5. Choice - Solutions

# 5. Choice - Solutions - Chapter 5 NAME Choice Introduction...

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Chapter 5 NAME Choice Introduction. You have studied budgets, and you have studied prefer- ences. Now is the time to put these two ideas together and do something with them. In this chapter you study the commodity bundle chosen by a utility-maximizing consumer from a given budget. Given prices and income, you know how to graph a consumer’s bud- get. If you also know the consumer’s preferences, you can graph some of his indiFerence curves. The consumer will choose the “best” indiFerence curve that he can reach given his budget. But when you try to do this, you have to ask yourself, “How do I ±nd the most desirable indiFerence curve that the consumer can reach?” The answer to this question is “look in the likely places.” Where are the likely places? As your textbook tells you, there are three kinds of likely places. These are: ( i ) a tangency between an indiFerence curve and the budget line; ( ii ) a kink in an indiFerence curve; ( iii ) a “corner” where the consumer specializes in consuming just one good. Here is how you ±nd a point of tangency if we are told the consumer’s utility function, the prices of both goods, and the consumer’s income. The budget line and an indiFerence curve are tangent at a point ( x 1 ,x 2 )ifthey have the same slope at that point. Now the slope of an indiFerence curve at ( x 1 2 )i sthera t io MU 1 ( x 1 2 ) /M U 2 ( x 1 2 ). (This slope is also known as the marginal rate of substitution.) The slope of the budget line is p 1 /p 2 . Therefore an indiFerence curve is tangent to the budget line at the point ( x 1 2 )when 1 ( x 1 2 ) 2 ( x 1 2 )= p 1 /p 2 .Th i sg iv e s us one equation in the two unknowns, x 1 and x 2 . If we hope to solve for the x ’s, we need another equation. That other equation is the budget equation p 1 x 1 + p 2 x 2 = m . With these two equations you can solve for ( x 1 2 ). Example: A consumer has the utility function U ( x 1 2 x 2 1 x 2 .T h e price of good 1 is p 1 = 1, the price of good 2 is p 2 = 3, and his income is 180. Then, 1 ( x 1 2 )=2 x 1 x 2 and 2 ( x 1 2 x 2 1 h e r e - fore his marginal rate of substitution is 1 ( x 1 2 ) 2 ( x 1 2 2 x 1 x 2 /x 2 1 = 2 x 2 /x 1 . This implies that his indiFerence curve will be tangent to his budget line when 2 x 2 /x 1 = p 1 /p 2 = 1 / 3. Simplifying this expression, we have 6 x 2 = x 1 . This is one of the two equations we need to solve for the two unknowns, x 1 and x 2 . The other equation is the budget equation. In this case the budget equation is x 1 +3 x 2 = 180. Solving these two equations in two unknowns, we ±nd x 1 = 120 and Some people have trouble remembering whether the marginal rate of substitution is 1 2 or 2 1 . It isn’t really crucial to remember which way this goes as long as you remember that a tangency happens when the marginal utilities of any two goods are in the same proportion as their prices.

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50 CHOICE (Ch. 5) x 2 = 20. Therefore we know that the consumer chooses the bundle ( x 1 ,x 2 ) = (120 , 20).
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5. Choice - Solutions - Chapter 5 NAME Choice Introduction...

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