16- d. Algebraically derive how many hot dogs and mustard she will buy. Label this consumption bundle on the graph. Georgia’s budget constraint is: Since she always consumes then in equal proportions, the number of hot dogs will be equal to the ounces of mustard. Substituting in for the number of hot dogs we have: She will buy 5 hot dogs and 5 ounces of mustard. This consumption bundle is at the corner of the indifference curve shown in the graph above, where it touches on her budget constraint. 5. Georgia always eats hot dogs together with 2 oz. of mustard. Each hot dog eaten in this way provides 10 units of utility. If she has an excess of one of the ingredients than these proportions, she will just throw the extra amount away. a. Draw a couple of Georgia’s indifference curves for consumption of hot dogs and mustard, with the number of hot dogs on the vertical axis. Label your graph carefully. Let’s first draw the indifference curve that corresponds to the consumption of 2 ounces of mustard and a single hot dog. Adding more mustard gives no additional utility, so the indifference curve must be flat to the right of the (2,1) bundle. Similarly, since getting more hot dogs while keeping the level of mustard the same yields no additional utility, the indifference curve must be vertical above the (2,1) bundle. Repeating the process for the (4,2) bundle gives the figure below. -17- b. Suppose she has $30 to spend on hot dogs and mustard, and that hot dogs cost $1.5 each, while mustard costs $0.20 per ounce. Draw her budget line, with the number of hot dogs indicated on the vertical axis. Label all intercepts on your graph. If Georgia spends all her money on hot dogs she will be able to buy 20 of them. If she spends all her money on mustard she will be able to buy 150 ounces. Therefore, the budget constraint is given by the figure below. -18- c. Algebraically derive how many hot dogs and mustard she will buy. [Quantities do not have to be integers.] Label this consumption bundle on a graph that combines the budget constraint and the indifference curve that corresponds to the optimal consumption bundle. Georgia will always buy twice as many ounces of mustard as she will buy hot dogs. Therefore, The budget constraint of Georgia is given by Using the fact that she will consume twice as much mustard as hot dogs we get It immediately follows that she will consume Aron OH FTN NN 10 5 67 8 9 4 -22- b. Draw two indifference curves for this consumer with L on the vertical axis and C on the horizontal axis. For this person, lattes and cappuccinos are perfect substitutes and the indifference curves would be straight lines. However, even though lattes and cappuccinos are perfect substitutes, he likes latter more than the former (they give him more utility per unit of consumption). In fact, 2 cappuccinos give him as much satisfaction as 3 lattes. Therefore, one indifference curve would be a straight line that connects the consumption bundle of C=2 and L=0 with the consumption bundle of C=0 and L=3. Moreover, any indifference curve would be parallel to this one. A couple are drawn below. Note: Mathematically, one could draw an indifference curve by fixing utility to some number, then solving the utility function for L and plotting the resulting equation. -23- c. How many lattes and how many cappucinos will this consumer purchase? The consumer has linear utility and is facing a linear budget constraint. Therefore, he will either consume only cappuccinos or he will consumer only lattes. Since the prices are the same and cappuccinos are more desirable, he will only drink cappuccinos. He can afford 10 of them. He will buy no lattes. Alternative solution: The solution of the utility maximization problem of a consumer with linear utility facing a linear budget constraint is a corner solution. The consumer will only purchase the product that yields the higher (marginal) utility per dollar. Since while he will only buy only cappucinos. d. Suppose the person visits Italy, and cappuccinos there cost $4, while lattes still cost only $3. Will the person switch to lattes? Why or why not? He will still drink only cappuccinos. The reason is that marginal utility of cappuccinos per dollar spent is equal to 3/4, which is still higher than the marginal utility per dollar of lattes (which remains equal to 2/3). 8. A consumer’s utility of hot dogs and burgers is given by . The price of a hot dog is 2 and the price of a burger is . The consumer has 12 to spend on hot dogs and burgers, that is, his (relevant) income is 12. a. Suppose . Draw this consumer’s budget constraint in a figure with H on the vertical axis and B on the horizontal axis