Properties Of Indifference Curve

We have seen the definition of Indifference Curve. Now we will discuss the the properties of Indifference Curve.

Property 1: Indifference Curves are Downward Sloping (Negative Slope)

An Indifference Curve always slopes downward from left to right, meaning it has a negative slope.

If the consumer gets more of Good X, they must give up some of Good Y to remain at the same level of utility. If both goods increase, utility would increase; which means moving to a higher Indifference Curve, not staying on the same indifference curve.

Mathematical Proof:

We know that along an IC, utility is constant:

U(X,Y)=k

Taking the total differential:

dU=MU_X \cdot dX +MU_Y \cdot dY =0

Solving:

\frac{dY}{dX}= – \frac{MU_X}{MU_Y}

By the assumption of non-satiation (more is better):

MU_X>0 \ \ \ and\ \ MU_Y>0

Therefore:

\frac{dY}{dX}= – \frac{MU_X}{MU_Y}<0

\frac{dY}{dX}<0

This confirms that the slope of Indifference Curve is negative i.e., it confirms that Indifference Curve is downward sloping.

Exception: If one of the goods is a “Bad” good i.e., something the consumer dislikes, then the Indifference Curve could be upward sloping. But in standard analysis, we deal with goods, so Indifference Curve is downward sloping.

Property 2: Higher Indifference Curve Represents Higher Level of Utility

A consumer always prefers a higher indifference curve to a lower one because higher Indifference Curve represents more utility.

Let there are two Indifference Curves $IC_1$ and $IC_2$ :

$IC_1 = U(X,Y)=k_1$

$IC_2 = U(X,Y)=k_2$ where $k_2>k_1$

Consider two points:

Point A on $IC_1 : (X_1, Y_2)$ with $U=k_1$

Point B on $IC_2 : (X_1, Y_2)$ with $U=k_2$

Since $k_2>k_1$ :

U(X_2, Y_2)>U(X_1,Y_1)

By the assumption of monotonicity (more is better), a bundle with more goods is preferred.

If we take a point $B = (X_2,Y_2)$ that lies to the upper right of point $A = (X_1,Y_1)$ , meaning $X_2 \ge X_1$ and $Y_2 \ge Y_1$ (at least one strict), then:

U(X_2, Y_2) > U(x_1,Y_1)

k_2>k_1 \implies IC_2 \succ IC_1

Higher Indifference Curve = More Satisfaction = More Preferred

Property 3: Indifference Curve Never Intersect Each Other

Two Indifference Curves cannot cut or intersect each other.

This is one the most important properties, and it follows from the logic of consistency and transitivity of preferences.

Mathematical Proof (by contradiction):

Suppose $IC_1(U = k_1)$ and $IC_2 (U=k_2)$ , where $k_2 > k_1$ intersect at point $P$ .

Let:

Point $P$ = Intersection point (on both $IC_1$ , and $IC_2$ )

Point $A$ = on $IC_1$ only

Point $B$ = on $IC_2$ only

From $IC_1 :$ Both $P$ and $A$ give the same utility:

U(P) = U(A) = k_1 …………………………….. (i)

From $IC_2 :$ Both $P$ and $B$ give the same utility:

U(P) = U(B) = k_2 …………………………….. (ii)

From $(i)$ and $(ii)$ :

U(A)=U(P)=U(B)

\implies U(A) = U(B)

But this means $A$ and $B$ give the same utility, which means $A$ and $B$ should be on the same indifference curve.

However, we said $B$ is on $IC_2$ (higher IC with more goods/higher utility) and $A$ is on $IC_1$ .

If $B$ has more of at least on e good than $A$ (which it does since it’s on a higher IC), then:

U(B)>U(A)

This contradicts our result that $U(A) = U(B)$ .

Contradiction! which implies $IC_1$ and $IC_2$ cannot intersect

This proves that indifference curves can never intersect.

