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: MU1 = \frac{\partial U(x1,x2)}{\partial x1}.
  • Similarly: MU2 = \frac{\partial U(x2,x1)}{\partial x2}.

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 PX and PY, 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 (X1, Y1), 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{PX}{PY}.
  • The slope of the indifference curve is equal to the Marginal Rate of Substitution (MRS).
  • Therefore, at the most preferred consumption bundle, MRS = -\frac{PX}{PY}.
  • Tangency condition: \frac{\partial U / \partial X}{PX} = \frac{\partial U / \partial Y}{PY}
  • (\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 PX and PY, 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.