Production I. Production and cost

 In this first LP on production, we will examine the decisions that lead to optimal levels of production. This is crucial, as it mirrors the same decisions that we saw consumers making: assigning our limited (and expensive!) resources in the best way possible in order to maintain optimal levels of production. This will ultimately lead us to the same dual problem: whether to minimise costs assuming an optimal, fixed level of production, or whether to hedge our bets and maximise production whilst fixing a budget for costs.

In order to do this, we will begin by looking at the basics, production and costs, with an entry on:

• Isoquants, or graphical representations of our production possibilities, and relate this to the
• MRTS, which shows the trade off between our inputs, and put all this together in our summary of the
• economic region of production, which represents our rational possibilities. We will then go on to examine
• production functions and their characteristics and finish this first section with
• isocosts, which closely relate to the concept of isoquants.

In the second half of this LP, we will go on to cover the problem of production duality, beginning with an entry on:

• Production maximisation, and then looking at the flip side of the coin with
• cost minimisation, before putting it together in
• production duality.

Production and cost. Isoquants

An isoquant shows the different combinations of K and L that produce a certain amount of a good or service. Mathematically, an isoquant shows:

f (K,L) = q0

Graphically, the shape of an isoquant will depend on the type of good or service we are looking at. The shape of isoquants is also in close relation with the terms marginal rate of technical substitution (MRTS) and returns to scale.

The first example of isoquant map showed in the adjacent graph is the most common representation. It shows four convex isoquants (green), showing each curve what amount of capital K the producer can stop applying when increasing the amount of labour L, while maintaining the quantity of output produced constant. This relation gives us the MRTS between these inputs, which is the slope of the curve in each of its points.

Our second example is an isoquant map with four parallel lines (cyan). This is the case for inputs which are perfect substitutes, since the lines are parallel and MRTS = 1, that is the slope has an angle of 45º with each axis. It can also be the case for inputs that are perfect substitutes but in different proportions. In that case, the slope will be different and the MRTS can be defined as a fraction, such as 1/2 ,1/3 , and so on. For perfect substitutes, the MRTS will remain constant.

Our third example shows an isoquants map with four isoquants (red) that represent perfect complementary inputs. This is, there will not be an increase on the amount produced unless both inputs increase in the required proportion. The best example of complementary inputs are shovels and diggers, since the amount of holes will not increase when there are extra shovels without diggers. Notice that the elbows are collinear, and the line crossing them defines the proportion in which each input needs to increase in order to have an increase in the production. In this case the horizontal fragment of each isoquant has a MRTS = 0 and the vertical fractions a MRTS = ∞.

Isoclines are lines which ‘join up’ the different production regions. Having defined  and decided the optimal levels of K and L we need to produce the different quantities, the line that passes through these optimal levels is an isocline (cyan). In other words, it is the line that joins points where the MRTS of each isoquant is constant:

Video – Isoquants:

s we all know, we need inputs in order to be able to produce goods and services. But how much of each? Let’s look at this in our entry on the MRTS, or Marginal Rate of Technical Substitution.

Production and cost. Marginal rate of technical substitution

The marginal rate of technical substitution (MRTS) can be defined as, keeping constant the total output, how much input 1 have to decrease if input 2 increases by one extra unit. In other words, it shows the relation between inputs, and the trade-offs amongst them, without changing the level of total output. When using common inputs such as capital (K) and labour (L), the MRTS can be obtained using the following formula:

The MRTS is equal to the slope of isoquants. In the adjacent figure you can see three of the most common kinds of isoquants.

The first one has a MRTS that changes along the curve, and will tend to zero when diminishing the quantity of L and to infinite when diminishing the quantity of K.

In the second graph, both inputs are perfect substitutes, since the lines are parallel and the MRTS = 1, that is the slope has an angle of 45º with each axis. When considering different substitutes inputs, the slope will be different and the MRTS can be defined as a fraction, such as  1/2 ,1/3, and so on. For perfect substitutes, the MRTS will remain constant.

Lastly, the third graph represents complementary inputs. In this case the horizontal fragment of each indifference curve has a  MRTS = 0 and the vertical fractions a MRTS = ∞.

Not to be confused with: marginal rate of substitution and marginal rate of transformation.

Video – Marginal rate of technical substitution:

We’re now in a position to study our production level possibilities globally, and determine which production regions make sense from an economic perspective. Understanding this is crucial if we don’t want to go bankrupt… This is what we’ll examine in our entry on the economic region of production.

Production and cost. Economic region of production

The economic region of production shows the combinations of factors at a certain cost that make economic sense. Areas outside the economic region of production mean that at least one of the inputs has negative marginal productivity. This region is marked by what are called ridge lines, which are simply the boundaries beyond which one of the two factors is being overused. Therefore, outside the economic region of production, there is clear inefficiency, and the company would be better off using less of one of the two factors, bringing costs down whilst maintaining equal production output. Graphically:

Video – Economic region of production:

In order to analyse these production possibilities, let’s have a look at production functions and their main characteristics.

Production and cost. Production function

A production function shows how much can be produced with a certain set of resources. Generally, when looking at production, we assume there are two factors involved in production: capital (K) and labour (L), as this allows us graphical representations of isoquants. However, any analysis made with 2 factors can mathematically be extended to n factors.

Therefore, a production function can be expressed as q = f(K,L), which simply means that q (quantity) is a function of the amount of capital and labour invested. In the adjacent figure, qx is function of only one factor, labour, and it can be graphically represented as shown (green).

It is well worth introducing here another concept: marginal productivity, which is how much more quantity we could produce by adding one unit more of a factor. As is logical, this will depend on how we are employing the factors we already have. The marginal product is the partial derivative of the production function with respect to the factor we are examining:

Marginal productivity decreases with each additional unit, as it can be seen in the above figure (cyan). At a certain point, the more workers we have, for example, the more each additional worker will be redundant if we do not invest in other necessary factors. This is the same as saying that the second derivative is negative. At that point, A, production is as efficient as possible.

Video – Production function:

And, to finish this first section, a quick mention on the subject of isocosts, which will tie in when we examine costs.

Production and cost. Isocosts

Isocost lines show combinations of productive inputs which cost the same amount. They are the same concept as budget restrictions when looking at consumer behaviour. Mathematically, they can be expressed as:

rK + wL = C

Where r is the cost of capital and w is the cost of labour. Generally, we think of r as the interest rate the financial markets offer, as capital requires investment. Even if the capital can be paid for using a company’s own resources, r is still equivalent to the opportunity cost of having the money tied up in investments rather than in liquid assets which offer a return (r) by lending it to the markets. The cost of labour (w) is the salary paid to employees per unit of time.

Isocosts are usually represented graphically together with isoquant lines (which are combinations of productive inputs which produce a fixed quantity of outputs). The two have a tangency point, which determines the optimal production (where production is maximised or cost minimised).

Video – Isocosts:

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