Utility Functions, Budget Lines and Consumer Demand Overheads

Conditions for the consumer optimum a. The indifference curve and the budget line are tangent at the optimum combination of q1 and q2 b. The slope of the budget line and the slope of the indifference curve are equal at the optimum c. The price ratio equals the ratio of marginal utilities d. The marginal utility per dollar for each good is equal

Example problem Initial prices and income p1 = 5, p2 = 10, I0 = 50 Initial equilibrium 16 q1 14 Budget Line 0 12 u = 8 u = 10.99 10 8 6 4 2 1 2 3 4 5 6 7 8 q2

Change in income Subsequent prices and income p1 = 5, p2 = 10, I1 = 75. Increase in Income 16 q1 14 Budget Line 1 Budget Line 0 u = 10.99 u = 8 12 10 8 6 4 2 1 2 3 4 5 6 7 8 q2

Move to a higher indifference curve Initial equilibrium Subsequent equilibrium 2 4 6 8 10 12 14 16 1 3 5 7 q u = 10.99 Budget Line 1 u = 15.4188 u = 8 Budget Line 0

Another Example Initial equilibrium Initial prices and income p1 = 6, p2 = 4, I0 = 58 Budget Line 0 u=9.614 Initial equilibrium 16 q 14 1 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

Raise income so we have p1 = 6, p2 = 4, I0 = 78 Initial equilibrium Budget Line 0 u=9.614 Initial equilibrium Subsequent equilibrium u=12.018 16 q 14 1 12 Budget Line 1 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

So both goods are normal goods In these examples, we consume more of both goods as income rises So both goods are normal goods So what is a Normal Good? A Normal Good is a good that someone will buy more of as their income increases

What About an Inferior Good? An Inferior Good is a good that someone will buy less of as their income increases

Inferior Good With A Rise in Income Good 2 is inferior u = 7.7958 7 Budget Line 2 u=12.204 u=19.216 q 6 1 5 Budget Line 1 4 Budget Line 0 3 2 1 1 2 3 4 5 6 7 8 q 2

Inferior Good With A Rise in Income I1 = p1 q1 + p2 q2 q1 Good 1 is inferior I0 = p1 q1 + p2 q2 q2

Changes in Price Budget line will rotate rather than shift Budget line will rotate around a fixed point on the axis of the good whose price does not change

Consider a decrease in the price of good 2 from $10 to $6 Initial equilibrium Subsequent equilibrium 2 4 6 8 10 12 14 16 1 3 5 7 q u = 10.99 u = 14.198 u = 8 Budget Line 0 Budget Line 1

The quantity of good 2 increases with a fall in its price The quantity of good 1 does not change with a fall in the price of good 2

Different utility function with different prices and income Initial prices and income p1 = 6, p2 = 4, I0 = 58 Initial equilibrium u=9.614 16 q 14 1 12 10 Budget Line 0 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

Now decrease p1 from $6 to $4 Initial prices and income p1 = 6, p2 = 4, I0 = 58 u=9.614 Budget Line 0 Initial equilibrium Subsequent equilibrium u=10.586 16 q 14 Budget Line 1 1 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

The quantity of good 1 increases with a fall in its price The quantity of good 2 decreases with a decrease in the price of good 1 The two goods are clearly substitutes in consumption

Using indifference curves to determine demand Initial prices and income p1 = 6, p2 = 3, I0 = 58. Initial Optimum q1 = 2.80, q2 = 13.733 u = 11.171 16 q 14 1 12 10 Budget Line 0 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

p2 q2 3.00 13.733 Raise the price of good 2 to 4.00

Using indifference curves to determine demand Next prices and income p1 = 6, p2 = 4, I0 = 58. Next Optimum q1 = 3.00, q2 = 10.00 u = 11.171 16 u=9.614 q 14 1 12 10 Budget Line 0 Budget Line 1 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

p2 q2 3.00 13.733 4.00 10.00 Raise the price of good 2 to 5.00

Using indifference curves to determine demand Next prices and income p1 = 6, p2 = 5, I0 = 58. Next Optimum q1 = 3.20, q2 = 7.76 u = 11.171 16 u=9.614 q 14 1 u=8.635 12 10 Budget Line 0 Budget Line 1 Budget Line 2 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

p2 q2 3.00 13.733 4.00 10.000 5.00 7.760 Raise the price of good 2 to 6.00

Using indifference curves to determine demand Next prices and income p1 = 6, p2 = 6, I0 = 58. Next Optimum q1 = 3.40, q2 = 6.267 u = 11.171 16 u=9.614 q 14 u=7.962 1 u=8.635 12 10 Budget Line 0 Budget Line 2 Budget Line 1 Budget Line 3 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

p2 q2 3.00 13.733 4.00 10.000 5.00 7.760 6.00 6.267

p2 q2 3.00 13.733 4.00 10.000 5.00 7.760 6.00 6.267 Demand for good 2 10 p 2 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

p2 q2 3.00 13.733 4.00 10.000 5.00 7.760 6.00 6.267 Demand for good 2 10 p 2 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

p2 q2 3.00 13.733 4.00 10.000 5.00 7.760 6.00 6.267 Demand for good 2 10 p 2 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

p2 q2 3.00 13.733 4.00 10.000 5.00 7.760 6.00 6.267 Demand for good 2 10 p 2 8 6 D2 4 2 2 4 6 8 10 12 14 16 18 20 q 2

