Utility: Derivation of an Individual Demand Curve

This section details the process of deriving an individual consumer's demand curve, a fundamental concept in microeconomics. We will explore the relationship between quantity consumed and the utility gained, leading to the concept of marginal utility and its role in shaping consumer choices.

1. Understanding Utility

Utility represents the satisfaction or happiness a consumer derives from consuming a good or service. It is subjective and varies from person to person. Economists often use the concept of marginal utility to analyze consumer behavior.

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

2. The Law of Diminishing Marginal Utility

A core principle in economics is the Law of Diminishing Marginal Utility. This law states that as a consumer consumes more and more units of a good or service, the additional satisfaction gained from each additional unit will eventually decrease.

Mathematically, this can be represented as:

$$ \frac{dMU}{dQ} < \frac{dMU}{dQ} < ... < \frac{dMU}{dQ} $$

Where Q represents the quantity consumed.

3. Deriving the Individual Demand Curve

To derive an individual demand curve, we need to understand how a consumer makes choices based on utility maximization. A consumer will allocate their budget to maximize their total utility.

1. Budget Constraint: A consumer's budget is limited. The budget constraint represents all possible combinations of goods and services the consumer can afford given their income and prices.
2. Utility Maximization: Consumers aim to maximize their total utility subject to their budget constraint.
3. Optimal Consumption Bundle: The optimal consumption bundle is the combination of goods and services that provides the highest level of utility given the budget constraint.

By analyzing the consumer's choices at different price levels, we can derive their individual demand curve.

4. Illustrative Example: A Simple Utility Maximization Problem

Consider a consumer who consumes two goods: X and Y. The utility derived from consuming X (U_X) and Y (U_Y) depend on the quantities consumed. We can assume a simple utility function for each good.

Let:

• U_X = $ \sqrt{x}$ (Square root utility of X)
• U_Y = $ y$ (Linear utility of Y)

The consumer's objective is to maximize their total utility: $U = \sqrt{x} + y$ subject to a budget constraint.

Assume the prices of X and Y are $P_X$ and $P_Y$ respectively, and the consumer's income is $I$. The budget constraint is:

$$ P_X x + P_Y y = I $$

To derive the demand curve, we can use the method of Lagrange multipliers or solve the budget constraint for one of the variables and substitute it into the utility function. For simplicity, let's solve for y:

$$ y = \frac{I - P_X x}{P_Y} $$

Substitute this into the utility function:

$$ U = \sqrt{x} + \frac{I - P_X x}{P_Y} $$

Now, differentiate U with respect to x and set the derivative equal to zero to find the optimal quantity of x (x*):

$$ \frac{dU}{dx} = \frac{1}{2\sqrt{x}} - \frac{P_X}{P_Y} = 0 $$

Solving for x*:

$$ \frac{1}{2\sqrt{x}} = \frac{P_X}{P_Y} $$ $$ \sqrt{x} = \frac{P_Y}{2P_X} $$ $$ x = \frac{P_Y^2}{4P_X^2} $$

Substitute x* back into the budget constraint to find the optimal quantity of y (y*):

$$ y* = \frac{I - P_X (\frac{P_Y^2}{4P_X^2})}{P_Y} = \frac{I - \frac{P_Y^2}{4P_X}}{P_Y} = \frac{4IP_X - P_Y^2}{4P_X P_Y} $$

Therefore, the individual demand curve for X is:

$$ x = \frac{P_Y^2}{4P_X^2} \quad \text{and} \quad y = \frac{4IP_X - P_Y^2}{4P_X P_Y} $$

This derived demand curve shows the relationship between the price of X and the quantity of X the consumer is willing and able to purchase, given their income and the price of Y.

5. Graphical Representation

6. Conclusion

Deriving an individual demand curve involves understanding the concept of utility maximization and the limitations imposed by a consumer's budget. The law of diminishing marginal utility plays a crucial role in shaping consumer choices. The resulting demand curve illustrates the relationship between the price of a good and the quantity consumers are willing to purchase.

Variable	Description
Utility	Satisfaction derived from consuming a good or service.
Marginal Utility (MU)	Additional satisfaction gained from consuming one more unit.
Law of Diminishing Marginal Utility	The additional satisfaction gained from each additional unit decreases.
Budget Constraint	All possible combinations of goods and services a consumer can afford.
Demand Curve	Relationship between the price of a good and the quantity consumers are willing to purchase.