Describe the significance of production function in business decision-making. How do firms achieve optimal employment of two variable inputs? Explain.

The production function, which expresses the technical relationship between inputs and output as Q = f(L, K), is a central tool in business decision-making, and a firm achieves the optimal employment of two variable inputs at the tangency of its iso-cost line and the highest attainable iso-quant.

The significance of the production function:

1. Shows the input–output relationship: it tells the manager exactly how much output each combination of labour and capital will yield, which is the starting point of every production decision.
2. Helps find the least-cost input combination: together with input prices it identifies the cheapest way to produce a given output, controlling cost.
3. Reveals returns to scale: it shows whether expanding all inputs gives increasing, constant or decreasing returns, guiding decisions about the scale of operation.
4. Guides the choice of technology: through factor intensity (e.g. the ratio α/β in a Cobb-Douglas function) it shows whether a capital-intensive or labour-intensive technique suits the firm's input prices.
5. Helps in output planning and forecasting: knowing the function, the firm can predict how much output a planned input level will produce.
6. Aids cost estimation: since costs depend on input quantities and their prices, the production function underlies the firm's cost curves and profit planning.
7. Supports expansion decisions: by tracing the expansion path it shows how the input mix should change as the firm grows.
8. Identifies the efficient stage of operation: through the law of variable proportions it shows that a rational producer should operate in Stage II, where output is maximised and marginal product is still positive.

How firms achieve optimal employment of two variable inputs:

9. Goal: the firm seeks to produce the maximum output at minimum cost, i.e. the optimal (least-cost) combination of labour and capital.
10. Tools used: the iso-quant (equal-output curve) and the iso-cost line (C = wL + rK) are combined.
11. Slopes to compare: the slope of the iso-quant is MRTS = MPL/MPK and the slope of the iso-cost line is the price ratio w/r.
12. Necessary condition: optimal employment occurs where the iso-cost line is tangent to the highest attainable iso-quant, so MRTS = MPL/MPK = w/r.
13. Restated condition: this is equivalent to MPL/w = MPK/r, meaning the last rupee spent on each input yields the same marginal product.
14. Sufficient condition: the iso-quant must be convex to the origin at the tangency, ensuring it is truly a least-cost point.
15. Result and example: at tangency point E the firm employs the input mix that minimises cost for its output; for instance, a garment workshop chooses the balance of tailors and machines where the extra output per rupee spent on labour equals that per rupee spent on machinery.

Thus, the production function is significant because it links inputs to output and underlies cost, technology, scale and forecasting decisions, and a firm reaches the optimal employment of two variable inputs where MRTS equals the input-price ratio (MPL/w = MPK/r).