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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 and Transportation Modes 1. (a) What is the formula for his demand functions for x1 and x2 (see lecture on 1 - 29 - 13)? u = π‘₯ 1 0 . 33 βˆ— π‘₯ 2 0 . 67 We derived then the utility function by x1 and x2 : πœ•π‘’ πœ•π‘₯ 1 = 0 . 33 π‘₯ 1 βˆ’ 0 . 67 π‘₯ 2 0 . 67 πœ•π‘’ πœ•π‘₯ 2 = 0 . 67 π‘₯ 1 0 . 33 π‘₯ 2 βˆ’ 0 . 33 Let P1 and P2 the prices of x1 and c2, and M the income. We have : πœ•π‘’ πœ•π‘₯ 1 𝑃 1 = πœ•π‘’ πœ•π‘₯ 2 𝑃 2 And we have the following budget constraint : π‘₯ 1 𝑃 1 + π‘₯ 2 𝑃 2 = 𝑀 Hence : 0 . 33 π‘₯ 1 βˆ’ 0 . 67 π‘₯ 2 0 . 67 𝑃 2 = 0 . 67 π‘₯ 1 0 . 33 π‘₯ 2 βˆ’ 0 . 33 𝑃 1 𝑃 2 = 2 π‘₯ 1 π‘₯ 2 𝑃 1 Then : π‘₯ 1 𝑃 1 + 2 π‘₯ 1 𝑃 1 = 𝑀 π‘₯ 1 = 𝑀 3 𝑃 1 And π‘₯ 2 = 2 𝑀 3 𝑃 2 (b) What is the income elasticity of demand for x1 with this demand function? (Hint: if income increases by 10%, say, by how much does the demand for x1 increase?)

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    The demand function for x1 is : π‘₯ 1 = 𝑀 3 𝑃 1 Then : βˆ† π‘₯ 1 = βˆ† 𝑀 3 𝑃 1 The income elasticity demand is defined as follow : πœ€ = βˆ† π‘₯ 1 βˆ† 𝑀 βˆ— 𝑀 π‘₯ 1 = 1 3 𝑃 1 βˆ— 3 𝑃 1 = 1 (c) Is there an β€œincome effect” with these demand functions? There is an β€œincome effect” : there is an increase of demand by an 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 each good? What is his expenditure on each good? π‘₯ 1 = 𝑀 3 𝑃 1 = 1200 3 βˆ— 3 β‰ˆ 134 π‘₯ 2 = 2 𝑀 3 𝑃 2 = 2 βˆ— 1200 3 βˆ— 1 = 800 The expenditure on good is defined as : 𝑒 = ( 𝑃 1 𝛼 ) 𝛼 βˆ— ( 𝑃 2 1 βˆ’ 𝛼 ) 1 βˆ’ 𝛼 βˆ— 𝑒 Where Ξ± = 0.33 Then : 𝑒 = ( 3 0 . 33 ) 0 . 33 βˆ— ( 1 0 . 67 ) 0 . 67 βˆ— 134 0 . 33 βˆ— 800 0 . 67 = $1200 The expenditure of good 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 it portray?

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    The indirect utility function U* shows the maximal utility of the consumer according to a price level P and 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 as before? Explain the reason for your answer. If we consider the following diagram that represents the trend of the indirect utility functio n in function of the prices P1 and P2 at a constant income M : If the price of P1 decreases to 1 the indirect utility function curve is shifted to the left : therefore there is an increase of U*, which means the consumer is better of as before as a result of this price change. (g) How does his expenditure on x1 change? The expenditure on good x1 is : 133 βˆ— 1 = $133 : there is a reduction of 400 - 133 = $267 of the expenditure on x1 (h) Draw a diagram showing the change in Marshallian consumer’s surplus associated with this particular 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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