Linear Programming and Shadow Prices

Origin. Leonid Kantorovich developed linear programming (1939) for plywood production optimization; later extended to economy-wide planning. Nobel Prize in Economics (1975, shared with Koopmans).

Mechanism. Express the planning problem as maximizing a linear objective (total value) subject to linear constraints (resource limits, production requirements). The solution identifies the optimal production plan and, crucially, the shadow prices — the marginal value of each constrained resource. Shadow prices guide decentralized decisions: if each unit maximizes profit at shadow prices, the result is globally optimal.

Procedure. Formulate the objective function: what are you maximizing (output, value, utilization)? Formulate the constraints: resource availability, capacity limits, demand requirements. Solve using simplex or interior-point methods. Read the shadow prices from the dual solution. Shadow prices indicate which constraints are binding (positive shadow price) and how much the objective would improve if that constraint were relaxed. Use shadow prices to guide resource allocation decisions.

Applies to. Production planning, resource allocation, logistics, diet problems, any optimization with linear structure.

Limitations. Linearity is a strong assumption: no economies of scale, no indivisibilities, no diminishing returns. Shadow prices are meaningful only at the optimal solution; away from optimum they can be misleading. Also: shadow prices assume the constraints are accurate; if constraints are misspecified, the prices optimize the wrong problem. Kantorovich's prices were never implemented in Soviet planning because they threatened bureaucratic control.