A difference between the short run and the long run is that a firm in the short run faces a constrained cost minimization problem, whereas the firm is unconstrained in the long run.

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In the short run, a firm faces limitations on the factors of production, such as capital and labor, which influence its cost minimization decisions. In contrast, the long run allows the firm to adjust its production processes and scale of operations without such constraints, leading to an unconstrained cost minimization problem. In the short run, a firm is constrained by its fixed inputs and cannot adjust them to minimize costs, whereas in the long run all inputs are variable and the firm can adjust them to minimize costs. Therefore, the firm faces a constrained cost minimization problem in the short run and an unconstrained cost minimization problem in the long run.

A Difference Between the Short Run and the Long Run