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How to Derive a Demand Function Fallaw Sowell Tepper School of Business Carnegie Mellon University August 2006

Motivation An Alaskan oil pipeline is shutdown. What will be the impact on the demand for “our” product? It would be nice to have a function that gave the answer -- quantity demanded for good i -- per unit price for good j -- consumer’s income to be spent -- Demand Function

Basic Application of Derivatives You have just learned to use derivatives to determine critical values of a function (typically maxima or minima). Same basic approach is used to derive demand functions. Make the connection between these today. A preview of how these tools will be used. More QSRP, Mini One Quant class(es), before … problem sets in Managerial Economics during mini 2.

Where a Parabola Takes its Maximum Find the value of X where the parabola takes its maximum.

Where a Parabola Takes its Maximum Find the value of X where a parabola takes its maximum. First derivative First derivative set equal to zero. One equation in one unknown. The Solution.

Where a Parabola Takes its Maximum

The Common Structure Parabola Demand Function Objective Function First Order Conditions one equation in one unknown N+1 equations in N+1 unknowns Solutions

Derive the Demand Function New Math Tools to Derive the Demand Function Derivatives of a multivariate function. Partial derivative – Derivative holding all the other variables fixed. (Section 17.2 in your book) Optimal value of a multivariate function when a constraint must be satisfied. Technique of Lagrange Multiplier. (Section 17.8 in your book) More QSRP, Mini One Quant class(es), before … problem sets in Managerial Economics during mini 2.

Derive the Demand Function New Economic Models to Derive the Demand Function Utility function – Gives the level of happiness for a give bundle of goods. Typically denoted -- quantity demanded for good i -- per unit price for good i -- consumer’s income to be spent Budget Constraint – The consumer will spend all of their income purchasing goods to consume.

The Basic Consumer Problem (N=2) Go buy things to make yourself as happy as possible. Select quantities of goods to maximize utility subject to the constraint that you cannot spend more than your income max s.t. To solve this form the function.

The Objective Function This function is called a Lagrangian. The new variable is called the Lagrange multiplier. Just like for the parabola, take derivatives with respect to each of the variables and set them equal to zero. and

The First Order Conditions Partial derivatives – derivative holding the other variables constant. Solve this system of 3 equations in 3 unknowns.

The Solutions Demand Functions – very useful in business. Lagrange multiplier at the optimal consumption bundle. Not that important in Economic business applications. Very important in other business applications.

Example with CES Utility Function

First Order Conditions (FOC’s) 3 equations in 3 unknowns

The Demand Functions

Demand function for good

Cobb-Douglas Utility function Additional Example Cobb-Douglas Utility function

A quasilinear utility function Additional Example A quasilinear utility function