Optimal Input Combination Quiz

The optimal input combination minimizes the cost of producing a target output. It is found where the isoquant is tangent to the lowest possible isocost line, satisfying the condition that MPL divided by w equals MPK divided by r. At this point, no reallocation of spending between inputs can reduce total cost. The firm has equalized the productive value of every dollar spent across all inputs.

Equalizing marginal product per dollar means MPL divided by w equals MPK divided by r. Each dollar redirected from one input to another would produce the same additional output, leaving total output unchanged while keeping cost constant. This condition confirms no cost-reducing reallocation is possible. If any input provides more output per dollar than another, the firm could reduce cost by using more of the higher-value input and less of the lower-value one.

When MPL over w exceeds MPK over r, each dollar spent on labor generates more additional output than a dollar on capital. The firm should shift spending toward labor and away from capital. As more labor is used, its marginal product falls, and as less capital is used, its marginal product rises, until the per-dollar marginal products are equalized. This adjustment moves the firm along the isoquant toward the cost-minimizing tangency point.

This statement is true. A strictly convex isoquant and a straight isocost line can only be tangent at exactly one point. This unique tangency is the single cost-minimizing input combination for the given output and price ratio. Multiple tangency points would require either a non-convex or kinked isoquant, which does not hold under standard assumptions of diminishing MRTS. The uniqueness of the optimal input combination is therefore guaranteed by these standard conditions.

A change in relative input prices rotates the isocost line, changing its slope. The isoquant remains unchanged since it depends on technology. The tangency point shifts to a new location along the isoquant reflecting the new price ratio. The firm substitutes toward the relatively cheaper input, moving to a new cost-minimizing combination that aligns with the updated input price ratio while keeping output constant.

The optimal input combination changes with the output level. Each output level corresponds to a different isoquant, and the tangency between that isoquant and its lowest feasible isocost line will generally be at a different point than the tangency for a higher or lower output. As output changes, the optimal mix of labor and capital may also change depending on the shape of the isoquants and the input price ratio. The optimal combination is output-specific, not universal.

Technical efficiency means a firm is on its production frontier, not wasting inputs. The optimal input combination is also economically efficient because it achieves the target output at minimum cost. A firm can be technically efficient while still not being at the cost-minimizing combination. The optimal input combination satisfies both technical and economic efficiency simultaneously, combining productive and allocative efficiency in input use.

MPL divided by w equals 20 divided by 5, which is 4. MPK divided by r equals 25 divided by 10, which is 2.5. Capital generates 2.5 units of output per dollar while labor generates 4 units. Labor is more productive per dollar, so the firm should shift spending toward labor and reduce capital spending. As more labor is used and less capital, their per-dollar marginal products will converge until equalized at the optimal combination.

When marginal products per dollar are unequal, the firm can improve: shifting a dollar from the input with lower marginal product per dollar to the one with higher marginal product per dollar increases total output while keeping spending unchanged, or equivalently achieves the same output at lower cost. Only when all inputs have equal marginal products per dollar does no such beneficial reallocation exist, confirming this as the minimum-cost condition.

The optimal input combination is found at the tangency between the target isoquant and the lowest feasible isocost line. At this single tangency point, the slope of the isoquant equals the slope of the isocost line, meaning MRTS equals the wage-to-rental rate ratio. This tangency confirms both that the target output is being achieved and that no cheaper input bundle exists, satisfying both the production constraint and the cost-minimization objective simultaneously.

At the optimal input combination, marginal product per dollar is equalized across all inputs, the isoquant is tangent to the isocost line, and MRTS equals the input price ratio. Using the largest possible quantity of each input is not a cost-minimization criterion and would exhaust the budget without identifying the cheapest mix. These three conditions together define and confirm the cost-minimizing optimal input combination.

When MPL rises, the per-dollar marginal product of labor increases relative to capital. The existing tangency is disrupted because MRTS now exceeds the input price ratio. The firm moves along the isoquant toward more labor and less capital until a new tangency is reached where MRTS falls back to equal the price ratio. The optimal input combination shifts toward more labor, reflecting the improved productive efficiency of each worker.

Marginal products alone are insufficient to determine the optimal input combination. While marginal products reflect the technical productivity of each input, the cost-minimizing combination requires comparing the marginal product per dollar, which depends on both productivity and input prices. Without knowing wages and rental rates, the firm cannot calculate MPL divided by w or MPK divided by r and cannot identify the point where these ratios are equal. Both technical and price information are essential.

As a firm scales its output up or down, the optimal input combination changes. The expansion path connects all the tangency points between successive isoquants and their corresponding lowest isocost lines, holding input prices constant. It shows the firm's cost-minimizing path of input use as it expands production. For a firm with constant input prices, the expansion path is typically upward-sloping, reflecting greater use of both inputs as output grows.

Allocative efficiency in input use means resources are deployed where they generate the most value per dollar. When MPL divided by w equals MPK divided by r, every dollar is equally productive across inputs. Any reallocation would generate less output per dollar from the receiving input than it sacrifices from the other, raising cost without raising output. This condition confirms the most value-generating and cost-minimizing allocation of the firm's input budget.