The term "Shadow Price" or "Shadow Pricing" refers to monetary values assigned to currently unknowable or difficult to calculate costs. The origin of these costs is typically due to an externalization of costs or an unwillingness to recalculate a system to account for marginal production. For example, consider a firm that already has a factory full of equipment and staff. They might estimate the shadow price for a few more units of production as simply the cost of the overtime. In this manner some goods and services have near zero shadow prices, for example information goods.

Less formally, shadow price can be thought of as the cost of decisions made at the margin without consideration for the total cost. For instance, consider a trip in your car. You might estimate the shadow price of that trip by including the cost of gas; but you are unlikely to include the wear on the tires or the cost of the money you might have borrowed to purchase the car.

In constrained optimization in economics, the shadow price is the instantaneous change, per unit of the constraint, in the objective value of the optimal solution of an optimization problem obtained by relaxing the constraint. In other words, it is the marginal utility of relaxing the constraint, or, equivalently, 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.[1] 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.

## Illustration #1

Suppose a consumer faces prices ${\displaystyle \,\!p_{1},p_{2}}$ and is endowed with income ${\displaystyle \,\!m}$, then the consumer's problem is: ${\displaystyle \max\{\,\!u(x_{1},x_{2}){\mbox{ }}:{\mbox{ }}p_{1}x_{1}+p_{2}x_{2}=m\}}$. Forming the Lagrangian auxiliary function ${\displaystyle \,\!L(x_{1},x_{2},\lambda ):=u(x_{1},x_{2})+\lambda (m-p_{1}x_{1}-p_{2}x_{2})}$, taking first order conditions and solving for its saddle point we obtain ${\displaystyle \,\!x_{1}^{*}{\mbox{, }}x_{2}^{*}{\mbox{, }}\lambda ^{*}}$ which satisfy:

${\displaystyle \lambda ^{*}={\frac {\frac {\partial u(x_{1}^{*},x_{2}^{*})}{\partial x_{1}}}{p_{1}}}={\frac {\frac {\partial u(x_{1}^{*},x_{2}^{*})}{\partial x_{2}}}{p_{2}}}}$

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 ${\displaystyle \,\!\lambda ^{*}}$ as above, then the change in maximal utility per dollar of additional income will be equal to ${\displaystyle \,\!\lambda ^{*}}$ since at the optimum the consumer gets the same amount of marginal utility per dollar from spending his additional income on either goods. In this case the shadow price concept does not carry much importance because the objective function (utility) and the constraint (income) are measured in different units.

## Illustration #2

Holding prices fixed, if we define

${\displaystyle U(p_{1},p_{2},m)=\max\{\,\!u(x_{1},x_{2}){\mbox{ }}:{\mbox{ }}p_{1}x_{1}+p_{2}x_{2}=m\}}$,

then we have the identity

${\displaystyle \,\!U(p_{1},p_{2},m)=u(x_{1}^{*}(p_{1},p_{2},m),x_{2}^{*}(p_{1},p_{2},m))}$,

where ${\displaystyle \,\!x_{1}^{*}(\cdot ,\cdot ,\cdot ),x_{2}^{*}(\cdot ,\cdot ,\cdot )}$ are the demand functions, i.e. ${\displaystyle x_{i}^{*}(p_{1},p_{2},m)=\arg \max\{\,\!u(x_{1},x_{2}){\mbox{ }}:{\mbox{ }}p_{1}x_{1}+p_{2}x_{2}=m\}{\mbox{ for }}i=1,2}$

Now define the optimal expenditure function

${\displaystyle \,\!E(p_{1},p_{2},m)=p_{1}x_{1}^{*}(p_{1},p_{2},m)+p_{2}x_{2}^{*}(p_{1},p_{2},m)}$

Assume differentiability and that ${\displaystyle \,\!\lambda ^{*}}$ is the solution at ${\displaystyle \,\!p_{1},p_{2},m}$, then we have from the multivariate chain rule:

${\displaystyle \,\!{\frac {\partial U}{\partial m}}={\frac {\partial u}{\partial x_{1}}}{\frac {\partial x_{1}^{*}}{\partial m}}+{\frac {\partial u}{\partial x_{2}}}{\frac {\partial x_{2}^{*}}{\partial m}}=\lambda ^{*}p_{1}{\frac {\partial x_{1}^{*}}{\partial m}}+\lambda ^{*}p_{2}{\frac {\partial x_{2}^{*}}{\partial m}}=\lambda ^{*}\left(p_{1}{\frac {\partial x_{1}^{*}}{\partial m}}+p_{2}{\frac {\partial x_{2}^{*}}{\partial m}}\right)=\lambda ^{*}{\frac {\partial E}{\partial m}}}$

Now we may conclude that

${\displaystyle \,\!\lambda ^{*}={\frac {\partial U/\partial m}{\partial E/\partial m}}\approx {\frac {\Delta {\mbox{Optimal Utility }}}{\Delta {\mbox{Optimal Expenditure}}}}}$

This again gives the obvious interpretation, one extra dollar of optimal expenditure will lead to ${\displaystyle \,\!\lambda ^{*}}$ units of optimal utility.

## Control theory

In optimal control theory, the concept of shadow price is reformulated as costate equations, and one solves the problem by minimization of the associated Hamiltonian via Pontryagin's minimum principle.