PARSONS GWU ECONOMICS 2101 SPRING 2012

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YOUR NAME

ECONOMICS 2101

FIRST MIDTERM EXAMINATION

Answer All Questions. The test has one hundred points. You have one hour and 10 minutes to complete the examination. (1) Optimization (20 points total) A farmer has F feet of fencing and wants to choose the rectangular dimensions (W,L) that will enclose the maximum amount of space for his cattle.

(1A) (5 points) Specify the farmer’s objective function and his constraint. (1B) (10 points) Solve the farmer’s problem using calculus techniques. (1C) (5 points) Prove using calculus techniques that your solution in (1B) is a

maximum area, not a minimum area.

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(2) Demand Elasticities (20 points total) Suppose the market for air travel between Chicago and Dallas is served by just two airlines, United and American. An economist has studied this market and has estimated that the demand curves for round-trip tickets for each airline are as follows: ܳௗ = 10,000 െ 100 ܲ + 99 ܲ United’s Demand ܳௗ = 10,000 െ 100 ܲ + 99 ܲ American’s Demand

where ܲ denotes United’s price and ܲ denotes American’s price.

(2A) (10 points) Suppose both airlines are charging $300 for a round trip and United is considering a unilateral price increase (American is not expected to raise its prices). What is the price elasticity of demand for United flights?

(2B) (10 points) Now imagine that the two airlines collude and agree to raise

prices in concert (same price changes at the same time). What is the elasticity of demand for flights in this case?

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(3) Deriving a Demand Function (30 points)

Consider a consumer in a static two-good world (x, y). The consumer has a utility

function: =

, and has an income of I and faces a competitive product market

with prices xp and yp .

(3A) (10 points) Derive the individual’s marginal rate of substitution (MRS) algebraically? (MRS is the negative of the slope of the indifference curve).

(3B) (10 points) Is the MRS increasing or diminishing in x? Prove your assertion using calculus methods.

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(3C) (10 points) Derive the individual’s demand curve for x.

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(4) (30 points) Samantha Samantha purchases food (F) and other goods (Y) with the utility function: ܷ = ܻܨ Her income is I and the price of food is ிܲ and ܲ respectively.

(4A) (10 points) Derive algebraically her demand function for food. (4B) (10 points) Assume that the government decides that the level of food

consumption by the individual implied by the demand function is insufficient to maintain good health and wants to calculate the additional income ȟܫ it would have to provide the individual to induce her to demand a healthy level of food consumption Fୌ. What is the formula for ȟܫ as a function of the problem parameters—food consumption goal, income, prices?

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(4C) (10 points) Using the formula derived in (4B) calculate the additional income

ȟܫ needed to insure that individuals will consume Fୌ = 8 when ܫ= 12, ிܲ = 2 and ܲ = 1.

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