Plot the price and quantity combinations as we go Increase in the price of good 2 16 u = 11.171 q 14 1 u=9.614 12 10 Budget Line 1 Budget Line 0 8 6 4 2 p2 = 3, q2 = 13.733 2 4 6 8 10 12 14 16 18 20 q 2 p2 = 4, q2 = 10.000 Demand for good 2 10 p 2 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

Plot some more price and quantity combinations Increase p2 some more u = 11.171 16 u=9.614 q 14 1 12 u=8.635 10 Budget Line 0 Budget Line 1 Budget Line 2 8 6 4 2 2 4 6 8 10 12 14 16 18 20 p2 = 5, q2 = 7.760 q 2 Demand for good 2 10 p 2 8 6 4 2 2 4 6 8 10 12 14 16 18 20 q 2

Plot some more price and quantity combinations Increase p2 some more u = 11.171 16 u=9.614 q 14 u=8.635 1 12 u=7.962 10 Budget Line 0 Budget Line 3 8 Budget Line 1 6 Budget Line 2 4 2 2 4 6 8 10 12 14 16 18 20 p2 = 6, q2 = 6.267 q 2 Demand for good 2 10 p 2 8 6 D2 4 2 2 4 6 8 10 12 14 16 18 20 q 2

Consumer theory and the law of demand The law of demand states that when the price of a good rises, and everything else remains the same, the quantity demanded of the good will fall.

Consumer theory and the law of demand When a good’s price changes, we can identify two separate effects on the quantity demanded When the price of a good falls, it becomes relatively less expensive, as compared to goods whose prices have not fallen The consumer will demand more of this good This is called the substitution effect

What is the Substitution Effect? When the price of a good falls, consumers will substitute it for other goods, which are now relatively more expensive When the price of a good rises, consumers will substitute from it to other goods, which are now relatively less expensive

Why ? The substitution effect of a price change arises from a change in the relative price of a good, And it always moves the quantity demanded in the opposite direction of the price change. With only the substitution effect, demand curves would always have negative slope Why ?

Because indifference curves slope down If we hold real income (utility) constant the effect of a price change on quantity will always be negative Consider then a change in price but hold real income constant

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 u = 10.99 16 u = 14.198 q 14 1 Old P New U 12 Budget Line 0 10 Budget Line 1 8 6 4 2 1 2 3 4 5 6 7 q2

The substitution effect Lower the price of q2, find the new utility level Give the consumer enough income to get to the new utility level without the price change At the new utility level, lower the price of q2

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 u = 10.99 16 u = 14.198 q 14 1 Old P New U 12 Budget Line 0 10 Budget Line 1 8 6 4 2 1 2 3 4 5 6 7 q2

What is the substitution effect? After changing price, hold utility fixed at the new utility level, and then evaluate the effect of the price change Because indifference curves slope down, the substitution effect is always negative

Price and demand another way After changing price, hold utility fixed at the old utility level, and then evaluate the effect of the price change Because indifference curves slope down, the substitution effect is always negative

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 u = 10.99 16 u = 14.198 q 14 1 12 Budget Line 0 10 Budget Line 1 8 New P, Old U 6 4 2 1 2 3 4 5 6 7 q 2

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 u = 10.99 16 u = 14.198 q 14 1 Old P New U 12 Budget Line 0 10 Budget Line 1 8 New P, Old U 6 4 2 1 2 3 4 5 6 7 q 2

The income effect When the price of a good falls, there is a change When a good’s price changes, we can identify two separate effects on the quantity demanded When the price of a good falls, there is a change in the overall purchasing power of the consumer The consumer will demand more or less of this good This is called the income effect

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 u = 10.99 16 u = 14.198 q 14 1 Old P New U 12 Budget Line 0 10 Budget Line 1 8 Substitution 6 4 Income 2 1 2 3 4 5 6 7 q 2

What is the Income Effect of a change in price? The income effect is the change in quantity demanded that arises because of the change in purchasing power caused by a price change

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Budget Line 1 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 u = 10.99 16 u = 14.198 q 14 1 Old P New U Income 12 Budget Line 0 10 8 Substitution 6 4 2 1 2 3 4 5 6 7 q 2

