Consumer Choice, Demand Functions, And Welfare Analysis: A Case Study Of Jellybear Park And Transportation Modes

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Consumer Choice, Demand Functions, and Welfare Analysis: A Case Study of Jellybear Park andTransportation Modes1.(a) What is the formula for his demand functions for x1 and x2 (see lecture on 1-29-13)?u =𝑥10.33∗𝑥20.67We derived then theutility function by x1 and x2 :𝜕𝑢𝜕𝑥1=0.33𝑥1−0.67𝑥20.67𝜕𝑢𝜕𝑥2=0.67𝑥10.33𝑥2−0.33Let P1 and P2 the prices of x1 and c2, and M the income.We have :𝜕𝑢𝜕𝑥1𝑃1=𝜕𝑢𝜕𝑥2𝑃2And we have the following budget constraint :𝑥1𝑃1+𝑥2𝑃2=𝑀Hence :0.33𝑥1−0.67𝑥20.67𝑃2=0.67𝑥10.33𝑥2−0.33𝑃1𝑃2=2𝑥1𝑥2𝑃1Then :𝑥1𝑃1+2𝑥1𝑃1=𝑀𝑥1=𝑀3𝑃1And𝑥2=2𝑀3𝑃2(b) What is the income elasticity of demand for x1 with this demand function? (Hint: if incomeincreases by 10%, say, by how much does the demand for x1 increase?)

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The demand function for x1 is :𝑥1=𝑀3𝑃1Then :∆𝑥1=∆𝑀3𝑃1The incomeelasticity demand is defined as follow :𝜀=∆𝑥1∆𝑀∗𝑀𝑥1=13𝑃1∗3𝑃1=1(c) Is there an “income effect” with these demand functions?There isan“income effect” :there is an increase ofdemand byan increase of the income : a 10%increase of income leads to a 10% increase in demand.(d) If his income is $1200 and prices are p1 = $3 and p2 = $1, how much does he buy of eachgood? What is his expenditure on each good?𝑥1=𝑀3𝑃1=12003∗3≈134𝑥2=2𝑀3𝑃2=2∗12003∗1=800The expenditure on goodis defined as :𝑒=(𝑃1𝛼)𝛼∗(𝑃21−𝛼)1−𝛼∗𝑢Whereα= 0.33Then :𝑒=(30.33)0.33∗(10.67)0.67∗1340.33∗8000.67=$1200Theexpenditureofgood x1 is : 133.33*3 = $400 and for the good x2 : 1*800 = $800.(e)Give a definition of the indirect utility function: what relationship does itportray?

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The indirect utility functionU*shows the maximal utility of the consumer according to a price level Pand an income amount M. It portrays the preferences of the consumer over market conditions.(f) Suppose the price of good 1 becomes p1 = $1 while, as before his income is $1200 and p2 =$1. Is the consumer better off as a result of this price change, worse off, or equally well off asbefore? Explain the reason for your answer.If we consider the following diagram that represents thetrend of the indirect utility function infunction of the pricesP1and P2at a constant income M :If the price of P1 decreases to 1 the indirect utility function curveis shifted to the left : thereforethere is an increase of U*, which means the consumer is better of as before as a result of this pricechange.(g) How does his expenditure on x1 change?The expenditure on good x1 is :133∗1=$133: there is a reduction of 400-133 = $267 of theexpenditure on x1(h) Draw a diagram showing the change in Marshallian consumer’s surplus associated with thisparticular price change (you need not actually calculate the change in consumer’s surplus)We need to plot the Marshallian Demand curve that passes through the points (133, $3) and (400,$1) :

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