Indifference Curves with Utility

  • All of the bundles on one indifference curve should have the same output from the utility function
  • Similarly, bundles on different indifference curves should have different values
    • These values/outputs are known as utility levels
  • The collection of all indifference curves with their assigned utility levels for a given preference relation is known as an indifference map
    • In other words, this map maps each curve to a utility level which creates a one-to-one (injective) relationship which leads to a utility function
    • There are an infinitely many number of ways to create a utility function for a preference relation
      • For example, adding or subtracting any number from the function will still maintain the required ordering
Consider a consumer who values bundles in the following way: (2,3)>(4,1) (2,2)Suppose that u(x1,x2)=x1x2 represents a preference relation.Consider (4, 1), (2, 3), and (2, 2):u(2,3)=6u(4,1)=u(2,2)=4u(2,3)>u(4,1)=u(2,2)Thus, this utility function correctly represents the consumer’s preferences.Consider a new function v=u2. Then, v(x1,x2)=x12x22u(2,3)=36u(4,1)=u(2,2)=16u(2,3)>u(4,1)=u(2,2)Therefore, v is also a valid utility function.Another valid utility function could be w=u+10Thus, there an infinite amount of valid utility functions.\text{Consider a consumer who values bundles in the following way: } (2, 3) > (4, 1) ~ (2, 2) \\ \text{Suppose that } u(x_1, x_2) = x_1 x_2 \text{ represents a preference relation.} \\ \text{Consider (4, 1), (2, 3), and (2, 2):} \\ u(2, 3) = 6 \\ u(4, 1) = u(2, 2) = 4 \\ u(2, 3) > u(4, 1) = u(2, 2) \\ \text{Thus, this utility function correctly represents the consumer's preferences.} \\ \text{Consider a new function } v = u^2 \text{. Then, } v(x_1, x_2) = x_1^2 x_2^2 \\ u(2, 3) = 36 \\ u(4, 1) = u(2, 2) = 16 \\ u(2, 3) > u(4, 1) = u(2, 2) \\ \text{Therefore, v is also a valid utility function.} \\ \text{Another valid utility function could be } w = u + 10 \\ \text{Thus, there an infinite amount of valid utility functions.} \\
  • If u is a utility function that represents a preference relation p, and f is a strictly increasing function, then v = f(u) is also a utility function representing p

Goods, Bads, and Neutrals

  • A good is a commodity unit which increases utility
  • A bad is a commodity unit which decreases utility
  • A neutral is a commodity unit which doesn’t affect utility

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Examples of Utility Functions

The general form for a utility function of perfect substitutes is u(x1,x2)=x1+x2The general form for a utility function of perfect complements is u(x1,x2)=min(x1,x2)\text{The general form for a utility function of perfect substitutes is } u(x_1, x_2) = x_1 + x_2 \\ \text{The general form for a utility function of perfect complements is } u(x_1, x_2) = min(x_1, x_2)

Cobb-Douglas Utility Functions

Cobb-Douglas utility functions take the form of u(x1,x2)=x1a+x2bwhere a > 0 and b > 0\text{Cobb-Douglas utility functions take the form of } u(x_1, x_2) = x_1^a + x_2^b \\ \text{where a > 0 and b > 0}
  • Cobb-Douglas indifference curves are all hyperbolic and asymptote to but never touch the x or y axes

Marginal Utility

  • The marginal utility of any commodity i is the rate of change of total utility as the quantity of commodity i changes
    • To measure the rate of change, we find the partial derivative of the utility function with respect to i
mui=∂u∂iu(x1,x2)=x10.5+x22, mu1=12x1−0.5x22, mu2=2x10.5x2mu_i = \frac{\partial u}{\partial i} \\ u(x_1, x_2) = x_1^{0.5} + x_2^2 \text{, } mu_1 = \frac{1}{2} x_1^{-0.5} x_2^2 \text{, } mu_2 = 2 x_1^{0.5} x_2

Marginal Utility (MU) and Marginal Rate of Substitution (MRS)

  • Recall that the MRS is the slope of the indifference curve
u(x1,x2)≡k, where k is some constantAlong an indifference curve, a change in x1 perfectly offsets changes in x2such that the utility of the bundle remains the same. This means that the MU and MRS are related.This gives us the following identity:∂u∂x1dx1+∂u∂x2dx2=0∂u∂x1dx1=−∂u∂x2dx2dx2dx1=−∂u/x1∂u/x2MRS=−∂u/x1∂u/x2u(x_1, x_2) \equiv k \text{, where k is some constant} \\ \text{Along an indifference curve, a change in } x_1 \text{ perfectly offsets changes in } x_2 \\ \text{such that the utility of the bundle remains the same. This means that the MU and MRS are related.} \\ \text{This gives us the following identity:} \\ \frac{\partial u}{\partial x_1} dx_1 + \frac{\partial u}{\partial x_2} dx_2 = 0 \\ \frac{\partial u}{\partial x_1} dx_1 = - \frac{\partial u}{\partial x_2} dx_2 \\ \frac{dx_2}{dx_1} = - \frac{\partial u / x_1}{\partial u / x_2} \\ MRS = - \frac{\partial u / x_1}{\partial u / x_2} \\
  • As long as there is a slope, then the identity holds

Example: u(x1,x2)=x1x2∂u∂x1=x2, ∂u∂x2=x1MRS=−mu1mu2=−x2x1u(x_1, x_2) = x_1 x_2 \\ \frac{\partial u}{\partial x_1} = x_2 \text{, } \frac{\partial u}{\partial x_2} = x_1 \\ MRS = - \frac{mu_1}{mu_2} = -\frac{x_2}{x_1}