# ECN312 Consumer Theory and Supply and Demand

ECN312: Intermediate Microeconomic TheoryProblem Set 1: Consumer Theory and Supply andDemand1. The demand function for a good is:Qd = a ? bpand the supply function is:Qs = c + epwhere a, b, c, and e are positive constants.(a) Calculate the current market equilibrium price (p) and quantity (Q).(b) Illustrate this equilibrium on a well annotated graph.(c) Illustrate graphically how and why the equilibrium would change if the price of acomplementary good decreased. (Please do this on a graph of its own)(d) Illustrate graphically how and why the equilibrium would change if the price offactor inputs increased. (Please do this on a graph of its own)(e) How would you expect a to change (if at all), given an overall decrease in consumerwealth (or income). Illustrate graphically any change in the equilibrium price andquantity. (Please do this in a graph of its own)2. Given a utility function, U (x1 , x2 ) = x1 + Ax?1 x?2 + x2 . What is the marginal rate ofsubstitution?3. Bruce has the quasi log-linear utility function,U (x1 , x2 ) = x1 + 2 ln(x2 ). Bruce has an income of $100 and faces prices p1 = p2 = 20.(a) What is the marginal rate of substitution for this utility function?(b) Solve for Bruceâs optimal bundle.1(c) Suppose Bruceâs income falls to $20. What will happen to his optimal bundle? Isthe M RS = M RT at the new optimal bundle?4. Dilbert spends his entire income of $50 a month on hours of internet access (x1 ) andother goods (x2 ). Draw his budget constraint for each of the following situations.Please use a separate graph for each part and label all axes and intercepts.(a) The price of all other goods is $1 and the price of internet access is $5 per hour.(b) The prices are the same as in part (a), but Dilbertâs grandmother sends him anextra $20 per month.(c) Dilbertâs income is once again $50.00. However, the internet service providercharges the following rates:â¢ First four hours: $5.00 per hourâ¢ Next six hours: $2.50 per hourâ¢ All hours thereafter: $1.00 per hour5. A consumerâs utility is defined by the function11u(x1 , x2 ) = x13 x22Assume the prices of x1 and x2 are respectively defined by p1 and p2 , and the consumerhas m dollars in disposable income.11(a) What type of utlity function is u(x1 , x2 ) = x13 x22(b) Using Lagrangeâs method of constrained optimization, find the consumerâs individual demand curve for both goods, x?1 (p1 , p2, w) and x?2 (p1 , p2, w). Show yourwork!6. Tess derives all her utility from eating ham and cheese sandwiches, but she will eatonly sandwiches made with two pieces of ham and one piece of cheese (she is a verypicky eater!)(a) Draw Tessâ indifference map over ham and cheese, and illustrate her utility maximizing choice of the two products.(b) Find Tessâ demand for ham and cheese as a function of the price of cheese (pc ) andthe price of ham (ph ). (Hint: You will not be able to use constrained optimizationto solve for demand in this case. Normally, the optimal ratio of two goods thatmaximizes the consumerâs utility is determined where the slope of the budgetconstraint is equal to the slope of the indifference curve. In this case, the slope ofthe indifference curve is undefined. Instead, use the description of the consumerâspreferences to infer what ratio of ham and cheese will maximize her utility.)27. Suppose that the average household in a state consumes 500 gallons of gasoline a year.A 10 cent tax per gallon of gasoline is introduced, and couples are given a $50 annualtax rebate per household. Will the household be better or worse off after the newprogram is introduced? Use a well annoted graph in your explanation. (Hint: Thinkif this in terms of a Slutsky pivot.)