Some of the benefit of a price decline is that there is more money to spend on other goods u = 10.99 16 u = 14.198 q 14 1 Old P New U Income 12 Budget Line 0 10 Budget Line 1 8 6 4 2 1 2 3 4 5 6 7 q 2

this will increase the quantity demanded Some of the benefit of a price decline is that there is more money to spend on other goods For normal goods, this will increase the quantity demanded

The substitution and income effects work in the same direction (more q2) u = 10.99 16 u = 14.198 q 14 1 Old P New U Income 12 Budget Line 0 10 Budget Line 1 8 Substitution 6 4 2 1 2 3 4 5 6 7 q 2

What about an Inferior Good ? Consider a family of indifference curves 10 20 30 40 50 60 q2 q1 u1 u2 u3 u4 u5

q1 = 40 q2 = 15 60 50 40 30 20 10 Consider a set of prices and income Income = 66 p1 = 1.25 p2 = 1 q1 = 40 q2 = 15 Budget Line 0 10 20 30 40 50 60 q2 q1 52.8 u1 u2 u4

q1 = 34 q2 = 31 60 50 40 30 20 10 Consider a drop in p1 Income = 66 p1 = 1.25 p2 = 1 q1 = 40 q2 = 15 Income = 66 p1 = 1 p2 = 1 q1 = 34 q2 = 31 66 10 20 30 40 50 60 q2 q1 Budget Line 1 Budget Line 0 u2 u4

What has happened? Income = 66 p1 = 1.25 p2 = 1 q1 = 40 q2 = 15 Drop p1 Income = 66 p1 = 1 p2 = 1 q1 = 34 q2 = 31 Price falls and demand falls Is the law of demand false?

Let’s look closer 5 10 15 20 25 30 35 40 45 50 55 60 60 p1 falls q1 50 Budget Line 0 60 Budget Line 1 p1 falls q1 Old P New U 50 u = 2527.5 u = 2755.5 40 Income Substitution 30 20 10 5 10 15 20 25 30 35 40 45 50 55 60 q2

Again Budget Line 0 60 Budget Line 1 p1 falls q1 Old P New U 50 u = 2527.5 u = 2755.5 40 Income Substitution 30 20 10 5 10 15 20 25 30 35 40 45 50 55 60 q2

What is the Income Effect of a change in price? Some of the benefit of a price decline is that there is more money to spend on other goods For normal goods, this will increase the quantity demanded The substitution and income effects work in the same direction

What is the Income Effect of a change in price? For inferior goods, a decrease in income will decrease the quantity demanded The substitution and income effects work in opposite directions

If the income effect is large enough, a law of demand may not be satisfied for inferior goods

Example where the income effect is not so large I0 = p1 q1 + p2 q2 q2 q1 Drop p1 Substitution Income

Another Example Budget Line 0 u0 18 q1 Budget Line 1 Old P New U u1 16 14 12 Raise p1 10 Income Substitution 8 6 4 2 35 40 45 50 55 60 65 q2

Inferior goods could disobey the law of demand For normal goods, both substitution and income effects work together, causing the quantity demanded to move in the opposite direction of price. Normal goods always obey the law of demand For inferior goods, the substitution and income effects of a price change work against each other. The substitution effect moves quantity demanded in the opposite direction as price, while the income effect moves it in the same direction as price. Inferior goods could disobey the law of demand

Most goods are such a small proportion But, the substitution effect almost always dominates Most goods are such a small proportion of income that the income effect of a price change is usually small So most, if not all, inferior goods obey the law of demand

Consumers and markets The market demand curve is found by horizontally summing the individual demand curves of every consumer in the market

P D 20 0 16 2 12 4 8 10 4 20 Demand for Consumer 1 2 4 6 8 D1 5 10 15 20 2 4 6 8 D1 P D 20 0 16 2 12 4 8 10 Demand for Consumer 2 2 4 6 8 10 12 16 20 D2 4 20

P D 20 0 16 2 12 4 8 10 4 20 0 30 D12 Aggregate Demand for 2 Consumers 20 0 16 2 12 4 8 10 4 20 0 30 20 D12 15 10 5 5 10 15 20 25 30

P D1 D2 D 20 0 0 0 18 1 0 1 16 2 0 2 14 3 0 3 12 4 0 4 10 5 0 5 8 6 4 10 6 7 8 15 4 8 12 22 2 9 16 25 0 10 20 30

Three Consumers P D1 D2 D3 D 20 0 0 0 0 18 1 0 0 1 16 2 0 0 2 14 3 0 0 3 12 4 0 0 4 10 5 0 0 5 8 6 4 2 12 6 7 8 4 19 4 8 12 6 26 2 9 16 8 33 0 10 20 10 40

