Machina Math Handout

Objective function to obtain

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Unformatted text preview: optimal value function. Sometimes we will be optimizing subject to a constraint on the control variables (such as the budget constraint of the consumer). Since this constraint may also depend upon the parameter(s), our problem becomes: max ƒ( x1 ,..., xn ; α ) x1 ,..., xn subject to g ( x1 ,..., xn ; α ) = c (Note that we now have an additional parameter, namely the constant c.) In this case we still define the solution functions and optimal value function in the same way – we just have to remember to take into account the constraint. Although it is possible that there could be more than one constraint in a given problem, we will only consider problems with a single constraint. For example, if we were looking at the profit maximization problem, the control variables would be the quantities of inputs and outputs chosen by the firm, the parameters would be the current input and output prices, the constraint would be the production function, and the optimal value function would be the firm’s “profit function,” i.e., the highest attainable level of profits given current input and output prices. In economics we are interested both in how the optimal values of the control variables, and the optimal attainable value, vary with the parameters. In other words, we will be interested in differentiating both the solution functions and the optimal value function with respect to the parameters. Before we can do this, however, we need to know how to solve unconstrained or constrained optimization problems. Econ 100A 7 Mathematical Handout First Order Conditions for Unconstrained Optimization Problems The first order conditions for the unconstrained optimization problem: max ƒ( x1 ,..., xn ) x1 ,..., xn are simply that each of the partial derivatives of the objective function be zero at the solution * * values (x1 ,..., xn ), i.e. that: ∗ ∗ ƒ1 ( x1 ,..., xn ) = 0 ∗ ∗ ƒ n ( x1 ,..., xn ) = 0 The intuition is that if you want to be at a “mountain top” (a maximum) or the “bottom of a bowl” (a minimum) it must be the case that no small change in any control variable be able to move you up or down. That means that the partial derivatives of ƒ(x1,..., xn) with respect to each xi must be zero. Second Order Conditions for Unconstrained Optimization Problems If our optimization problem is a maximization problem, the second order condition for this solution to be a local maximum is that ƒ(x1, ..., xn) be a weakly concave function of (x1,..., xn) (i.e., a mountain top) in the locality of this point. Thus, if there is only one control variable, the second order condition is that ƒ ″(x*) < 0 at the optimum value of the control variable x. If there are two control variables, it turns out that the conditions are: ƒ11(x1*, x2*) < 0 ƒ22(x1*, x2*) < 0 and ∗ ∗ ∗ ∗ ƒ11 ( x1 , x2 ) ƒ12 ( x1 , x2 ) * * * * ƒ 21 ( x1 , x2 ) ƒ 22 ( x1 , x2 ) >0 When we have a minimization problem, the second order condition for this solution...
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This document was uploaded on 09/18/2013.

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