Least Cost Input Combination Quiz

The least cost input combination is the point at which the firm achieves a given output target at the minimum possible cost. Graphically, it is the tangency point between the target isoquant and the lowest possible isocost line. At this point, the MRTS equals the input price ratio, meaning the firm's technical substitution rate between inputs matches the market's price-based substitution rate, ensuring no cheaper input bundle exists.

The least cost combination is identified at the tangency between the isoquant and the lowest-cost isocost line that still reaches the target output. At this single tangency point, the slopes of both curves are equal, meaning MRTS equals the wage-to-rental rate ratio. Any other point on the isoquant either lies on a higher isocost line, representing greater total cost, or is unaffordable with the minimum budget needed to produce the target output.

At the least cost input combination, MRTS equals w divided by r. Rearranging this gives MPL divided by w equals MPK divided by r. This condition states that the marginal product per dollar spent must be equal across both inputs. If one input provides more output per dollar, the firm can cut costs by reallocating toward it. Only when both inputs yield equal output per dollar is no cost-saving reallocation possible.

When MPL divided by w exceeds MPK divided by r, each dollar spent on labor generates more output than a dollar spent on capital. The firm can reduce cost without reducing output by reallocating spending from capital to labor. As more labor is hired, its marginal product falls due to diminishing returns, and as capital is reduced its marginal product rises, until the two ratios equalize at the new least cost input combination.

At a tangency, the isocost line just touches the isoquant at one point, meaning it is the lowest-cost line that reaches the desired output level. At an intersection, the isocost line crosses the isoquant, meaning part of the isoquant lies below the isocost line. The firm could use a lower isocost line that is still tangent to the isoquant, achieving the same output at lower cost. The tangency is therefore the unique cost-minimizing point.

Minimizing cost does not mean minimizing total quantity of inputs. The optimal input combination depends on both the marginal products of inputs and their prices. A relatively cheap input should be used in greater quantities even if it increases total input volume, because doing so lowers total expenditure. The least cost combination is determined by equalizing marginal product per dollar across all inputs, not by minimizing the total number of input units used.

A rise in wages increases the relative cost of labor compared to capital. The isocost line rotates inward around the capital axis, changing its slope. The original tangency point becomes infeasible or suboptimal. To maintain the same output at minimum cost, the firm moves to a new tangency on the same isoquant that uses less labor and more capital, adjusting the input mix in response to the changed price ratio.

A technological improvement shifts the isoquant inward, meaning the firm can produce the same output with fewer inputs. The isocost line that is tangent to this new isoquant is lower than before, representing a reduced total cost. The firm reaches the same output target more cheaply. The new least cost combination involves fewer of both inputs, and the tangency condition MRTS equals w over r continues to identify the optimal mix at the new isoquant position.

Total cost equals the sum of spending on all inputs. Labor cost equals 5 workers multiplied by 10 dollars per worker, which is 50 dollars. Capital cost equals 4 units multiplied by 20 dollars per unit, which is 80 dollars. Total minimum cost is therefore 50 plus 80, which equals 130 dollars. This calculation confirms the least cost combination by applying input quantities directly to their respective prices.

A change in relative input prices alters the slope of the isocost line. The isoquant remains unchanged because it depends on technology, not prices. The tangency point between the isocost line and the isoquant shifts to a new location along the isoquant, reflecting the new optimal mix. The firm substitutes toward the relatively cheaper input and away from the relatively more expensive one until the new tangency condition MRTS equals the new price ratio is satisfied.

At the least cost input combination, MRTS equals the wage-to-rental rate ratio, the marginal product per dollar is equalized across all inputs, and the firm is at the unique tangency between the isoquant and the lowest reachable isocost line. Minimizing total input quantity regardless of prices is not a condition for cost minimization and can actually lead to higher total expenditure if cheaper inputs are underused.

The expansion path is derived by finding the least cost input combination for every possible output level while keeping input prices constant. Each point on the expansion path is a tangency between an isoquant and an isocost line. Connecting these tangency points traces the path of optimal input choices as output expands. The expansion path shows how the firm adjusts its input mix when growing production, holding the price ratio constant.

These are two distinct concepts. The least cost input combination answers how to produce a given output at the lowest cost, solving a cost-minimization problem. The profit-maximizing output level answers how much to produce by comparing marginal revenue and marginal cost. A firm must first find the least cost way to produce each output level and then determine which output level maximizes profit. Cost minimization is a necessary step toward profit maximization but is not equivalent to it.

When the firm has a specific output target, it must reach the corresponding isoquant using as little expenditure as possible. This means finding the isocost line with the smallest intercepts that still touches the target isoquant. The tangency point of this lowest feasible isocost with the isoquant is the least cost input combination. The firm achieves its required output while minimizing total input expenditure at current prices.

At any point on the isoquant other than the tangency, the marginal product per dollar is unequal across inputs. If MPL over w exceeds MPK over r, shifting a dollar from capital to labor raises output without raising cost, meaning the same output can be achieved more cheaply. This reallocation opportunity shows the current point is not cost-minimizing. Only at the tangency, where MRTS equals the price ratio, does no such profitable reallocation exist.