Consumer's_Problem(- 2A%2b2B

Consumer's_Problem(- 2A%2b2B - CONSUMERS PROBLEM (2A)...

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CONSUMER’S PROBLEM (2A) Perfect Substitute & Corner Solution Consider the utility function U= 11 22 ax ax + . Note that in this case =0 implies that the MRS is 11 2 2 du a dx a dx =+ 2 12 tan o n s t a a = 1 uc dx dx = . Hence the two goods are perfect substitutes. Now set up the usual Lagrangian 1 2 2 2 (, ) [ ] [ ] Lu xx mp x p x a x x λ = + + The first order conditions are 1 00 L up ap x λλ =− =⇒− = (1) 2 L x =− = ⇒− = (2) 0 L x =− − = (3) Note that (1) &(2)± together imply 1 = 1 (4)± But are all given numbers. Hence ± is either true or false. There is no way the consumer can set the two sides equal through appropriate choice of 12 1 2 ,,, aa pp & x x . What happens when uap =≠ 1 2 ? The typical indifference curve in this case is 0 u + = or 0 1 21 a u x x aa . This is a downward sloping straight line with slope 1 2 a a . At the same time, the budget line is p xp xm += or 1 p m x x p p Since 1 > 1 , the indifference curve is steeper than the budget line as shown in the figure below.
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In this case the highest indifference curve is attained at the corner point B where 2 x =0 and 1 1 m x p = , because the consumer spends the entire income on good 1. The opposite would be true if 1 2 a a (i.e. 1 2 u u ) is less than 1 2 p p . This can be explained intuitively as follows. The marginal utility of $1 spent on good 1 is 1 1 u p , similarly for good 2 it is 2 2 u p . Thus 11 22 up > 12 uu p p > . In other words, marginal utility of a $1 spent on good 1 exceeds the marginal utility of a $1 spent on good 2. In that case, the consumer would be better off spending a $1 on good 1 rather than on good 2. But because are constant numbers, 1 & 2 1 2 p p > no matter how much is already spent on good 1. Hence the consumer spends all
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This note was uploaded on 10/19/2010 for the course ECON 1202 taught by Professor Matel during the Fall '08 term at UConn.

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Consumer's_Problem(- 2A%2b2B - CONSUMERS PROBLEM (2A)...

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