Marginal Utility (MU) and Marginal Rate of Substitution (MRS) Microeconomic Principles (ECON201 PDF

Preview

Summary

Marginal Utility (MU) and Marginal Rate of Substitution (MRS) Microeconomic Principles (ECON201) Dr. Fernando Aragon Summer 2013 These notes review two key concepts in consumer theory: marginal utility and marginal rate of substitution. 1 Marginal Utility (MU) The marginal utility is the change in u...

Description

Marginal Utility (MU) and Marginal Rate of Substitution (MRS) Microeconomic Principles (ECON201) Dr. Fernando Aragon Summer 2013

These notes review two key concepts in consumer theory: marginal utility and marginal rate of substitution.

1

Marginal Utility (MU)

The marginal utility is the change in utility associated with a small change in the amount of one of the goods consumed holding the quantity of the other good fixed. Given the above definition, we can write the marginal utility with respect to good 1 (M U1 ) as the ratio:

∆U ∆x1 U (x1 + ∆x1 , x2 ) − U (x1 , x2 ) = ∆x1

M U1 =

(1) (2)

where ∆ represents the change of a variable i.e., ∆U means the change in Utility, while ∆x1 the change in x1 . Note that M U1 measures the rate of change of Utility given a change in x1 , keeping all the other variables (x2 ) constant. This definition is easier to understand using calculus. Let us assume that the change ∆x1 is very small , so small that in the limit converges to zero (lim ∆x1 → 0). In that case, expression (2) becomes the definition of the partial derivative of U (x1 , x2 ) with respect to x1 . Hence:

1

∂U (x1 , x2 ) ∂x1 ∂U (x1 , x2 ) M U2 = ∂x2

M U1 =

Examples: 1. Perfect substitutes: U (x1 , x2 ) = αx1 + βx2 Taking partial derivatives we obtain that: M U1 = α and M U2 = β xβ2 and M U2 = βxα1 xβ−1 2. Cobb-Douglas: U (x1 , x2 ) = xα1 xβ2 . So, M U1 = αxα−1 2 1 3. Perfect complements: U (x1 , x2 ) = min(αx1 , βx2 ). Note that in this case the utility function is not differentiable. For that reason, we cannot take first derivatives. But, we can use the definition of marginal utility to find out that in the kink of the indifference curve (i.e. when αx1 = βx2 ) the marginal utilities are: M U1 = M U2 = 0.

2

Marginal Utility and Marginal Rate of Substitution (MRS)

Marginal utility has a serious problem: it changes when applying a monotonic transformation to the utility function. For example, consider an individual with preferences given by U (x1 , x2 ) = x1 + x2 . Note that both goods are perfect substitutes for the individual and that M U1 = M U2 = 1. Now, let us take a monotonic transformation. In particular, let us multiply the utility times 2. We obtain that preferences are now represented by 2x1 + 2x2 . This new function represents the same ranking of consumption bundles, but now M U1 = M U2 = 2. Even though the marginal utilities per se are useless, their ratio is still informative of consumer behavior. In particular, it represents the slope of the indifference curve, and can tell us the rate at which a consumer is willing to substitute between two goods M RS = −

M U1 M U2

To see this consider an indifference curve that give the consumer a level of utility of k. The equation that describes this indifference curve is, by definition,: U (x1 , x2 ) = k 2

We can take total derivatives of this expression, i.e. the change in U when both x1 and x2 change.1 dU =

∂U ∂U dx1 + dx2 ∂x1 ∂x2

Since the change in the consumption is such that the consumer remains on the same indifference curve, then dU = 0. Hence ∂U ∂U dx1 + dx2 = 0 ∂x1 ∂x2 Re-arranging, we obtain: ∂U

dx2 1 = − ∂x ∂U dx1 ∂x2 = −

M U1 M U2

Recall that the slope of the indifference curve is

dx2 dx1

NOTE: This procedure of taking total derivatives to obtain the slope of a curve is very useful, we will use it several times in this course.

1

In general, the total derivative of a function of several variables f (x, y) is equal to df =

3

∂f ∂f ∂x dx + ∂y dy....