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