Optimal Employment of Two Inputs

The producer can select an input combination using the isoquants theory that maximizes output at the lowest cost or minimizes cost while producing a given input. When a producer hires such a pair of inputs that he has no inclination to rearrange inputs, he is said to be in equilibrium. In terms of the distribution of his entire cost expenditure among two inputs, he is then in a balanced position. The equilibrium point represents either the maximum output at a given cost or the minimum cost for a given production. There are two criteria in regard to defining producer’s equilibrium. They are as follows:

• Production maximization given a total cost budget.
• Cost reduction within a certain production quota.

Assumptions

• Output maximization given a total cost budget.
• An isoquant map and isocost line are necessary for a producer.
• A producer needs to be logical.
• A producer must make a certain total outlay.
• Two inputs' prices are still set in stone.
• Producer must invest a certain total cost outlay in two inputs in order to optimize output (i.e. Labour and Capital).

Conditions

• Necessary Condition: Isocost line is tangent to the isoquant [ or the slope of isoquants equals to the slope of isocost line, i.e. ( MRTSkl = MPk/MPl) = r/w ].
• Sufficient Condition: Iso-quant is convex to the origin.

Why does producer not attain equilibrium at higher isoquant, i.e. IQ3?

The idea of an isoquant map states that IQ3 produces a higher degree of output than IQ2, and so on. In order to achieve equation at such IQ, the producer also makes an effort. However, any points residing on such an IQ are unreachable (or beyond the investment capacity of the producer ). As a result, he is unable to achieve equilibrium at higher isoquant, IQ3.

Why does producer not attain equilibrium at any points lying on IQ2 except E?

Any points on IQ2 will produce the same amount of output. All other combinations outside E, however, are either impossible to get or lie outside the iso-cost line. Therefore, he is unable to reach equilibrium at any other places on IQ2 than E.

Why does producer not attain equilibrium at the intersecting point between IQ1 and AB isocost line? Why is tangency needed?

The AB isocost line and IQ1 cross at R and S locations in the illustration, respectively. R point has more labor units and fewer capital units. In this case, the producer uses capital instead of labor. In a similar vein, S point has fewer units of labor and more units of capital. The producer will here replace capital with labor. Up to the point where the slope of the isoquant equals the slope of the isocost line, the substitution process will be carried out. Therefore, tangency between the isoquant and isocost lines must exist in order to maximize production at a given total cost of investment.

Minimization of Cost at Given Production Quota

Assume the business owner has already agreed on the volume of output to be produced (or has already taken production order from a wholesaler). The next issue is whatever combination of factors the entrepreneur will use to attempt to create a specific amount of output. The entrepreneur will select the set of components to create a particular level of output that minimizes his cost of production since only in this way will he be maximizing his earnings. As a result, a producer will use the least expensive combination of components or the best pairing of two inputs to achieve a certain amount of output. Under this criterion, this point is also known as the producer's equilibrium point.

Assumptions

• Producers must own isoquant and the isocost line family.
• Producer needs to be sensible.
• Two inputs' prices are still set in stone.
• Each producer has a varied level of financial resources or overall cost commitment.
• Producers are required to produce at the lowest possible cost for a level of output.

Conditions

• Necessary condition: The slopes of the isoquant and isocost lines are equal.
• Sufficient condition: IQ curve convex to origin.

What is an expansion path?

An expansion path is the location of the points of tangency between isoquants and isocost lines (or a scale line).

Reference

Koutosoyianis, A (1979), Modern Microeconomics, London Macmillan