ECON5000 Micro - Lecture 4: Marginal Utility and Consumer Optimisation

Utility Functions and Monotonic Transformations

• Utility functions are not unique representations.
• If $U$ represents preferences $U(x,y)$, then there are many different utility functions that represent the same preferences.
• If $F$ is a (positive) monotonic transformation $F(U)$, then $V(x,y) = F(U(x,y))$ represents the same preferences.
• Monotonic transformation preserves the ordering of a set of numbers.

Marginal Utility

• Marginal utility of commodity $i$ is the rate-of-change of total utility as the quantity of commodity $i$ consumed changes.
• Marginal Utility of good 1: $MU<em>1 = \frac{\partial U(x</em>1,x<em>2)}{\partial x</em>1}$.
• Similarly: $MU<em>2 = \frac{\partial U(x</em>2,x<em>1)}{\partial x</em>2}$.

Marginal Rate of Substitution (MRS)

• MRS = slope of the indifference curve at a point.
• If we are given a utility function $U(x,y)$, we can derive a mathematical formula for the marginal rate of substitution.
• Along an indifference curve, we have $U(x,y) = c$, where $c$ is some constant.
• $MRS_{YX} = -\frac{(\partial U / \partial X)}{(\partial U / \partial Y)}$ = - (marginal utility of X) / (marginal utility of Y).

Consumer Optimization

• Given preferences, prices, and wealth, we consider what an individual would actually choose.
• Basic Problem: An individual has preferences over two goods X and Y, with prices $P<em>X$ and $P</em>Y$, and wealth $M$. The goal is to determine how much of X and Y they would buy.

Graphical Representation of the Problem

• At an optimum, the consumption bundle must lie on the budget line due to monotonicity.
• The indifference curve passing through the chosen consumption bundle just touches the budget line (at only one point).
• At the optimal point $(X<em>1, Y</em>1)$, the budget line is tangent to the indifference curve.
• The slope of the indifference curve is equal to the slope of the budget line.

Mathematical Solution to the Problem

• The slope of the budget line is equal to $-\frac{P<em>X}{P</em>Y}$.
• The slope of the indifference curve is equal to the Marginal Rate of Substitution (MRS).
• Therefore, at the most preferred consumption bundle, $MRS = -\frac{P<em>X}{P</em>Y}$.
• Tangency condition: $\frac{\partial U / \partial X}{P<em>X} = \frac{\partial U / \partial Y}{P</em>Y}$
• $(\partial U / \partial Y) / P_Y$ = Marginal Utility (MU) per unit of money. It means “The marginal utility from an additional pound spent on good Y equals the marginal utility from an additional pound spent on X.”

Algorithm for computing the optimal bundle

• Given $U(X,Y)$, prices $P<em>X$ and $P</em>Y$, and wealth $M$.
• Check if preferences are convex.
• Check if we have an interior solution.
• If you have verified both convexity and an interior solution, then use
  • I) the tangency condition along with
  • II) the budget equation to find the solution.