What do you understand by the term least-cost solution is derived from technical and economic relationships in production? Provide a discussion of the diagrams you have presented. Make explicit any assumptions you are making. ‘Least-cost solution’ is found when marginal rate of technical substitution (MRTS) between factors is equal to the ratio of factor prices. It is a solution in a sense that it helps a firm (a price taker) in perfect competition to decide what the optimal level of output (Q) to produce, so that it can minimize its costs. In other words, a firm usually has to produce subject to how much costs it can afford, ‘least-cost solution’ gives answer to what the least cost way of producing a particular level of output is.

In the following essay, it explains the technical and economic relationships between inputs and outputs. The solution is then obtained by finding the equilibrium of these relationships. The answer might differ depends on the nature of a production, i.e. short run and long run. Before getting into further explanations, there are a number of assumptions to be made. Firstly, we assume that we are in a two-factor world, i.e. L (labour) and K (capital). The production function is therefore Q = f (K, L). Secondly, these inputs are hired in perfectly competitive markets so that firms can sell or buy them at the prevailing rental rates.

MRTS refers to the technical relationship between input and output. It can be shown by an isoquant, which indicates the different combination of the factors that produce the same output. MRTS is equal to change in K/change in L (K/L). It could also be an isoquant map that shows different level of Q. The following isoquant map (diagram 1) gives us information about the dolls production in a toy factory. The isoquant Q1 produces 500 dolls, Q2 produces 600 dolls, and Q3 produces 700 dolls. A firm expands its production by moving from one isoquant to another.

According to diagram 2, if the factory is producing 500 dolls at point S, it is capital-intensive technique because 30 capitals (K) are used while only 10 workers (L) are employed. If it is producing 500 dolls at point T, it is more labour-intensive technique compare with point S. As moving down the isoquant, it substitutes more and more labour for capital. From point S to point U, MRTS is equal to (30 – 20)/(12 – 10), which is 5. From point R to point T, MRTS is equal to (15 – 10)/(30 – 16), which is 0.357. This illustrates diminishing marginal rate of factor substitution that causes the shape of the isoquant.

Diminishing MRTS can also be related to the law of diminishing returns. Any point on the same isoquant produces the same output by definition. That is, output remains the same when we use less K and more L as moving down the isoquant. This means the loss in output by using less K (marginal output of K (MPk) x ?K) is completely compensated by the gain in output by using more L (marginal output of L (MPl) x L). Therefore, we can say that MPk x ?K = MPl x ?L and ?K/?L = MPl/MPk if we rearrange the equation. As one moves down an isoquant, the production will get more and more labour-intensive because it is substituting more labour for capital. MPl will decrease relative to MPk. Given that MRTS = ?K/?L = MPl/MPk, MRTS must fall when MPl/MPk decreases.

The ratio of factor prices can be deduced from an isocost that demonstrates the economic relationship between input and output. It is assumed previously that there are only two factors in production and prices of factors are fixed as r and w. An isocost shows all the combinations of two factors that cost the same to employ. Kmax and Lmax are the maximum amount of K and L the firm is able to purchase with its outlay. The formula of an isocost is C = wL+ rK, given that C is the firm’s costs. Kmax = C/r and Lmax = C/w. Joining Kmax and Lmax forms the isocost. Let the outlay be \$1000, r be \$10 and w be \$50. Kmax is equal to 100 and Lmax is equal to 20. The slope of the isocost line is given by the quantity of K divided by that of labour. This also gives us the inverse price ratio, which is w/r. When the firm increases its outlay, the whole isocost line shifts to the right as Kmax and Lmax increases.

In diagram 4, isoquant Q1 shows the combination of factors to produce 500 dolls. Both of isocosts TC and TC1 (which have same slope) intersect Q1. However, at point C or point D, the outlay will be more than that at point E. The least costly point on isoquant Q1 is therefore E where isocost touches isoquant, i.e. MRTS = w/r. At E, the firm achieves ‘least-cost solution’. The cost-minimizing input combination is L and K. To expand production, the firm simply moves from E to E1 on Q2, and still spending least cost.

As for the short run, there is at least one fixed cost. Typically, K is kept constant. In diagram 5, it shows that K is a horizontal straight line. It is impossible for the firm to move from E1 to E2 even if it increases L from L1 to L2, which means the firm will not be at equilibrium if it produces more. Therefore, a firm in short run cannot expand its production and still achieve economic efficiency.

In conclusion, a ‘least-cost solution’ is derived from both economic and technical relationships between inputs and outputs of a firm. The equilibrium of the two gives us the least cost, economic efficient way of producing a certain quantity. The point where the isocost is tangent to the isoquant, that is, when marginal rate of technical substitution is equal to the ratio of factor prices, is the general least cost solution. However, this answer is only applicable when all factors, in this case, K and L, are variable. This means a short run firm, which has at least one fixed cost, will not be able to achieve economic efficiency and cost minimization at the same time while expanding production.

Reference:

Sloman J. (2000), Economics, Financial Times/Prentice Hall, New York. Nicholson W. (1998), Microeconomics Theory: basic principles and extensions, The Dryden Press, Fort Worth.