Utility: Derivation of an Individual Demand Curve

This section details the derivation of an individual consumer's demand curve, building upon the concept of utility. We will explore how changes in price affect the quantity demanded by an individual consumer, assuming utility maximization.

1. The Concept of Utility

Utility represents the satisfaction or happiness a consumer derives from consuming a good or service. It's a subjective measure and is often assumed to be increasing in the quantity consumed, although this isn't always the case (diminishing marginal utility). We assume consumers aim to maximize their total utility, given their budget constraint.

2. The Consumer's Budget Constraint

The budget constraint represents the limit of goods a consumer can purchase given their income and the prices of those goods. It's a fundamental concept in microeconomics.

Let:

• $P_1$ = Price of good 1
• $P_2$ = Price of good 2
• $I$ = Consumer's Income
• $Q_1$ = Quantity of good 1 consumed
• $Q_2$ = Quantity of good 2 consumed

The budget constraint can be represented as:

$$P_1Q_1 + P_2Q_2 = I$$

3. Utility Maximization and the Marginal Utility Concept

Consumers allocate their income to maximize their utility. This occurs when the ratio of the marginal utility of each good to its price is equal for all goods.

Marginal Utility (MU) is the additional utility gained from consuming one more unit of a good.

$$MU_1 = \frac{\Delta U}{ \Delta Q_1} \quad \text{and} \quad MU_2 = \frac{\Delta U}{ \Delta Q_2}$$

The optimal consumption bundle is found where:

$$\frac{MU_1}{P_1} = \frac{MU_2}{P_2}$$

4. Deriving the Demand Curve

To derive the individual demand curve for good 1, we need to consider how changes in the price of good 1 affect the quantity demanded, holding the consumer's income and the price of good 2 constant.

We can use the tangency condition (as described above) to express the optimal consumption levels of both goods in terms of their prices and the consumer's income.

From the tangency condition, we can derive the following relationship:

$$ \frac{MU_1}{P_1} = \frac{MU_2}{P_2} \implies MU_1 = \frac{P_1}{P_2} MU_2$$

5. Expressing Demand as a Function of Price

We can substitute the utility maximization condition into the budget constraint to solve for $Q_1$ in terms of $P_1$, $P_2$, $I$, and $MU_1$ and $MU_2$. This will give us the demand function for good 1.

The process involves the following steps:

1. Express $MU_1$ and $MU_2$ in terms of $Q_1$ and $Q_2$ respectively.
2. Substitute these expressions into the tangency condition.
3. Substitute the expressions for $MU_1$ and $MU_2$ into the budget constraint.
4. Solve for $Q_1$ in terms of $P_1$, $P_2$, $I$, $MU_1$ and $MU_2$.

The resulting equation will be a function of $P_1$ and $I$, representing the individual demand curve for good 1.

6. Demand Curve Properties

The derived demand curve will typically exhibit the following properties:

• **Downward Sloping:** As the price of good 1 increases, the quantity demanded of good 1 decreases (assuming the price of good 2 remains constant).
• **Relationship to Marginal Utility:** The demand curve reflects the diminishing marginal utility of good 1. Consumers are willing to buy less of a good as the marginal utility they derive from it decreases.
• **Influence of Income:** Changes in income will affect the quantity demanded of good 1. Whether the demand increases or decreases depends on whether good 1 is a normal good or an inferior good.

Variable	Description
$P_1$	Price of good 1
$P_2$	Price of good 2
$I$	Consumer's Income
$Q_1$	Quantity of good 1 consumed
$Q_2$	Quantity of good 2 consumed
$MU_1$	Marginal Utility of good 1
$MU_2$	Marginal Utility of good 2

7. Limitations

This derivation assumes rational consumers who aim to maximize utility. In reality, consumer behavior can be influenced by various factors, including cognitive biases and incomplete information. Therefore, the derived demand curve is a simplification of real-world consumer behavior.