# Introduction to Economic Analysis, v. 1.0

by R. Preston McAfee and Tracy R. Lewis

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### Learning Objective

1. If a firm faces constraints on its behavior, how can we measure the costs of these constraints?

When capital K can’t be adjusted in the short run, it creates a constraint, on the profit available, to the entrepreneur—the desire to change K reduces the profit available to the entrepreneur. There is no direct value of capital because capital is fixed. However, that doesn’t mean we can’t examine its value. The value of capital is called a shadow valueThe value associated with a constraint., which refers to the value associated with a constraint. Shadow value is well-established jargon.

What is the shadow value of capital? Let’s return to the constrained, short-run optimization problem. The profit of the entrepreneur is

$π=pF(K,L)−rK−wL.$

The entrepreneur chooses the value L* to maximize profit; however, he is constrained in the short run with the level of capital inherited from a past decision. The shadow value of capital is the value of capital to profit, given the optimal decision L*. Because $0= ∂π ∂L =p ∂F ∂L (K,L*)−w,$ the shadow value of capital is $dπ(K,L*) dK = ∂π(K,L*) ∂K =p ∂F ∂K (K,L*)−r.$

Note that this value could be negative if the entrepreneur might like to sell some capital but can’t, perhaps because it is installed in the factory.

Every constraint has a shadow value. The term refers to the value of relaxing the constraint. The shadow value is zero when the constraint doesn’t bind; for example, the shadow value of capital is zero when it is set at the profit-maximizing level. Technology binds the firm; the shadow value of a superior technology is the increase in profit associated with it. For example, parameterize the production technology by a parameter a, so that aF(K, L) is produced. The shadow value of a given level of a is, in the short run,

$dπ(K,L*) da = ∂π(K,L*) ∂a =pF(K,L*).$

A term is vanishing in the process of establishing the shadow value. The desired value L* varies with the other parameters like K and a, but the effect of these parameters on L* doesn’t appear in the expression for the shadow value of the parameter because $0= ∂π ∂L$ at L*.

### Key Takeaways

• When an input is fixed, its marginal value is called a shadow value.
• A shadow value can be negative when an input is fixed at too high a level.
• Every constraint has a shadow value. The term refers to the value of relaxing a constraint. The shadow value is zero when the constraint doesn’t bind.
• The effect of a constraint on terms that are optimized may be safely ignored in calculating the shadow value.
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