## Demand function

• "Optimal bundle" for generic prices and income = Demand Function.

• The "optimal bundle" changes when income or prices change: that is why we call it demand function.

## Demand function

• For a specific utility (preferences) and given set of income and prices, we have learned how to find the optimal bundle.

• If you solve this, so called, consumer's problem for generic income and prices $(p_1,p_2,m)$, what you obtain is the demand function: $x_1^{*}(p_1,p_2,m)$.

• For example, if utility is Cobb-Douglas, then $x_1^{*} (p_1,p_2,m) = \frac{a m}{ (a+b) p_1}$

A consumer's demand function indicates the optimal choice for a given set of prices and income.

## Changes in income

$x_1 = x_1(p_1,p_2,m)$

• How does the optimal consumption of $x_1$ changes with changes in income? $\frac{\partial x_1}{ \partial m }$ Vs. $0$

• A normal good: its consumption increases when income increases.

• $\frac{\partial x_1}{ \partial m } > 0$
• Do graph
• An inferior good: its consumption decreases when income increases.

• $\frac{\partial x_1}{ \partial m } < 0$
• Do graph

## Engel Curve

• The Engel Curve maps each level of income to the optimal consumption of a good, holding prices constant.

## Changes in prices

• $\Delta p_1$ tilts or pivots the budget line.

• How does the optimal consumption of $x_1$ changes with changes in $p_1$? $\frac{\partial x_1}{ \partial p_1 }$ Vs. $0$

• Ordinary good: its consumption decreases when its price increases.

• $\frac{\partial x_1}{\partial p_1} < 0$
• Do graph
• Giffen good: its consumption increases when its price increases.

• $\frac{\partial x_1}{\partial p_1} > 0$
• Do graph

## Demand curve

• Demand curve: describes the relationship between the optimal choice of a good and its price, with income and other prices held constant.

## Substitutes and Complements

• Gross Substitutes: $\frac{\partial x_1}{ \partial p_2 } > 0$

• Gross Complements: $\frac{\partial x_1}{ \partial p_2 } < 0$