Finishing Up with the Advertising Elasticity of Demand

Chapter 3: Calculus, Optimization, and You

Constraining functions: What you can’t do

A constraint function represents a limitation on your behavior. The dependent variable in the constraint represents the limitation. Examples of constraint functions include the number of units you must produce in order to satisfy a contract, the production function for a given technology, and the budget available to a consumer.

Constructing the Lagrangian function

The technique for constructing a Lagrangian function is to combine the objective function and all constraints in a manner that satisfies two conditions. First, optimizing the Lagrangian function must result in the objective function’s optimization. Second, all constraints must be satisfied. In order to satisfy these conditions, use the following steps to specify the Lagrangian function.

Assume u is the variable being optimized and that it’s a function of the variables x and z. Therefore,

In addition, there are two constraints, c1 and c2, that are also functions of x and z;

The following steps establish the Lagrangian function:

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1. Respecify the constraints so that they equal zero.

2. Multiply the constraints by the factors lambda one and lambda two, λ1 and λ2, respectively (more on these in a moment).

3. Add the constraints with the lambda term to the objective function in

order to form the Lagrangian function £.

In this specification of the Lagrangian function, the variables are represented by x, z, λ1, and λ2. Taking the partial derivatives of the Lagrangian with respect to λ1 and λ2 and setting them equal to zero ensure that your constraints are