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prelims_Micro Prelim ANSWERS August 2007 - University of...

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University of California, Davis Date: August 30, 2007 Department of Agricultural and Resource Economics Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE ANSWER KEY Part 1. The present and the future 1.1. Optimization problem max u ( x 1 , x 2 ) subject to p 1 x 1 + p 2 x 2 < p 1 ω 1 + p 2 ω 2 , with solution function 112 212 ˆˆ ((,) ,(,) ) xpp xpp (the “Jevonsian” demand function), where p 1 , p 2 and r are related by the equality 1 2 1 p r p = + . (1) x 1 slope = 1 2 (1 ) p r p =− + ω 1 ω 2 x 2

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2 1.2. Slutsky equation ˆ ˆˆ () (,(,() ) ) (,() ) kk k j j jj xh x pp v p w p p w p x w ∂∂ =− ω % , j , k = 1, 2, (2) where: p = ( p 1 , p 2 ), , h 11 2 2 ˆ wp p p ≡ω k is the Hicksian demand function and v is the indirect utility function. 1.3(a) Without further information, what can we say about the sign of 112 1 ˆ (, ) xpp p when the consumer is a borrower? A saver? Autarkic? From (2) [ 111 ˆ xhx x ppw ∂∂∂ ] = −− % ω . (3) By the Slutsky Theorem, 1 1 0 h p (the substitution effect is nonpositive), and in fact 1 1 0 h p < under our assumptions (in particular, the utility function is differentiable with nonzero gradient). Moreover, because good 1 is normal, 1 0 x w > % . If the consumer is a borrower, [ ] 0 x −ω> , hence the wealth effect [] 1 x x w % ω is negative. The total effect is then the sum of two negative terms, and is therefore negative. Intuitively, if you are a borrower you are a (net) buyer of present consumption (good 1). An increase in the price of a good that you buy reduces your purchasing power, which tends to decrease your demand for all normal goods. The wealth effect then reinforces the substitution effect, and the demand for good 1 unambiguously decreases. If the consumer is autarkic, then [ ] 0 x −ω= , hence the wealth effect is zero. The total effect is then equal to the substitution effect, negative. If the consumer is a saver, then [ ] 0 x −ω< , hence the wealth effect is positive: Now you are selling good 1, and an increase in its price raises your purchasing power. The substitution effect and the wealth effect move in opposite directions, and nothing can be said on the sign of 1 ˆ p in the absence of further information. Table 1 summarizes.
3 Borrower Autarkic Saver 112 1 ˆ (, ) xpp p < 0 < 0 ? Table 1. Good 1 normal; 1 ˆ p . 1.3(b) Without further information, what can we say about the direction of the change in present consumption as the interest rate r rises when the consumer is a borrower? A saver? Autarkic? By (1) and the 0-homogeneity of ˆ , we can write the demand for present consumption as a function of r as 11 ˆ ˆˆ () ( 1 ,1 ) x rx r =+ , i. e., an increase in r is equivalent to an increase in p 1 . Then 1 ˆ ( ) dx r x r dr p ∂+ = , with sign given by (3). Hence, it follows from 1.3(a) that an increase in the interest rate reduces the demand for present consumption if the consumer is a borrower or autarkic, whereas the effect is indeterminate when the consumer is a saver. Table 2 summarizes.

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prelims_Micro Prelim ANSWERS August 2007 - University of...

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