PS6_solutions - Intermediate Microeconomics 2008 Problem...

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Intermediate Microeconomics, 2008 Problem Set No 6: Solutions Problems 1) We considered preferences represented by the following utility function in the last homework: U ( q 1 , q 2 ) = q 1 q 2 We assume that income is fixed at Y = 2 and the price of good two is fixed as p 2 = 1 . 1)a) Suppose p 1 increases. Illustrate graphically how the change in demand can be decomposed into a substitution effect and an income effect. 1
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The original budget line is green and the initial optimal bundle is A . The new budget line when p 1 increases is red, and new optimal choice is C . The total effect is therefore ( C - A ) (measure either q 1 or q 2 at these points to get the corresponding total effect for either good). The blue budget line represents the income required to remain on the initial indifference curve at the new prices, and corresponding optimal bundle B . Since B and A are on the same indifference curve, and difference between them reflects only the difference in relative prices, hence ( B - A ) gives the substitution effect. Since C and B are on budget lines with the same relative prices but different income levels, the difference between them ( C - B ) is the income effect. 1)b) Suppose p 1 increases from 1 4 to 1 2 . Derive the total change of the demand for good 1. Decom- pose that change into the change due to the substitution effect and the change due to the income effect. Note: The total change in demand is D 1 ( 1 2 , Y ) - D 1 ( 1 4 , Y ) . The change due to the substitu- tion effect is H 1 ( 1 2 , ¯ U 0 ) - H 1 ( 1 4 , ¯ U 0 ) . The change due to the income effect is the change in demand D 1 ( 1 2 , Y ) when income changes from Y = E ( 1 2 , ¯ U 0 ) to Y = E ( 1 2 , ¯ U N ) . ¯ U 0 and ¯ U N are the utility levels at the old and at the new price. To solve this problem, we will need to derive the uncompensated demand functions D 1 ( p 1 , Y ) and D 2 ( p 1 , Y ) for arbitrary values of Y , and the compensated demand functions H 1 ( p 1 , ¯ U ) and H 2 ( p 1 , ¯ U ) and expenditure function E ( p 1 , ¯ U ) for arbitrary values of ¯ U . We proceed as in problem 6 of Problem Set 5, and hold p 2 = 1 throughout. Begin with the uncompensated demand functions. Using the substitution method , rearrange the budget constraint to solve for q 2 as a function of q 1 : p 1 q 1 + p 2 q 2 = Y q 2 = Y - p 1 q 1 p 2 q 2 = Y - p 1 q 1 Substitute what you found for q 2 ( q 1 ) into the utility function, to get utility as a function of only one variable, q 1 : U ( q 1 , q 2 ) = q 1 q 2 U ( q 1 ) = ( q 1 )( Y - p 1 q 1 ) = Y q 1 - p 1 q 1 2 Find the value of q * 1 that maximizes this function by setting the derivative, dU dq 1 , equal to zero: 2
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dU dq 1 = 0 Y - 2 p 1 q 1 = 0 2 p 1 q 1 = Y q * 1 = Y 2 p 1 The uncompensated demand curves are: D 1 ( p 1 , Y ) = q * 1 = Y 2 p 1 D 2 ( p 1 , Y ) = q * 2 = Y - p 1 Y 2 p 1 = Y 2 Now let’s find the compensated demand functions H 1 ( p 1 , ¯ U ) and H 2 ( p 1 , ¯ U ). We’ll minimize costs, X ( q 1 , q 2 ), subject to achieving a specific level of utility, ¯ U . Using the substitution method , rearrange the utility function at ¯ U , to solve for q 2 as a function of q 1 : ¯ U = q 1 q 2 q 2 = ¯ U q 1 Substitute what you found for q 2
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