In the long run all inputs are variable. Isoquants show all possible combinations of labor and capital capable of producing a given level of output. Isoquants are downwardsloping to reflect the fact that if larger amounts of labor are used, less capital is required to produce the same output level. The marginal rate of technical substitution (MRTS) is the absolute value of the slope of an isoquant and measures the rate at which the two inputs can be substituted for one another while maintaining a constant level of output: MRTS = ΔK / ΔL.
The isocost curves show the various combinations of inputs that may be purchased for a given dollar outlay. The equation of an isocost curve is given by
where 
A manager minimizes the total cost of producing a given level of output or maximizes output for a given level of cost (expenditure on inputs) by choosing an input combination at the point of tangency between the relevant isoquant and isocost curves. The point of tangency indicates the lowest isocost curve that includes an input combination that is capable of producing the desired output level. Alternatively, the point of tangency indicates the largest output (the highest isoquant) that is attainable from any combination on the given isocost curve.
Since the cost-minimizing or output-maximizing input combination occurs at the point of tangency between the isoquant and the isocost curve, the slopes of the two curves are equal at the optimal input combination. The optimization condition may be expressed as
Thus the marginal product per dollar spent on the last unit of each input is the same. Equating marginal product per dollar spent on all variable inputs is the rule managers should follow both in the long run when all inputs are variable and in the short run when two or more inputs are variable.
The expansion path shows the equilibrium (or optimal) input combination for every level of output. An expansion path shows how input usage changes when output changes, input prices remaining constant. All points on the expansion path are both cost-minimizing and output-maximizing combinations of labor and capital.
The long-run cost curves are derived from the expansion path. Since the expansion path gives the efficient combination of labor and capital used to produce any particular level of output, the long-run total cost of producing that output level is the sum of the optimal amounts of labor and capital times their prices. Long-run average cost (LAC) is defined as
and is 
and is also
The long-run average cost curve gives the lowest possible unit costs of producing various output levels because in the long run all inputs are adjusted optimally and the firm operates on the (long-run) expansion path. Once the firm installs the optimal input combination for its planned production level, the firm then operates in the short run, facing the set of short-run cost curves determined by the amount of the fixed input selected from the long-run planning horizon. Because managers have the greatest flexibility in choosing inputs in the long run, long-run costs are lower than short-run costs for all output levels except the output level for which the fixed input is at its optimal level. Unless the firm is operating at the point of intersection between the long-run expansion path and its current short-run expansion path, a firm's short-run costs can be reduced by adjusting the fixed inputs to their optimal long-run levels when the opportunity to adjust fixed inputs arises in the long run.