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I have recently read a text which deals with the dual variables attached to constraints. In an economic sense, one can interpret them as shadow variables indicating market clearing for resource constraint and zero-profits for cost equations. But I could not really follow the discussion.

So from my understanding, the variables attached to the constraints in LP are used as coefficient to build a linear combination in order to get from the primal to the dual problem. If a constraint is binding, then this coefficient will be positive, and zero if non-binding.

Thus I conclude that people economically interpret those coefficients which relate to binding constraints. I can imagine that if we have a demand problem, then the shadow price is (maybe?) some sort of market clearing price of the constraint. And if we are in a cost scenario, then a binding constraint would indicate that costs equal profit and thus output is positive but economic profit is zero?

I am also a bit puzzled since I know shadow variables in the context of Lagrangian multipliers. Is that just another approach of solving LP's and its interpretation is therefore similar to the dual variable (as both are called shadow variables)?

Any input is appreciated!

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The expression shadow price usually refers to the price of a good or service calculated by an economic mathematical model (mathematical programming). The shadow price, also known as the accounting price, indicates the value of the goods which derives from their scarcity, i.e. from the limited availability of the quantities of goods compared to needs. In a technical sense, the shadow price is the term used to call Lagrange multipliers in economics. In economic optimization problems, the notion of scarcity is reflected in the constraint equations of productive resources or a basket of goods where a function of revenues, costs or a utility function constitutes the function to be optimized.

Today, the expression shadow price finds a wider use than that originally attributed by mathematical programming. In economic evaluations, the shadow price of a good, be it an input or an output, constitutes its economic value whenever the market is not able to incorporate into the price the true opportunity cost of the good itself to which it is connected. In perfectly competitive and efficient markets or in conditions of optimal planning, market prices and shadow prices coincide, however in reality it is observed that markets can be distorted by taxes, duties, subsidies, rigid exchange rates, production or consumption, regulated tariffs, oligopoly or monopoly price setting and imperfect information. These elements mean that there is a difference between the market price and the shadow price. The main reasons for the distortion are: the absence of perfect competition following the presence of monopolies or oligopolies and imperfect information between economic agents; state intervention or other factors external to the economic sphere that disrupt the economic process (taxes, regulations, quotas and all types of economic policy measures).

The goal of using shadow prices is to correct distortions between market prices and true economic value.

Economic operators interested in evaluating and deciding on the optimal use of factors may find themselves faced with monetary prices recorded on the market which are poorly suited to being used as a decision-making tool. Since a good decision-making process requires measuring the value of goods/services in commensurable units, whenever the market price is inadequate or absent, the concept of shadow price is used with the aim of carrying out assessments of economic convenience between different purchasing decisions, investment.

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