**Perloff 5.1-3; WB 7, 8 and 9.**

Using the solution to the consumer's problem, we will derive the Demand curve from consumer choices and re-examine a few key Demand shifters in greater detail: the good's price, other goods' prices and income. Importantly, we will see that a consumer's price response can be decomposed into two “effects.”

For the consumer's income response, we will look at two new curves — the Income Consumption Curve (ICC) and the Engel curve. Finally, we can look at cross-price and own-price responses simultaneously using the Price Consumption Curve (PCC).

# Market Demand from Consumer Choice

### Deriving individual Demand

Finding a consumer's individual Demand X^{*} for a good as a function of its price P_{X} is an easy application of the Consumer Choice model.

- Derive a formula for the consumer's Demand of one good given a fixed price for the other good, fixed income and their preferences/utility (for any of the three preference types discussed earlier).

### Aggregating Demand

The market Demand curve seen in the supply-and-demand model — also called the *aggregate Demand* — is the sum of individual Demand curves. See Perloff, p20 and the Problems referred to in the margin on that page.

- Add two given Demand functions.
- Add two given inverse Demand functions.

# Price response

We can decompose the price response in X^{*} as the sum of two effects: the substitution effect and the income effect. Why would we want to do this?

“By decomposing the change in demand into two effects, economists gain extra information that they can use to answer questions about whether inflation measures are accurate and whether an increase in tax rates will raise tax revenue.” —Jeffrey M. Perloff

Given a Cobb-Douglas utility function, income and a price for Y, evaluate the effect of a change in the price of X:

- What is the total effect on the quantity of X demanded?
- What part of this effect is due to the substitution effect; and what part to the income effect?

The substitution and income effects add up to the total effect. The substitution effect is always negative. The income effect is negative for inferior goods and positive for normal goods.

- Know what it means for the income effect to be positive/negative.

The price elasticity of demand can be decomposed in a similar way: $\varepsilon = \varepsilon^*-\theta\cdot\xi$. The “elasticity of compensated demand” $\varepsilon^*$ captures the substitution effect; and $\theta\cdot\xi$, the share of income spent on the good^{1} times the income elasticity $\xi$, captures the income effect. This famous decomposition is known as the *Slutsky equation*.

### Giffen goods

A Giffen good violates the Law of Demand. This occurs because the income effect is negative and stronger than the substitution effect.

- Know what a Giffen good is, and why it is possible in the Consumer Choice model.

# Income and cross-price responses

Change in… | x-axis | y-axis | name of curve |
---|---|---|---|

I | X^{*} |
I | Engel for X |

I | X^{*} |
Y^{*} |
ICC |

P_{X} |
X^{*} |
P_{X} |
Demand curve |

P_{X} |
X^{*} |
Y^{*} |
PCC for P_{X} |

P_{X} |
Y^{*} |
P_{X} |
XY cross-price response |

Curves representing the solution to the consumer's problem in the Consumer Choice model.

Classification of goods based on income and own-price responses.

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The curves described in the table on the right are also straightforward applications of the Consumer Choice model.

### ICC

The ICC shows the income responses of both goods simultaneously. The path tells us if each good is *inferior* or *normal*; see Perloff, Figure 5.3. It is impossible for both to be inferior, and Giffen goods are always inferior.

We can also see if a normal good is a *necessity* or a *luxury* by comparing the path with a line through the origin. If one good is inferior, the other must be a luxury.

A good can change from one of these to another as income rises.

- Find a formula for the ICC given fixed prices and preferences/utility. Also, graph it.__
- Given a graph of an ICC, distinguish when each good is inferior/normal and a necessity/luxury. This might be multiple choice with statements like "X is normal then inferior, while Y is always a luxury."

- Know which such statements cannot be true. For example, "X is inferior, while Y is a necessity."

### Engel curve

These isolate the effect of income on a single good. The Engel curve bends backwards when the good is inferior. We can again distinguish necessities from luxuries using a line through the origin.

- Find a formula for the Engel curve given fixed prices and preferences/utility. Also, graph it.
- Given a graph of an Engel curve, distinguish when the good is inferior/normal and a necessity/luxury. This might be multiple choice with statements like "X is normal then inferior."

# More resources

- Perloff, Chapter 5 ``Applying Consumer Theory''
- Quizzes
- Applications

- Sloman's illustrative graphs
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