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The Utility-Maximizing Solution

25 April, 2016 - 09:12

We assume that each consumer seeks the highest indifference curve possible. The budget line gives thecombinations of two goods that the consumer can purchase with a given budget. Utility maximization is therefore a matter of selecting a combination of two goods that satisfies two conditions:

  1. The point at which utility is maximized must be within the attainable region defined by the budget line.
  2. The point at which utility is maximized must be on the highest indifference curve consistent with condition 1.

Figure 7.11 combines Janet Bain’s budget line from Figure Figure 7.7 with her indifference curves from Figure 7.9. Our two conditions for utility maximization are satisfied at point X, where she skis 2 days per semester and spends 3 days horseback riding.

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Figure 7.11 The Utility-Maximizing Solution
 

Combining Janet Bain’s budget line and indifference curves from Figure 7.7 and Figure 7.9, we find a point that (1) satisfies the budget constraint and (2) is on the highest indifference curve possible. That occurs for Ms.Bain at point X.

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Figure 7.12 Applying the Marginal Decision Rule
Description
 

Suppose Ms. Bain is initially at point S. She isspending all of her budget, but she is notmaximizing utility. Because her marginal rate ofsubstitution exceeds the rate at which themarket asks her to give up skiing for horseback riding, she can increase her satisfaction by moving to point D. Now she is on a higher indifference curve, E. She will continueexchanging skiing for horseback riding until she reaches point X, at which she is on curve A,the highest indifference curve possible.

The highest indifference curve possible for a given budget line is tangent to the line; the indifference curve and budget line have the same slope at that point. The absolute value of the slope of the indifference curve shows the MRS between two goods. The absolute value of the slope of the budget line gives the price ratio between the two goods; it is the rate at which one good exchanges for another in the market. At the point of utility maximization, then, the rate at which the consumer is willing to exchange one good for another equals the rate at which the goods can be exchanged inthe market. For any two goods X and Y, with good X on the horizontal axis and good Y on the vertical axis,

EQUATION 7.10

MRSX,Y=\frac{PX}{PY}