EEP101_Review2a - EEP 101/ECON 125 Review 2 DavidZilberman...

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EEP 101/ECON 125 Review 2 David Zilberman
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Question 1 The production function of a product with an old technology is  Y  = 10 X  - .5 X 2 . The cost of input  X  is $2 and the cost of the output is  $1. The input is polluting and its damage is $1.5 X 2  What is the input use without regulation? The profit in this case is 10 X  -.5 X  - 2 X . The optimal condition  determining the marginal condition is 10 -  - 2 = 0, so that the input is X  = 8. The profit in this case is 80 - 32 - 16 = 32. The social welfare is 32 - 96 = -64 (too much production).  What is the socially optimal input use? What is an optimal           pollution tax?
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The social welfare is 10 - .5 X  - 2 - 1.5 X 2.     The first-order  condition is  10 - X - 2 - 3X =  0,  so that the socially optimal X is  = 2,  the profit is 14, and the social welfare is 20 - 2 - 4 - 6 = 8.  The optimal tax is 3 X  at  = 2, namely 6. With this tax, the  optimization problem is to maximize 10 - .5 X  - 2 - 6 X,  and  the first-order condition is 10 - X - 2 - 6 = 0. Thus,  X = 2.
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Adoption Suppose that   a new technology with  Y = 20X - 1.5 X   is available.  What are  the profit-maximizing and socially optimal input        use levels? The profit maximizing input use is where 20 - 3 - 2 = 0 and  = 6. The  social optimum is where 20 - 3 - 2 - 3 = 0 so that  = 3.  Now suppose that this technology costs $10.    Will it be optimal under competitive or optimal outcome?    Profit under competitive outcome is 120 - 54 - 12 = 54 and with the  traditional technology competitive profits is 36 so it is  worthwhile to adopt,  since 54 - 36  > 10.  Social benefits under the optimal outcome is 60 - 13.5 - 6 = 40.5.  The gain  from adoption here is 40.5 - 8 = 32.5. If the price of the technology is $20, is it socially or privately optimal to 
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This note was uploaded on 04/04/2010 for the course C 125 taught by Professor Zilberman during the Spring '09 term at University of California, Berkeley.

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EEP101_Review2a - EEP 101/ECON 125 Review 2 DavidZilberman...

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