Indifference Curve - Economics Live

“An Indifference Curve is a curve that shows all possible combinations of two goods that give a consumer the same level of satisfaction, so that the consumer is indifferent between any two combinations on the same curve.”

In other words:

“The consumer does not prefer one point over another on the same indifference curve as they are equally happy at every point on it”.

Suppose a student likes good X and Y.

Combination	Good X	Good Y	Utility
A	1	10	Same
B	2	7	Same
C	3	5	Same
D	4	4	Same

All the four combinations above gives same level of satisfaction to the consumer. If we plot both the goods (X on horizontal axis and Y on vertical axis), we will get an Indifference Curve.

Assumption of Indifference Curve

1. Ordinal Utility: The consumer can rank their preferences like A over B or B over C.

2. Rationality: The consumer always tries to maximize their satisfaction.

3. Two Goods: We consider only two goods X and Y to keep this analysis simple.

4. Transitivity: If a consumer prefers bundle A over B, and B over C, then they prefer A over C.

“Mathematically: if $A \succ B$ and $B \succ C$ , then $A \succ C$ “

5. Non-Satiation (More is better): The consumer always prefers more of both goods. This is also called the Monotonicity assumption.

6. Diminishing Marginal Rate of Substitution (MRS): As the consumer gets more of one good, they are willing to give up less and less of the other good.

Mathematical Interpretation of Indifference Curve

Lets assume consumers utility depends on quantity of good X and good Y:

U = f(X,Y)

Where:

$U$ = Total Utility which shows level of satisfaction

$X$ = Quantity of good X

$Y$ = Quantity of good Y

As we know, an Indifference Curve represents all combinations $(X,Y)$ that give the same level of satisfaction; Lets say $k$

f(X,Y) = k

Where, $k$ is a constant.

This is essentially a level curve or contour line of the utility function.

So mathematically:

IC = \{(X,Y) \in \mathbb{R}^{2}_{+} : U(X,Y)=k\}

We can read this as: “The set of all bundles (X,Y) in the positive quadrant such that utility equals ‘k’.“

Indifference Curve in case of Cobb-Douglas Utility Function

As we know, Cobb-Douglas Utility Function is written as;

U=f(X,Y)= X^{\alpha} \cdot Y^{\beta}

Where $\alpha$ and $\beta$ are positive constants $(\alpha, \beta > 0)$

For an Indifference Curve, we set $U=k$ (constant):

X^{\alpha} \cdot Y^{\beta} = k

Solving for $Y$ (to express IC as $Y$ in terms of X):

Y^{\beta} = \frac{k}{X^{\alpha}}

Y = \left(\frac{k}{X^{a}}\right)^{\frac{1}{\beta}}

= k^{\frac{1}{\beta}} \cdot X^{\frac{-\alpha}{\beta}}

This equation tells us that:

– As X increases, Y decreases which shows the negative relationship and confirms the downward sloping nature of Indifference Curve.

– Different values of $k$ give different curves which forms Indifference Map.

Lets take a Special Case, When $\alpha = \beta = 1/2$

U = (X,Y) = X^{\frac{1}{2}} \cdot Y^{\frac{1}{2}} = \sqrt{XY}

For Indifference Curve;

\sqrt{XY} = k

XY = k^{2}

Y=\frac{k^{2}}{X}

This is a Rectangular Hyperbola, which is the typical shape of an Indifference Curve.

The Marginal Rate of Substitution (MRS)

This is one of the most important concepts in IC analysis.

“Marginal Rate of Substitution (MRS) is the rate at which a consumer is willing to give up on one good to get one more unit of another good, while remining on the same indifference curve i.e., same level of utility.“

Mathematically:

MRS_{XY} = -\frac{dY}{dX}\bigg|\ _{U=\text{constant}}

The negative sign is used to make MRS a positive number (since $\frac{dY}{dX}$ is negative along the IC).

How to calculate MRS using Partial Derivatives?

Since $U=f(X,Y) = k$ , we take the total differential:

dU=\frac{\partial U}{\partial X} \cdot dX+ \frac{\partial U}{\partial Y} \cdot dY = 0

Since utility is constant along Indifference Curve, $dU=0$ :

\frac{\partial U}{\partial X} \cdot dX+\frac{\partial U}{\partial Y} \cdot dY =0

\frac{\partial U}{\partial Y} \cdot dY = – \frac{\partial U}{\partial X} \cdot dX

\frac{dY}{dX}= – \frac{\partial U/\partial X}{\partial U/ \partial Y} = – \frac{MU_{X}}{MU_{Y}}

Therefore:

MRS_{XY} = – \frac{dY}{dX} = \frac{MU_X}{MU_Y}

Where:

$MU_X$ = Marginal Utility of X = $\frac{\partial U}{\partial X}$

$MU_Y$ = Marginal Utility of Y = $\frac{\partial U}{\partial Y}$

Key Insight: MRS equals the ratio of Marginal utilities of the two goods

MRS for Cobb-Douglas Function:

Let,

U=X^{\alpha} \cdot Y^{\beta}

MU_X = \frac{\partial U}{\partial X} = \alpha X^{\alpha -1}Y^{\beta}

MU_Y=\frac{\partial U}{\partial Y}=\beta X^{\alpha}Y^{\beta-1}

MRS_{XY}=\frac{\alpha Y}{\beta X}

Which means, as X increases, Y decreases, so MRS decreases. This confirms the Diminishing MRS.

Indifference Map

A single indifference curve represents one level of satisfaction or utility. But a consumer has many levels of satisfaction. The collection of all indifference curves for a consumer is called an Indifference Map.

U_1 < U_2 <U_3 \implies IC_1<IC_2<IC_3