Property 4: Indifference Curves are Convex to the Origin

An Indifference Curve is convex to the origin, which means it is “bowed inward” toward the origin. This is due to the Diminishing Marginal Rate of Substitution (DMRS).

When a consumer has a lot of Y and little X, they are willing to give up more units of Y to get some one more unit of X. But as the consumer gets more and more of X, they become less willing to give up Y. This declining willingness to substitute is called Diminishing MRS.

Mathematically;

A curve is convex to the origin if its second derivative is positive:

\frac{d^{2}Y}{dX^{2}} >0

We know that;

\frac{dY}{dX} = -MRS = – \frac{MU_X}{MU_Y}

For convexity (diminishing MRS):

\frac{d(MRS)}{dX}<0

This means as X increases, MRS falls:

\frac{d}{dX}\Big( \frac{MU_X}{MU_Y} \Big) < 0

In case of Cobb-Douglas Utility Function:

MRS = \frac{\alpha Y}{\beta X}

\frac{d(MRS)}{dX} = \frac{\alpha}{\beta} \cdot \frac{d (Y/X)}{dX}

Since Y decreases as X increases along IC ( $\frac{dY}{dX} <0$ ), the ratio $Y/X$ falls as X increases:

\frac{d(Y/X)}{dX} <0

Therefore:

\frac{d(MRS)}{dX} = \frac{\alpha}{\beta} \cdot \frac{d(Y/X)}{dX} <0

\frac{d(MRS)}{dX}<0 \implies \text{Diminishing MRS} \implies \text{IC is convex to the Origin}

Special Cases

Shape of IC	Type of Goods	MRS
Convex (Normal Shape)	Normal Goods (Imperfect Substitutes)	Diminishing
Straight Line (Linear)	Perfect Substitutes	Constant
Right Angle (L-Shaped)	Perfect Complements	Zero (Kinked)
Concave	Goods with increasing MRS	Increasing (Unusual)

Property 5: Indifference Curves Do Not Touch the Axes (Generally)

In standard analysis, an Indifference Curve does not touch either axis because we assume the consumer consumes positive quantities of both goods.

The domain of the Indifference curve is:

\{(X,Y): X>0,Y>0\}

If the indifference curve touches the X-axis, Y = 0, meaning the consumer consumes none of good Y, which violates the assumption that both goods are needed to give utility (as in Cobb-Douglas where $U = X^{\alpha}Y^{\beta} = 0$ ) if Y = 0.

There is an Exception: If goods are perfect substitutes, the Indifference curve is a straight line that can touch both axis.

Property 6: Indifference Curves are Smooth and Continous

Indifference Curves are drawn so smooth, continuous curves assuming that goods are perfectly

The utility function $U(X,Y)$ must be:

1. Continuous i.e., no sudden jumps.

2. Differentiable i.e., partial derivatives $\frac{\partial U}{\partial X}$ and $\frac {\partial U}{\partial Y}$ must exist.

This ensures that $MRS = \frac {MU_X}{MU_Y}$ is defined at every point along the curve.

U(X,Y) \in C^2 \ \ \text{(Twice continuously differentiable)}

All the properties can be summarized in a table below:

\frac{dY}{dX}= - \Big( \frac{MU_X}{MU_Y} \Big) < 0

Property	Statement	Mathematical Condition
Downward Sloping	IC has negative slope	$\frac{dY}{dX}= - \Big( \frac{MU_X}{MU_Y} \Big) < 0$
Higher IC = Higher Utility	Upper IC is preferred	$k_2 > k_1 \implies IC_2 \succ IC_1$
ICs Never Intersect	No two ICs can cross	By contradiction and transitivity
Convex to Origin	Bowed inward shape	$\frac{d^{2}Y}{dX^{2}} > 0$ , DMRS
Do Not Touch Axes	Consumer needs both goods	$X > 0 \ \text{and} \ Y>0$
Smooth and Continuous	No kinks or breaks	$U(X,Y) \ \text{is} \ C^2 \ \text{differentiable}$

Properties of Indifference Curve