Give one uncertainty problem where the premium is not actuarially fair – warn them that this is so

Spring 2001
Dr. Charles M. North

Answer each of the following questions in the test booklet provided. SHOW ALL WORK. Only answers showing work will receive any credit, partial or full. If you get stuck on a particular question and time is running out, carefully describe the steps you would take in order to solve the problem. This exam is worth 100 points, of which I am giving you 50 points for your assumed knowledge of basic economic principles and definitions. Your task is to earn as many of the remaining 50 points as you can. 1. A consumer’s preferences are represented by the utility function u(x , x ) = x x . (4 points) Find the consumer’s generalized demand functions, x * ( p , p , m) and x * ( p , p , m) . (1 point) Find the consumer’s optimal bundle if ( p , p , m) = (1 point) Find the consumer’s optimal bundle if p increases to 8, while m and p (2 points) Calculate the Slutsky income and substitution effects of the change in p on (2 points) Another consumer’s preferences are represented by the utility function u(x , x ) = −xx . What are the consumer’s generalized demand functions, x *( p , p , m) x * ( p , p , m ? Michael Johnson is headed to the Olympics in Barcelona. He knows that there is a 10 percent probability that he might unwittingly eat some bad fish, which would give him food poisoning. If he does not get food poisoning, then he will win a gold medal, which will be worth $9,000,000 in future income. If he gets food poisoning, he will not be able to win a gold medal, thus reducing his future income to only $1,000,000. Michael’s utility from any given state of nature is u(I ) = I , where I represents Michael’s future income in a given state. (1 point) What is Michael’s expected utility from future income in light of the risk of food poisoning? (2 points) What is Michael’s certainty equivalent? (3 points) Suppose Michael can buy $x of insurance at a premium price of $.12x. What is the optimal amount of insurance for Michael to buy? (NOTE: this insurance policy is NOT actuarially fair.) (1 point each) A consumer’s demand for good x is given by x = What is the price elasticity of demand for x ? Is demand price elastic or price inelastic? What is the cross price elasticity of demand for x with the price of x ? Are these goods What is the income elasticity of demand for x ? Is it a normal or inferior good? ECO 3306 – SPRING 2001 – FINAL EXAM – Page 1
(2 points each) The inverse demand function for chicken-fried steak in Waco is given by p and the supply function is S( p) = 3000 p − What is the equilibrium price and quantity in the Waco chicken-fried steak market? Suppose the Texas legislature imposes a sales tax on this Lone Star State icon by charging a value tax of 25% of the price charged by the seller. What is the new equilibrium quantity sold in the market, and what are the sellers’ and buyers’ prices? In part (b), how much revenue will the tax generate? What is the amount of the deadweight loss? (1 point each) A firm’s technology is represented by the following production function: f (x , x ) = x Answer each of the following questions, SHOWING ALL WORK. a. Is the marginal product of input 1 increasing, decreasing or constant? Is the marginal product of input 2 increasing, decreasing, or constant? Does this production function exhibit increasing, decreasing, or constant returns to scale? (2 points each) A monopolist produces its output with technology represented by the production function } f (x , x ) = min{x ,2x Find the monopolist’s cost-minimizing conditional factor demand functions, x * (w , w , y) and x * (w , w , y) . w = 6 and w = 8 , what is the monopolist’s cost function, c(y)? The monopolist faces inverse demand of P( y) = 75 − y , where y is the quantity demanded. Calculate the monopolist’s optimal price, output level, and profit. Schering Corporation manufactures the patented antihistamine Claritin. Suppose that the marginal cost of a prescription of Claritin is $10, that daily inverse demand for Claritin prescriptions in the United States is given by P (4 points) U.S. law prohibits importation of pharmaceutical products from other countries, which enables Schering to price discriminate. What price will be charged in each country, how many prescriptions will be sold each day in each country, and how much profit will Schering generate in each country? (3 points) If the U.S. law were changed, making price discrimination impossible, what price would Schering charge, how many prescriptions would be sold, and what would Schering’s profit be? ECO 3306 – SPRING 2001 – FINAL EXAM – Page 2
(2 points each) A perfectly competitive firm produces its output with technology represented by the production function f (x , x ) = x Find the firm’s generalized profit-maximizing factor demand functions, x * (w , w , p) x * (w , w , p . 20 , how much profit will the firm generate? The City of Hewitt seeks to deter speeding on its major thoroughfares. Sometimes, a policeman will run radar to detect and give citations to speeders. Sometimes, the city will park an unoccupied police car in various strategic locations along major thoroughfares. We’ll use game theory to analyze this practice. Suppose that the payoffs for a typical driver and for the city are given by the following payoff matrix. Payoffs are listed (Driver, City) and are measured in units of utility. (2 points) What are the pure strategy Nash equilibria in this game? (2 points) Is there a mixed strategy equilibrium in this game? (1 point) What mixed strategy could the City use to assure that (1) no driver would speed, and (2) the City’s expected utility would be as high as possible given that no driver will speed? BONUS QUESTION: DO NOT ATTEMPT THIS QUESTION UNTIL YOU HAVE COMPLETED ALL OTHER QUESTIONS ON THE EXAM!!! (5 points) A soccer player has been awarded a free kick, in which she takes a shot with only the goalie allowed to defend the shot. The player can kick the ball to either the left or right side of the goal. The goalie must also decide prior to the shot whether she will jump to the left or to the right in an attempt to block the shot. Assume for simplicity that the goalie will always block the shot if she jumps the correct direction. Assume also that the kicker has a stronger kick to the right and will always score in that direction (if the goalie does not jump right), but that the kicker’s probability of scoring on a kick to the left (if the goalie does not jump left) is p, where 0 < p < 1. A made goal is worth 1 point to the kicker, and a miss is worth 0. Similarly, a blocked goal is worth 1 point to the goalie, and a made shot is worth 0 points to the goalie. Draw a 2x2 payoff matrix of this game and solve for all Nash equilibria. ECO 3306 – SPRING 2001 – FINAL EXAM – Page 3



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