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PERIOD ANALYSIS

Period analysis show the inter-temporal dimension of production theory. It was developed by Alfred Marshallin his “Principles of Economics”, 1890, and has remained practically unaltered since. It tries to explain howequilibrium is achieved and explains the adjustment processes to reach it, going from the short-run equilibrium to the long-run equilibrium. His method is able to classify forces with references to the length of time needed for the production process; and then confine in ceteris paribus those forces that are of less importance with regards to the specific time we are considering.

Four periods can be distinguished: very short-run, short-run, long-run and very long-run.

The very short-run period is normally not studied as it is considered irrelevant. It is a period characterized by having all its production factors or inputs, fixed; thus implying there can be no change in their quantities.Costs are therefore considered to be fixed and logically there is inexistence of technological progress in this period.

The short-run period is a period in which some factors remain fixed while others may vary in their quantity. Capital elements, such as equipment, are considered fixed factors while labour is considered to be a variable factor. During this period, cost analysis has a large dependence on marginal cost and average total costs. But, since not all factors can be increased, this will result in diminishing returns, which will imply that marginal costs will increase. A firm should stop production until marginal costs reach average total costs in order to maximize profits. In this period technological progress is still non-existing.

The long-run period is formed by the succession of several short-run periods and it is considered to be like an envelope of many of these. As inputs can be increased, economies of scale can be achieved, so the cost of producing an extra unit of output will decrease, and thus reduce the average costs. Inputs and outputs should be increased until the point where the optimum size of the firm is achieved and from whereon an increase in size will only mean diseconomies of scale. All inputs may be varied but the basic technology of production still cannot be changed.

Finally, it is in the very long-run period when new production processes and new products come as a result of the existence of technological changes within this period, which allow for new ways of producing and different input requirements.
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PRODUCTION IN THE LONG RUN

When dealing with long run production, the main change from short run production is that we can vary the levels of fixed inputs we use (capital, K), as well as variable inputs (labour, L). Our levels of production will be determined by our returns to scale. It’s worth introducing here the concept homogenous functions. A function is considered homogenous if, when we have a multiplier, λ:

That is, we can reduce a production function to its common multiples multiplied by the original function. This is important to returns to scale because it will determine by how much variations in the levels of the input factors we use will affect the total level of production. Also, an homothetic production function is a function whose marginal rate of technical substitution is homogeneous of degree zero.

All this becomes very important to get the balance right between levels of capital, levels of labour, and total production. We can measure the elasticity of these returns to scale in the following way:

That is, the sum of the partial derivatives of production with regards to each factors multiplied by the proportion each input makes up of the whole. Graphically:

If µ is greater than one, we say that we have positive returns to scale (green). The more we produce, the less our costs will be per unit. If it is between one and zero, we have constant returns to scale (blue), and if it is negative, we will have negative returns to scale (red). When dealing with Cobb-Douglas functions, we can also determine which returns of scale are present, since α+β=µ.

All this will determine the average size of a business in the sector. Positive returns to scale, the most common scenario, will naturally attract a concentration of very large businesses, as is often the case in industrial, capital intensive sectors. Getting it right is crucial for another reason: in the long term, market equilibrium will determine that the total production of a certain good will be such to eliminate economic profits.
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LONG RUN COST ANALYSIS

In the long run, no cost is fixed. We can determine our production level and adjust plant sizes, investment in capital and labour accordingly. As we can see in the diagrams below, this gives us unlimited options. Depending on the scale we choose to implement, each level of production will be associated to new, short run cost curves. When we exhaust the infrastructure these provide us, we can upgrade to a new production level and so forth. The actual long run cost curve is made up of all of these individual scenarios, built up year after year.

If we look at average costs, the curve these draw is also the build up of the individual short run curves. These form a U shape, as we can see in the diagram. When average cost is decreasing with each additional investment, we are enjoying economies of scale, but not yet working at maximum efficiency. We reach this maximum efficiency at the minimum point, before the average cost per unit begins growing again.

Marginal costs only really make sense in the long run for each individual level of production. At optimal levels of production for each level, they intersect the long run average cost curve where this meets the short run average cost curve. This is what determines our optimal level of production. In summary, at our optimal level of production:

which is known as Le Châtelier’s principle. The other important lesson to take away from this is the fact that, when we reach the optimal point where efficiency is at its best, average and marginal costs are the same, as happened in the short run (see detailed graph below these lines). At this point, the slope of the tangent line, that is, the derivative, of the short run total cost (TCSR) is equal to zero, which is consequent with the fact that we are operating at maximum efficiency. What we can also glean from the graph is the fact that TCSR and TCLR are tangent where short run average cost (ACSR) and ACLR also meet, which is not necessarily an optimum. In fact, the only time where these junctions do suppose an optimum is in case 2.

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PRODUCTION I

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. -------------------------------------------------

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Production I: 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:

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:

As 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.

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Production I: Marginal rate of technical substitution

The marginal rate of technical substitution (MRTS) can be defined as, keeping constant the total output, how much should input 1 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.