Demand for Consumer 1 5 10 15 20 2 4 6 8 D1 Demand for Consumer 2 2 4 6 8 10 12 16 20 D2 P D1 D2 D3 D 8 Demand for Consumer 3 2 4 6 8 10 D3 6 4 2 12 4 8 12 6 26 2 9 16 8 33 0 10 20 10 40

D Aggregate Demand P D1 D2 D3 D 20 0 0 0 0 16 2 0 0 2 10 5 0 0 5 20 0 0 0 0 16 2 0 0 2 10 5 0 0 5 8 6 4 2 12 4 8 12 6 26 2 9 16 8 33 0 10 20 10 40 5 10 15 20 4 8 12 16 24 28 32 36 40 D

End of Presentation

Extra Stuff

Decrease in the price of good 2 u = 8 2 4 6 8 10 12 14 16 1 3 5 7 q u = 10.99 u = 14.198 Budget Line 1 u = 15.4188 Budget Line 2 Budget Line 0

Decrease in the price of good 2 u = 8 2 4 6 8 10 12 14 16 1 3 5 7 q u = 10.99 u = 14.198 Budget Line 1 u = 15.4188 Budget Line 2 Budget Line 0

Different utility function with different prices and income Initial prices and income p1 = 6, p2 = 4, I0 = 58 Initial equilibrium u = 8 16 u=9.614 q 14 u=10.586 1 12 u=12.018 10 Budget Line 0 8 Budget Line 1 6 Budget Line 2 4 2 2 4 6 8 10 12 14 16 18 20 q 2

q q u = 11.171 16 u=9.614 14 u=7.962 u=8.635 12 10 Budget Line 0 2 4 6 8 10 12 14 16 18 20 q 2

q q u = 11.171 16 u=9.614 14 u=7.962 u=8.635 12 10 Budget Line 0 2 4 6 8 10 12 14 16 18 20 q 2

Demand for good 2 D2 Increase in the price of good 2 q q 16 14 12 10 8 u = 11.171 16 u=9.614 q 14 u=8.635 1 12 u=7.962 10 Budget Line 0 8 Budget Line 1 6 Budget Line 2 4 Budget Line 3 2 2 4 6 8 10 12 14 16 18 20 q 2 Demand for good 2 2 4 6 8 10 12 14 16 18 20 q p D2

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 u = 10.99 16 u = 14.198 q 14 1 Old P New U 12 Budget Line 0 10 Budget Line 1 8 New P, Old U 6 4 2 1 2 3 4 5 6 7 q 2

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 u = 10.99 16 u = 14.198 q 14 1 Old P New U 12 Budget Line 0 10 Budget Line 1 8 New P, Old U 6 4 2 1 2 3 4 5 6 7 q 2

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 14 u = 10.99 q 12 1 u = 14.198 10 Budget Line 1 Budget Line 0 8 Budget Line 3 6 4 2 1 2 3 4 5 6 7 8 q 2

1 2 3 4 5 6 7 8 Decrease in p holding u fixed at 10.99 14 u = 10.99 q 12 1 u = 14.198 10 Budget Line 0 Budget Line 1 8 Budget Line 3 6 4 2 1 2 3 4 5 6 7 8 q 2

Adjust income to keep utility at 10.99 with the new prices

Initial prices and income - p1 = 5, p2 = 10, I0 = 50 Initial equilibrium - q1 = 4, q2 = 3 Lower p2 from 10 to 5 New equilibrium - q1 = 4, q2 = 5 u = 10.99 16 u = 14.198 q 14 1 Old P New U 12 Budget Line 0 10 Budget Line 1 8 New P, Old U 6 4 2 1 2 3 4 5 6 7 q 2

Demand for Consumer 3 2 4 6 8 10 D3

Budget Line 0 60 Budget Line 1 q1 Budget Line 2 50 u = 2527.5 u =2620.5 u = 2755.5 u = 2471.5 u = 2842.5 40 30 20 10 5 10 15 20 25 30 35 40 45 50 55 60 q2

Budget Line 0 60 Budget Line 1 q1 Budget Line 2 50 u = 2527.5 u =2620.5 u = 2755.5 u = 2471.5 u = 2842.5 40 30 20 10 5 10 15 20 25 30 35 40 45 50 55 60 q2

60 50 40 30 20 10 q1 q2 Budget Line 0 Budget Line 1 Budget Line 2 u1 10 20 30 40 50 60 q2 q1 u1 u2 u3 u4 u5

What about an Inferior Good ? I0 = p1 q1 + p2 q2 q2 q1 Drop p1

u1 u0 18 q1 Budget Line 0 16 u2 Budget Line 1 14 Budget Line 2 12 10 8 6 4 2 35 40 45 50 55 60 65 q2