Definitions

Loosely, the shadow price is the change in the objective value of the optimal solution of an optimization problem obtained by relaxing the constraint by one unit. 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, what is the price of keeping a production line operational for an additional hour if the production line is already operated at its maximum 40 hour limit? That price is the shadow price.

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.

The value of the shadow price can provide decision makers powerful insight into problems. For instance if you have a constraint that limits the amount of labor available to 40 hours per week, the shadow price will tell you how much you would be willing to pay for an additional hour of labor. If your shadow price is \$10 for the labor constraint, for instance, you should pay no more than \$10 an hour for additional labor. Labor costs of less than \$10/hour will increase the objective value; labor costs of more than \$10/hour will decrease the objective value. Labor costs of exactly \$10 will cause the objective function value to remain the same.

## Illustration #1

Suppose a consumer faces prices $,! p_1,p_2$ and is endowed with income $,!m$, then the consumer's problem is: $max \left\{,!u\left(x_1,x_2\right)mbox\left\{ \right\} :mbox\left\{ \right\} p_1x_1+p_2x_2=m\right\}$. Forming the Lagrangian auxiliary function $,! L\left(x_1,x_2,lambda\right):= u\left(x_1,x_2\right)+lambda\left(m-p_1x_1-p_2x_2\right)$, taking first order conditions and solving for its saddle point we obtain $,! x^*_1mbox\left\{, \right\}x^*_2mbox\left\{, \right\}lambda^*$ which satisfy:
$lambda^*=frac\left\{frac\left\{partial u\left(x^*_1,x^*_2\right)\right\}\left\{partial x_1\right\}\right\}\left\{p_1\right\}= frac\left\{frac\left\{partial u\left(x^*_1,x^*_2\right)\right\}\left\{partial x_2\right\}\right\}\left\{p_2\right\}$
This gives us a clear interpretation of the Lagrange Multiplier in the context of consumer maximization- if the consumer is given an extra dollar (the budget constraint is relaxed) at the optimal consumption level where the marginal utility per dollar for each good is equal to $,! lambda^*$ as above, then the change in maximal utility per dollar of additional income will be equal to $,! lambda^*$ since at the optimum the consumer gets the same amount of marginal utility per dollar from spending his additional income on either good. In this case the shadow price concept does not carry much import-- the objective function (utility ) and the constraint (income) are measured in different units.

## Illustration #2

Holding prices fixed, if we define
$U\left(p_1,p_2,m\right):= max \left\{,!u\left(x_1,x_2\right)mbox\left\{ \right\} :mbox\left\{ \right\} p_1x_1+p_2x_2=m\right\}$,
then we have the identity
$,! U\left(p_1,p_2,m\right)=u\left(x_1^*\left(p_1,p_2,m\right),x_2^*\left(p_1,p_2,m\right)\right)$,
where $,! x_1^*\left(cdot,cdot,cdot\right),x_2^*\left(cdot,cdot,cdot\right)$ are the demand functions, i.e. $x_i^*\left(p_1,p_2,m\right):= argmax \left\{,!u\left(x_1,x_2\right)mbox\left\{ \right\} :mbox\left\{ \right\} p_1x_1+p_2x_2=m\right\} mbox\left\{ for \right\} i=1,2$

Now define the optimal expenditure function
$,! E\left(p_1,p_2,m\right):=p_1x_1^*\left(p_1,p_2,m\right)+p_2x_2^*\left(p_1,p_2,m\right)$
Assume differentiability and thaton $,! lambda^*$ is the solution at $,! p_1,p_2,m$, then we have from the multivariate chain rule:
$,! frac\left\{partial U\right\}\left\{partial m\right\} =frac\left\{partial u\right\}\left\{partial x_1\right\}frac\left\{partial x_1^*\right\}\left\{partial m\right\} + frac\left\{partial u\right\}\left\{partial x_2\right\}frac\left\{partial x_2^*\right\}\left\{partial m\right\} =lambda^* p_1frac\left\{partial x_1^*\right\}\left\{partial m\right\} + lambda^* p_2 frac\left\{partial x_2^*\right\}\left\{partial m\right\}=lambda^* left\left(p_1frac\left\{partial x_1^*\right\}\left\{partial m\right\} + p_2 frac\left\{partial x_2^*\right\}\left\{partial m\right\} right\right) =lambda^* frac\left\{partial E\right\}\left\{partial m\right\}$
Now we may conclude that
$,! lambda^* = frac\left\{frac\left\{partial U\right\}\left\{partial m\right\}\right\}\left\{frac\left\{partial E\right\}\left\{partial m\right\}\right\} approx frac\left\{Delta mbox\left\{Optimal Utility \right\}\right\}\left\{Delta mbox\left\{Optimal Expenditure\right\}\right\}$
This again gives the obvious interpretation, one extra dollar of optimal expenditure will lead to $,! lambda^*$ units of optimal utility.