10HWS2 - Economics 104A A Solution for Problem Set #2...

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Economics 104A A Solution for Problem Set #2 Winter 2010 1. A consumer spends all his money on good 1 and good 2. The consumer’s utility function is: u ( x 1 ,x 2 ) = min { 4 x 1 , 2 x 1 + x 2 } . The price of good 1 is p 1 = 10, the price of good 2 is p 2 = 5, and the consumer’s income is m = 200. (a) Find the consumer’s optimal bundle. Answer: Notice that u ( x 1 ,x 2 ) = ( 2 x 1 + x 2 , if 2 x 1 x 2 ; 4 x 1 , if 2 x 1 x 2 . Thus, MRS( x 1 ,x 2 ) = 2 whenever 2 x 1 > x 2 . This shows that the slope of the IC’s is always equal to 2 at bundles where 2 x 1 > x 2 . Since p 1 /p 2 = 2, it follows that the budget line with 2 x 1 > x 2 is part of an IC. Hence, all the bundles on that part of the budget line are optimal. In particular, x * = (20 , 0) is optimal. See Figure 1 for an illustration. (b) Graph the consumer’s indifference curve passing through the optimal bundle you found in part (a). Answer:
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10HWS2 - Economics 104A A Solution for Problem Set #2...

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