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Demand, Utility and Expenditure
Chapter 5, Frank and Bernanke
Key ConceptsLaw 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.
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”
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
Units of X
T.Ux M.Ux M.Ux per $
Units of Y
T.Uy M.Uy M.Uy per $
1 70 1 140
2 110 2 220
3 140 3 280
4 161 4 322
5 179 5 358
6 195 6 390
7 208 7 416
8 220 8 440
• 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.
Units of X
T.Ux M.Ux M.Ux per $
Units of Y
T.Uy M.Uy M.Uy per $
1 70 70 1 140 140
2 110 40 2 220 80
3 140 30 3 280 60
4 161 21 4 322 42
5 179 18 5 358 36
6 195 16 6 390 32
7 208 13 7 416 26
8 220 12 8 440 24
Units of X
T.Ux M.Ux M.Ux per $
Units of Y
T.Uy M.Uy M.Uy per $
1 70 70 7.0 1 140 140 4.67
2 110 40 4.0 2 220 80 2.67
3 140 30 3.0 3 280 60 2.00
4 161 21 2.1 4 322 42 1.4
5 179 18 1.8 5 358 36 1.3
6 195 16 1.6 6 390 32 1.07
7 208 13 1.3 7 416 26 0.87
8 220 12 1.2 8 440 24 0.80
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.
Units of X
T.Ux M.Ux M.Ux per $
Units of Y
T.Uy M.Uy M.Uy per $
1 70 70 7.0 1 140 140 4.67
2 110 40 4.0 2 220 80 2.67
3 140 30 3.0 3 280 60 2.00
4 161 21 2.1 4 322 42 1.4
5 179 18 1.8 5 358 36 1.3
What do you buy next??
Remember, go for the highest MU per dollar.
Units of X
T.Ux M.Ux M.Ux per $
Units of Y
T.Uy M.Uy M.Uy per $
1 70 70 7.0 1 140 140 4.67
2 110 40 4.0 2 220 80 2.67
3 140 30 3.0 3 280 60 2.00
4 161 21 2.1 4 322 42 1.4
5 179 18 1.8 5 358 36 1.3
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.
Units of X
T.Ux M.Ux M.Ux per $
Units of Y
T.Uy M.Uy M.Uy per $
1 70 70 7.0 1 140 140 4.67
2 110 40 4.0 2 220 80 2.67
3 140 30 3.0 3 280 60 2.00
4 161 21 2.1 4 322 42 1.4
5 179 18 1.8 5 358 36 1.3
6 195 16 1.6 6 390 32 1.07
7 208 13 1.3 7 416 26 0.87
8 220 12 1.2 8 440 24 0.80
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
• 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
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).
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 = 1.96The difference is never too important in practice]
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