Intermediate Microeconomics - Problem Set
Please use graph paper or computer for all graphs, and clearly print all written answers.
Problem 1) Consumer Choice Model.
(a) Spano has a monthly income of $1,000 which he spends solely on his two favorite activities eating restaurant meals and drinking beers at bars. Each restaurant meal costs $20 and each beer costs $4. Give the algebraic formula for Spano's budget constraint.
(b) With beer on the x-axis and restaurant meals on the y-axis draw Spano's budget constraint and identify his opportunity set.
(c) Draw a set of indifference curves (3 or 4) with the budget constraint line. Show the point where Spano optimizes his utility. (remember to label)
(d) Suppose the price of a beer increases to $5. Show what happens to Spano's optimal level of consumption of restaurant meals and beer.
(e) Using the graph for problem (d) derive a demand curve.
(f) Suppose the bar owner gives Spano a gift certificate for 10 free beers per month. Draw Spano's new budget constraint line, assuming that the price of restaurant meals is still $20 and beers are $5 each.
Problem 2) Substitution and Income Effects: Using the consumer choice model, start from a consumer optimum position. Label the graph with hamburger on the x-axis and jeans on the y-axis. Show what happens when the price of hamburger is increased.
a) In the first graph, assume that hamburger is a normal good relative to jeans. Indicate the new optimum and show the substitution effect and the income effect of the price increase.
b) In the second graph assume the same price increase, but now assume that hamburger is an inferior good.
c) Observing the difference between the first graph and the second graph, what can you conclude about the slope of the Price Consumption Curve between normal goods and inferior goods.
d) Again observing the difference between the two results what can you conclude about the elasticity of normal goods versus inferior goods.
Problem 3) Consumer Choice and Free Goods: Suppose poor people spend their money on two goods: booze and food. The government doesn't want low-income welfare recipients to spend the money it gives them on booze, so it gives them food stamps to be used only on food. Using the consumer choice model, illustrate below how, paradoxically, giving people food stamps could lead to an increase in booze-buying. (Hint: First draw the budget constraint without food stamps and then with food stamps.)
Problem 4) Consumer Choice Model: Suppose you have a budget of $116 to spend of pretzels and beer. Your utility function is U = 30p + 36b+ 2pb - 0.5p2 - 0.5b2. The price of a small bag of pretzels is $2, and the price of a pint of beer is $4.
a) Using calculus, set up a constrained optimization problem using the Lagrangian multiplier that derives the optimal consumption of pretzels and beer. Find the optimal values.
b) At the optimum, what are the marginal utilities of both pretzels and beer? Does the Utility-Maximizing Rule hold? i.e. MUp/Pp = MUB/PB. (Show your work).
c) Does the utility function indicate a relationship between pizza and beer?
Problem 5) Time Allocation Sergei is a computer programmer with a utility function equal to U = Y + 120l - l2. He contracts with Uber that allows him to set his own work hours; they pay him $50/hour. He has allocated himself a work-hour time budget of a maximum of 250 hours per month, so he may work less than this, depending on how lucrative his work is.
a) Using Lagrangian multiplier techniques, solve for Sergei's optimal level of work hours per month, and optimal level of leisure (out of his 250 possible work hours).
b) Illustrate Sergei's optimal time allocation in a diagram with income per month on the y-axis and leisure hours on the x-axis.
c) Sergei gets a raise. His wage is now $100 per hour. Find Sergei's new optimum mathematically and show the change graphically in your diagram above. Be sure to solve for Sergei's new optimal level of leisure, work hours, and income and illustrate your answer on the graph.
d) Draw and properly label Sergei's labor supply curve for a wage increase from $50 to $100 per hour.
Problem 6) Uncertainty and Expected Utility: Suppose your current wealth is $160,000. A friend of yours has a start-up company and is looking for "angel" investors. He calls you and asks that you invest $70,000. If things go great, the company will go public and your investment in the company will be worth $160,000. If things go ok, then the company will be bought out by a competitor and your investment in the company will be worth $112,500. Finally, if things go bad, you will lose all that you invested. There is a 20 percent chance that the company will go public, a 40 percent chance that the company will be purchased by a competitor and a 40 percent chance that they will file for bankruptcy. Your other option is to use your $70,000 to buy a risk-free government bond that will guarantee 5% interest for one year. ("W" stands for wealth.)
(a) What is the expected return on the "angel" investment?
(b) Which investment will you choose if your utility function is U= W1/2
(c) Which investment will you choose if your utility function is U= W2
(d) In two separate graphs, plot the utility functions in parts (b) & (c). On each graph draw the chord that represents the expected utility of the investment. Explain the significance when the chord is above the arc and vice versa.
Problem 7) Uncertainty and Insurance: You own a house and have a utility function equal to U = (1 - pf)ln(x1)+ pf ln(x2) in which x1 represents your total wealth in a state of nature in which your house is still standing, and x2 represents your total wealth in a state of nature in which your house has burned down. Assume all of your total wealth is related to the value of your house (and its property) in each state of nature. If your house is still standing, it is worth $1,500,000. If it burns down, it is worth $500,000 (land value). The probability of a fire is pf = .02.
a) What is your current level of utility without fire insurance?
b) If the insurance company offers "fair insurance" how much coverage will you buy and what will be your insurance premium? (show work)
c) What is your level of utility with insurance?
d) What is the highest price you would be willing to pay for the insurance policy?