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Demand, Utility and Expenditure Chapter 5, Frank and Bernanke
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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.
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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”
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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
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Units of X T.UxM.UxM.Ux per $ Units of Y T.UyM.UyM.Uy per $ 1701140 21102220 31403280 41614322 51795358 61956390 72087416 82208440
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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.
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Units of X T.UxM.UxM.Ux per $ Units of Y T.UyM.UyM.Uy per $ 170 1140 2110402220 80 3140303280 60 4161214322 42 5179185358 36 6195166390 32 7208137416 26 8220128440 24
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Units of X T.UxM.UxM.Ux per $ Units of Y T.UyM.UyM.Uy per $ 170 7.01140 4.67 2110404.02220 802.67 3140303.03280 602.00 4161212.14322 421.4 5179181.85358 361.3 6195161.66390 321.07 7208131.37416 260.87 8220121.28440 240.80
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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.
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Units of X T.UxM.UxM.Ux per $ Units of Y T.UyM.UyM.Uy per $ 170 7.01140 4.67 2110404.02220 802.67 3140303.03280 602.00 4161212.14322 421.4 5179181.85358 361.3 What do you buy next?? Remember, go for the highest MU per dollar.
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Units of X T.UxM.UxM.Ux per $ Units of Y T.UyM.UyM.Uy per $ 170 7.01140 4.67 2110404.02220 802.67 3140303.03280 602.00 4161212.14322 421.4 5179181.85358 361.3 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.
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Units of X T.UxM.UxM.Ux per $ Units of Y T.UyM.UyM.Uy per $ 170 7.01140 4.67 2110404.02220 802.67 3140303.03280 602.00 4161212.14322 421.4 5179181.85358 361.3 6195161.66390 321.07 7208131.37416 260.87 8220121.28440 240.80
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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
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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
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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).
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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]
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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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