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Unformatted text preview: ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values that the independent variables may take. Example 1. A firms cost function is given by C = 0 . 05 Q 2 A + 0 . 01 Q A Q B + 0 . 03 Q 2 B + 10 Q A + 15 Q B + 12000 , (1.1) where Q A and Q B are the quantities of the firms product that are produced in the firms two facilities A and B, respectively. If the firm has a contract to produce 2000 units of output, then how many units should they produce in each facility to minimize their cost? The condition that total output should equal 2000, imposes the following restriction on the variables Q A and Q B Q A + Q B = 2000 , (1.2) together with the conditions Q A ,Q B 0, since neither output can be negative. The condition in equation (1.2) is called a constraint because it constrains (restricts) the possible values of the free variables Q A and Q B . The function to be optimized (the cost function, in the example above) is called the objective function in these types of problems. Example 2. A firms output is given by the Cobb-Douglas model Q = AK L , (1.3) where Q is the firms output, K is quantity of the firms capital input and L is the quantity of the firms labor input. The constants, , and A are all positive and we also assume that + = 1. If the prices per unit of capital and labor are p K and p L , respectively, and the firms production budget is B , then how should the firm allocate its budget to maximize its output? The constraint in this case is given by the equation p K K + p L L = B, (1.4) reflecting the facts that (a) it costs p K K + p L L to use K units of capital and L units of labor, and (b) the total cost must equal B . The objective function in this example is the output, Q . In what follows, well see two approaches to solving this type of optimization problem. One approach, substitution, is more elementary, and uses the constraint to reduce the 1 number of variables. The second approach, the method of Lagrange multipliers , is more sophisticated and actually introduces a new variable to the problem. On the other hand, this additional variable yields important information about the optimization problem that is more difficult to derive using the substitution method. Also, it is often the case, that the algebra involved in finding the relevant critical points is actually easier in the Lagrange multiplier approach. 2. Substitution One approach to solving a constrained optimization problem is to use the constraint (or constraints, if there is more than one) to reduce the number of variables, and transform the problem to an unconstrained optimization problem in fewer variables ....
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