obtain which satisfy This gives us a clear interpretation of the Lagrange multiplier in the context of consumer maximization. If the consumer is given an extra unit of income (the budget constraint is relaxed) at the optimal consumption level where the marginal utility per unit of income for each good is equal to as above, then the change in maximal utility per unit of additional income will be equal to since at the optimum the consumer gets the same amount of marginal utility per unit of income from spending his additional income on either good. Illustration #2 Holding prices fixed, if we define the indirect utility function as then we have the identity where are the demand functions, i.e. Now define the optimal expenditure function Assume differentiability and that is the solution at , then we have from the multivariate chain rule: Now we may conclude that This again gives the obvious interpretation, one extra unit of optimal expenditure will lead to units of optimal utility. Constrained optimization In constrained optimization in economics, the shadow price is the change, per infinitesimal unit of the constraint, in the optimal value of the objective function of an optimization problem obtained by relaxing the constraint. If the objective function is utility, it is the marginal utility of relaxing the constraint. If the objective function is cost, it is the marginal cost of strengthening the constraint. In a business application, a shadow price is the maximum price that management is willing to pay for an extra unit of a given limited resource. For example, if a production line is already operating at its maximum 40-hour limit, the shadow price would be the maximum price the manager would be willing to pay for operating it for an additional hour, based on the benefits he would get from this change. More formally, the shadow price is the value of the Lagrange multiplier at the optimal solution, which means that it is the infinitesimal change in the objective function arising from an infinitesimal change in the constraint. This follows from the fact that at the optimal solution the gradient of the objective function is a linear combination of the constraint function gradients with the weights equal to the Lagrange multipliers. Each constraint in an optimization problem has a shadow price or dual variable. Control theory In optimal control theory, the concept of shadow price is reformulated as