Shadow price

Updated: Aug 20, 2021

1. In general terms, a price which is imputed as the true marginal value of a good or opportunity cosT of a resource, and which may differ from the market price. For example, it is often argued that if a certain type of labour being used to produce a particular good would, if not producing that good, be otherwise unemployed, then society foregoes no alternative output by keeping it employed; its opportunity cost is zero (ignoring the value of the workers’ leisure time foregone), and so a shadow price of zero should be imputed to it. lts actual market price, on the other hand, may be whatever wage rate the relevant labour union can secure for it by using its bargaining power. Hence the market price exceeds the shadow price. To take a converse example: a number of firms located around a lake may dispose of their effluent by discharging it into the lake, using the ‘services’ of the lake as a waste disposal unit. The lake is being used as a free good, since they do not have to pay for its services and the market price is therefore zero. However, suppose that it is desired, on grounds of social cost, to restrict the extent of lake pollution. Corresponding to a specified degree of pollution will be a price which, if charged to the firms, would lead them to choose a rate of effluent discharge which pollutes the lake to that degree and no more. This price is then the shadow price of the lake’s service as an industrial waste-disposal unit. If controls are imposed on the physical quantity of pollution, the market price continues to be zero. The idea of the shadow price however suggests an alternative solution: if the firms were actually charged the shadow price, then they would adjust to this and the desired standards would again be met. There are often arguments made for using a pricing mechanism rather than quantitative controls in cases of this sort, on grounds of simplicity, flexibility and lower costs of administration and enforcement. These two examples suggest the importance of the shadow price concept in cost-benefit analysis and applications of welfare economics generally. 2. Specifically, in mathematical programming problems, e.g. linear programming, a solution, if it exists, will always have associated with it a set of so-called ‘dual variables’ which are often called ‘shadow prices’. For example, suppose the problem is to choose outputs of three goods, given the per-unit profit on each good, in such a way that total profit is maximized. Each good requires specified amounts of labour time, machine time and raw material per unit produced. There are fixed total amounts of labour time and machine time available, and so no set of outputs can be produced with total labour and machine time requirements in excess of these. By methods of mathematical programming, three output levels can be found which can be produced, and which yield at !east as much profit as any other output levels which could be produced. The solution will also produce money values, one for labour time and one for machine time, which measure the gain in total profit which would result if the fixed quantity of the corresponding resource were increased very slightly. These money values are the shadow prices of the resources. They can play an important role in planning. For example, if additional machine time can be leased at a price below the shadow price of machine time, then the firm will increase its profit by doing so. Similarly, if another branch of the firm wished to divert a small amount of labour away from production of these three products, it could be ‘charged’ a price equal to the shadow price of labour time, since this is the loss of profit caused by such a diversion.

The concept of a shadow price originated in the development of mathematical programming methods, and its extension to more general usage is closely associated with the recognition that all problems of resource allocation are essentially mathematical programming problems.

Reference: The Penguin Dictionary of Economics, 3rd edt.

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James Knight
Editor of Education
James is the Editor of Education for Invezz, where he covers topics from across the financial world, from the stock market, to cryptocurrency, to macroeconomic markets.... read more.