Shadow price

A shadow price is a monetary value assigned to currently unknowable or difficult-to-calculate costs in the absence of correct market prices. It is based on the willingness to pay principle – the most accurate measure of the value of a good or service is what people are willing to give up in order to get it. A shadow price is often calculated based on certain assumptions, and so it is subjective and somewhat inaccurate. 
The origin of these costs are typically costs that are external to a market 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 way, some goods and services have near zero shadow prices, for example information goods. 
Less formally, a shadow price can be thought of as the cost of decisions made at the margin without consideration for the total cost.
Although shadow pricing may be inaccurate, it is still often used in cost-benefit analyses. For example, before defining a project, businesses and governments may want to weigh the costs and benefits of the project to decide whether the project is worthwhile. While tangible costs and benefits such as the cost of labor are easy to quantify, intangible costs and benefits such as the number of hours saved is much more difficult to quantify. In this case, business owners and policymakers turn to shadow pricing to determine what these intangibles are. There are usually many tools to estimate monetary values of these intangibles. They include contingent valuation, revealed preferences, and hedonic pricing.
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.
In advance of adequate regulation or market pricing for some commodity items, conservative organizations will place on their balance sheets a value they believe to be an accurate reflection of the value of those items to their operations. This is common for companies with a large carbon footprint or water footprint. As an example Microsoft has placed a $27/ton price on its carbon emissions which is then billed to the P&L of its individual business units and used to fund the company's renewable energy and efficiency programs.
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.
Cost-benefit analysis.
Shadow pricing is frequently used to figure out the monetary values of intangibles which are hard to quantify factors during cost-benefit analyses. In the context of public economics, shadow pricing is very useful for governments and policymakers to evaluate whether a public project should be pursued. This is because public goods are very rarely exchanged in the market, making it difficult to determine its price. To help determine the monetary value of these goods, these three tools are often used. Take the example of a government determining whether it wants to undertake a freeway project that would save commuters 500,000 hours a year, save 5 lives a year, and reduce air pollution due to decreased congestion but with a present value cost of $250 million.
Contingent valuation.
Contingent valuation estimates the value a person places on a good by asking him or her directly. It is essentially surveys for individuals on how much they would be willing to pay for some intangible benefits or to avoid some intangible harms. Typically, these surveys contain detailed descriptions of hypothetical public goods or services, ask respondents how much they would pay for it, and collect relevant demographic data of these respondents. Some common types of these survey questions include: open-ended, referendum-type, payment-card type, and double-bounded referendum-type.
The advantage of contingent valuation is that it is sometimes the only feasible method for valuing a public good. This is especially the case when there is no obvious market price that one can use to determine the value. On the other hand, there are also many disadvantages of this method. For instance, how the survey is structured and how the questions are framed can lead to widely varying results and can induce bias into the results. Other times, the respondents may simply have no idea how much they value the public good in question.
In the freeway project example, policymakers can design a survey that asks respondents on how much they would pay to save a certain period of time or to spend less time in traffic. However, respondents may find it difficult or uncomfortable to put a value on a life.
Revealed preferences.
Revealed preferences are based on observations on real world behaviors to determine how much individuals place on non-monetary outcomes. In other words, observing individuals' purchasing behaviors is the best way to determine their preferences. It assumes that individuals have made their purchasing decisions over other alternatives – making their final purchases the preferred one. It also allows room for the preferred choice to vary depending on the prices and the budgetary constraints. As such, by varying prices and budgetary constraints, a schedule can be created of an individual's/individuals' preferred choices under certain prices and constraints.
The advantage of revealed preferences is that it reduces biases that contingent valuation may bring. As it is based on real-world behaviors, it is much harder for individuals to manipulate or guess-work their answers. On the other hand, this tool also has its limits. For example, it is difficult to control for other factors that may make one prefer a choice over another. It also fails to fully incorporate indifference between two equally preferred choices.
In the freeway project example, where contingent valuation may fall short in determining how much individuals value lives, revealed preferences may be better suited. For instance, policymakers can look at how much more individuals need to be paid to take on riskier jobs that increase the probability of fatality. However, the drawbacks with revealed preferences also arise – in this case, if the riskier jobs increase the probability of not only death but also injury, or are also unpleasant in other respects, the higher wages may incorporate the other factors, misrepresenting the result.
Hedonic pricing.
Hedonic pricing is a model that uses regression analysis to isolate the value of a specific intangible cost or benefit. It is based on the premise that that price is determined by both internal characteristics and external factors. It also assumes that individuals value the characteristics of a good rather than the good itself, which implies that price will reflect a set of internal and external characteristics. It is most often used to calculate variances in housing prices that reflect the value of local environmental factors. The model is based on widely-available and relatively accurate market data, making this method uncontroversial and inexpensive to use.
As such, one of hedonic pricing's main advantages is that it can be used to estimate values on actual choices. This method is also very versatile and can be adapted to incorporate multiple other interactions with other factors. However, one of its major downfalls is that it is rather limited – it can mostly only measure things that are related to housing prices. It also assumes that individuals have the freedom and power to select the preferred combination given their income but in actuality, this may not be the case as the market may be influenced by changes in taxes and interest rates.
In the freeway project example, hedonic pricing may be useful to value the benefits of reduced air pollution. It can run a regression of home values on clean air with a variety of control variables that can include home size, age of home, number of bedrooms and bathrooms, crime statistics, school qualities, etc. Hedonic pricing may also be considered in quantifying the monetary value of time saved. It can run a regression of home values on proximity to work with a similar set of control variables.
Illustration #1.
Suppose a consumer with utility function formula_1 faces prices formula_2 and is endowed with income formula_3 Then the consumer's problem is:
Forming the Lagrangian auxiliary function formula_5 taking first-order conditions and solving for its saddle point we obtain formula_6 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 formula_8 as above, then the change in maximal utility per unit of additional income will be equal to formula_8 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 formula_12 are the demand functions, i.e. formula_13
Now define the optimal expenditure function
Assume differentiability and that formula_15 is the solution at formula_16, 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 formula_8 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.