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Demand, Utility and Expenditure

Demand, Utility and Expenditure. Chapter 5, Frank and Bernanke. Key Concepts. Law of Demand – other things equal, when price goes up, the quantity demanded goes down.

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Demand, Utility and Expenditure

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  1. Demand, Utility and Expenditure Chapter 5, Frank and Bernanke

  2. Key Concepts Law of Demand – other things equal, when price goes up, the quantity demanded goes down. Utility maximization – consumers determine the quantity demanded of each of two goods by equating their marginal utility per dollar. Demand and expenditure – the Law of Demand makes no prediction on the relation of price and expenditure. When price goes up, expenditure may go up, go down or stay the same. Expenditure = price times quantity purchased. Elasticity = responsiveness of quantity demanded to price changes.

  3. Utility Maximization • Consumers apply the equimarginal principle and find the point at which the marginal benefit of spending another dollar on Good X equals the marginal cost of NOT spending another dollar on Good Y. MUx MUy Px Py • This equation is the “rational spending rule”

  4. Application of the Rational Spending Rule • Suppose good X (“pizza”) is at a price of $10 a pie, and good Y (“concert tickets”) is at a price of $30 a concert. • You have a budget of $ 130 for entertainment, and want to rationally allocate it among the two goods. • You know that the marginal utility of either good declines with the amount consumed (though TOTAL utility continues to increase) • You know your utility tables – see the next slide

  5. Finding marginal utility: • MU of X = change in total utility from X with 1 more X • MU of Y = change in total utility from Y with 1 more Y • Finding marginal utility per dollar: • MU per dollar of X is the MU of X divided by the price of X • And likewise for the MU per dollar of Y. • The last is the crucial step – we are changing our choices dollar by dollar, not unit by unit.

  6. Using the table • The problem can’t be easily solved by considering all possible choices. • Reduce it to the simpler problem: what do I buy next? • Since for the first unit MU of x per $ is 7.0, and MU of y per $ is 4.67, buy X first. • Now compare the MU per dollar of the second unit of X ( 4.0) with the MU per dollar of the first unit of Y (4.67). Buy Y next.

  7. What do you buy next?? Remember, go for the highest MU per dollar.

  8. Remember to ask yourself how much you have left – So far, you’ve spent $ 30 on X and $ 30 on Y, Leaving you with $ 70 from the budget of $ 130.

  9. Checking the rational spending rule At X = 4 and Y = 3, we’ve exhausted our budget ($ 40 on X, $ 90 on Y) Go back to the table to calculate our utility score: Utility of 4 X = 161 utils Utility of 3 Y = 280 utils Total utility = 441 utils

  10. Could we do better? If we bought one less Y, we would have $ 30 more and could buy 3 more X : • The new consumption bundle is 2 Y and 7 X • Utility of 7 X = 208 utils • Utility of 2 Y = 220 utils • Total utility = 428 utils (less than 441 utils) • Try buying one more Y and 3 less X: • Utility of 1 X = 70 utils • Utility of 4 Y = 322 utils • Total utility = 392 utils

  11. Utility maximization, functionally speaking • A common economic model for a utility function is the logarithmic function: • TUx = 100 ln X • TUy = 200 ln Y (it wasn’t an accident that the tables just used are almost the same as you would get from computing 100 ln 2, 100 ln 3, etc. The only slight difference is that ln 1 = 0, so the table was shifted back 1 level to avoid TU of 1 = 0).

  12. Marginal Utility, functionally speaking • It can be shown that if TUx = A ln X, MUx = A divided by X Full proof requires calculus, but you should be able to see that the formula works by a few examples: If TUx = 100 ln X, what is the marginal utility between 50 and 51 units of X? MUx = 100 ln 51 minus 100 ln 50 MUx = 393.1826 minus 391.2023 = 1.9803 Using the formula for marginal utility, MUx = 100 / X = 100 / 50.5 = 1.9802 [should you divide by 50 or 51? Dividing by 50 gets MUx = 2, by 51 gets MUx = 1.96 The difference is never too important in practice]

  13. Rational spending, functionally speaking • Let TUx = 150 ln X and TUy = 300 ln Y • Then MUx = 100 / X and MUy = 300 / Y • Hence the rational spending rule is: MUx / Px = MUy / Py or 150 / Px X = 300 Py Y or Py Y = 2 Px X

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