Log Linear Demand and Supply

Consider the following “log-linear” supply and demand model: Demand: ???????????????? = ???????? + ???????????????????? + ????????????????????, ???????? < ????, ???????? > ???? (1) Supply: ???????????????? = ???????? + ???????????????????? + ???????????????????? ???????? > ????, ???????? < ???? (2) where P and Q have their usual meaning, I denotes household income and W the wage rate the firm must pay to hire workers. (1A) (10 points) What is the own price elasticity of demand in this market? Prove your answer, showing your work. (1B) (5 points) Is this a normal good or inferior good? Prove your answer, showing your work.. PARSONS GWU ECONOMICS 2101 SPRING 2015 2 (1C) (15 points) You are interested in knowing how the equilibrium price of this product will vary with a change in the competitive wage (W). What is the equilibrium price function? How will equilibrium price vary with a variation in W? Be as precise as you can be, always showing your work. PARSONS GWU ECONOMICS 2101 SPRING 2015 3 (2) (20 points) Deriving demand functions David has a quasi-linear utility function in X and Y of the form: ???? = ???? 1 2 + ???? Income is I and the two prices are ???????? and ???????? respectively. (2A) (15 points) Derive algebraically his demand function for X. Show your work. PARSONS GWU ECONOMICS 2101 SPRING 2015 4 (2B) (5 points) Is David’s demand for x is independent of his income? Demonstrate your answer formally. PARSONS GWU ECONOMICS 2101 SPRING 2015 5 (3) (30 points) Taxing away the benefits of a price changes A consumer in a two good economy (????, ????) with fix prices has the following preference function: ???? = ???????? Income is I and the two prices are ???????? and ???????? respectively. (3A) (15 points) Derive algebraically his demand functions for X. Show your work. PARSONS GWU ECONOMICS 2101 SPRING 2015 6 (3B) (15 points) Continuing the problem in 3A, consider the following situation: Income I=120 and the two prices are ???????? = 4 and ???????? = 1 respectively. Suddenly the price of X falls to 3 (???????? = 3). The government is tempted to capture all the gains the consumer gets from this price fall to finance a worthy project. How much income can they take from the consumer and still leave the consumer no worse off than he was before the price change? PARSONS GWU ECONOMICS 2101 SPRING 2015 7 PARSONS GWU ECONOMICS 2101 SPRING 2015 8 (4) (20 points) The Corn Market and Farmer Prosperity You are a political planner for an incumbent president and you know that farmers tend to vote for the party in power when times are good (or at least better) and for the challenger if times are bad. Times are good if total farm income (farm revenue) increases. . Thanks to the many favors your boss has done for interest groups, your boss has campaign reserves for more advertising if campaign problems arise in corn country. You check the fundamentals of the corn market. Agriculture Department economists assure you the market is competitive and also report that the elasticity of demand is ???????? and the elasticity of supply is ????????. (They give you a number for each but we want the general form so we use symbols for those numbers.) Then you learn that growing conditions are fabulous and that the country will have a bumper crop of corn. Should you commit your advertising reserves to the corn states or not? Be as specific in your answer as you can