6 - California State University, Northridge



6.a Indifference curves

Bundles

·A "bundle" consists of a combination of goods that a consumer can utilize. Assuming there are only two goods, a bundle is comprised of certain amount of good X and a certain amount of good Y. For example, there may be two combinations of X & Y termed bundle A and bundle B. Bundle A has XA number of good X and YA number of good Y. Similarly, bundle B has XB number of good X and YB number of good Y.

Assumptions about consumer behavior

1. Completeness: A consumer can say that either A is preferred to B, B is preferred to A, or A and B are equally preferred. Shorthand notation for this will be A > B, B > A, A = B, respectively.

2. Dominance (more is preferred to less): A > B if XA is greater than XB and YA is greater than YB, OR if XA is greater than XB while YA equals YB.

3. Transitivity: if A > B and if B > C, then A > C.

Utility functions and Indifference curves

A utility function shows the level of utility a consumer derives from a particular bundle of X & Y. An example of a utility function is the Cobb-Douglas utility function:

U = Xa·Yb

U stands for a numerical value of utility and X & Y stand for the number of the goods in the bundle. Say U = X0.5·Y0.5 what combinations of X & Y (bundles) yield a utility level of 10?

Bundle X Y

A 1 100

B 2 50

C 3 33 1/3

D 4 25

All bundles satisfy the equation 10 = X0.5·Y0.5. Therefore, A=B=C=D, and the consumer is indifferent between the bundles because they all yield the same level of utility.

Showing these bundles on a graph forms an indifference curve.

Bundle N, shown above, has more of good X than bundle B and is therefore preferred since more is preferred to less. Since the consumer is indifferent between bundles A,B, C and D, bundle N is preferred to A,C and D as well. Bundle N must lie on a higher indifference curve because it is preferred to A,B,C and D.

Properties of IC's

1. Must have negative slope. If IC's had a positive slope, point B would have more of both X and Y than point A, and thus be preferred by our assumption that more is preferred to less. Therefore, they could not be on the same IC curve because the consumer would not be indifferent between the two.

2. IC's cannot intersect. If they were to intersect, we would have a bundle (at the intersection) that produces two different levels of utility.

3. There are infinite number of IC's. We can always add one more of good X or Y infinitely and the bundle would raise utility, moving to a higher IC curve.

Marginal Rate of Substitution (MRS): The maximum amount of Y a consumer is willing to give up to get one more unit of X. The MRS shows the number of good Y the consumer would trade to get one more of good X.

MRS = (X / (Y = slope of IC curve

A diminishing MRS implies that as X increases, the slope of the IC curve gets flatter. In other words, as we get more and more of X, we are willing to trade less and less of good Y to get the next unit of X. This means that IC curves (of the Cobb-Douglas form) are convex.

Marginal Utility(MU) & MRS

MUx is the marginal utility of good X; the utility gained from having one more unit of X. MUy is the marginal utility of good Y; the utility gained from having one more unit of Y. To find the marginal utility, take the partial derivative of the utility function with respect to X and Y.

MUx = (total utility / (X = (U / (X

MUy = (total utility / (Y = (U / (Y

The marginal rate of substitution (MRS) is the marginal utility of X divided by the marginal utility of Y.

MRS = MUx / MUy

Therefore, in order to find the marginal rate of substitution (MRS) we follow three steps. First, take the partial derivative of the utility function with respect to X to get MUx. Then take the partial derivative of the utility function with respect to Y to get MUy.

Specific Utility Functions

The first type of utility function is the Cobb-Douglas utility function. It has the form U = Xa·Yb. The marginal rate of substitution for the Cobb-Dpuglas utility function is MRS = (a/b)(Y/X). For example, say our function is U= X0.5Y0.5. First, we take the partial derivative of U with respect to X to get MUx.

(U/(X = 0.5X-0.5Y0.5.

Next, we take the partial derivative with respect to Y to get MUy.

(U/(Y = 0.5X0.5Y-0.5.

Dividing MUx by MUy we get

MRS =

The next form of utility function is called the perfect substitute form. This is when the two goods X & Y can be substituted interchangeably and would make no difference to the consumer. An example of perfect substitutes is Mobil gasoline and Exxon gasoline. Presuming the prices are the same and there is no unusual preference for gasoline name brand, one gallon of Mobil gas can be substituted for one gallon of Exxon gas and the consumer would be completely indifferent. The mathematical form of perfect substitutes utility function is

U = a·X + b·Y.

The marginal utility for X and Y are

MUx = a MUy = b.

The MRS = a/b which is a constant. Therefore, the indifference curve for perfect substitutes is a straight line. The indifference curve for perfect substitutes is shown in Diagram 6.a.2.

The next form of utility function is for goods that are called perfect compliments. An example of two goods that are perfect compliments is a right shoe and a left shoe. Having an equal number of each good raises utility. If you have one right shoe and one left shoe, your utility would be the same as having one right shoe and twenty left shoes, since those extra left shoes do not have a complimentary right shoe. The utility function for perfect compliments is in the form of

U = min(X,Y).

This means our value of utility is the minimum value between X and Y. For example, say we had 2 X and 1 Y; the utility function would look like U = min(2,1) = 1 since one is the minimum value. Our utility would not raise to two until we had two of both X and Y. The graph of the indifference curve for perfect compliments is shown in diagram 6.a.3.

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[pic]

Diagram 6.a.1

Diagram 6.a.1 shows bundles A,B,C and D which all produce a utility level of 10. Since they all produce the same utility, they are indifferent to the consumer, and comprise an indifference curve (IC). Bundle N has more of both X and Y, so it must be on a higher IC

0.5X-0.5Y0.5 Y

0.5X0.5Y-0.5 X

=

[pic]

Diagram 6.a.2 shows the IC for perfectly substitutable goods. The MRS for perfect substitutes is MRS= a/b which is a constant. Since MRS = slope of IC, a constant MRS means the IC curve is a straight line

[pic]

Diagram 6.a.3 shows the IC for perfectly complimentary goods. Adding one more right shoe without adding another left shoe does not raise utility and is on the same indifference curve

Diagram 6.a.2

Diagram 6.a.3

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