1 Economies and diseconomies of Size and Scale

From the perspective of the agricultural economist the terms economies and diseconomies of size and scale are not interchangeable and must be used very deliberately. I find that these are often misused and leads to a lot of confusion. I trust that you will really internalise the difference and use the terms correctly going forward.

1.1 Economies and diseconomies of Size

The term economies of size is used to describe a situation in which as the farm expands output, the cost per unit of output decreases. There are a number of reasons why costs per unit of output might decrease as output levels increase. The farm may be able to spread its fixed costs over a larger amount of output as the size of the operation increases. It may be possible to do more field work with the same set of machinery, equipment, management of administration capacity. A building designed for housing pigs might be used to house more animals than before, lowering the depreciation costs per unit of livestock produced. An expansion in output may reduce some variable costs. A farmer who previously relied on bagged fertilizer may be able to justify the additional equipment needed to handle fertiliser in bulk, while fixed costs for machinery and/or infrastructure may increase slightly, these increases may be more than offset by a reduction in the cost per unit of fertilizer.

The larger producer may be able to take advantage of pecuniary economies. As the size of the operation increases, the farmer might pay less per unit of variable input because inputs can be bought in larger quantities and thus this would result in a larger degree of competition among input suppliers. Such pecuniary economies might be possible for inputs such as seed, feeds, fertilizers, herbicides, and insecticides.

The term diseconomies of size is used to refer to increases in the per unit cost of production arising from an increase in output. There exist two major reasons why diseconomies of size might occur as the farm is expanded. One, a farmer who is successful in managing a 500 hectare farm in which does not make use of managers may not be equally adept at managing a 2000 hectare farm that includes a manager or two and several salaried employees. The skills of the managers will not necessarily be equivalent to the skills of the farm owner. A firm with many employees may not necessarily be as efficient as a firm with only few employees. Two, The farm may become so large that the assumptions of the purely competitive model are no longer met. This could result in the large firm to a degree determining the price paid for certain inputs or factors of production. Or the farm may no longer be able to sell all its output at the going market price. Although this may seem unlikely for a commodities such as maize and wheat, it is quite possible for speciality crops. For example, a friend of mine once flooded the market for prickly pears by accident to such an extent that the fruit could not be sold at any price.

1.2 Economies and diseconomies of Scale

The term scale of farm is substantially more restrictive than the term size of farm. If the scale of a farm is to increase, then each individual input must also increase proportionately. Therefore the term economy or diseconomy of scale refers to what happens when all input categories are increased proportionately. Assume that all input categories are doubled. If output doubles, neither economies or diseconomies of scale are said to exist, a situation that we refer to as constant returns to scale. If output more than doubles, economies of scale or increasing returns to scale exist. If output does not double, diseconomies of scale or decreasing returns to exist.

For economies or diseconomies of size to take place, all that is required is that the output level change, all inputs need not change proportionately. However, if economies or diseconomies of scale are to take place, not only must output change but each of the inputs must change in the same proportion to the others. For example, the term economies or diseconomies of scale could be used to describe what happens to per unit costs of production when all inputs are doubled, tripled, quadrupled, or halved. The term economies or diseconomies of size could be used to describe what happens to per unit costs of production when output is doubled, tripled, quadrupled, or halved when not all input levels changed in the same proportionate amounts.

The term scale is closely linked with the length of time involved since it is likely that it would not be possible to increase or decrease the quantity of all inputs proportionately within a short period. A proportionate increase in all variable inputs might be easier to achieve over the short term but much more time is likely to be needed to increase all fixed inputs proportionately.

The returns to scale can be illustrated using isoquants if we assume that we’re dealing with a production process that only requires two inputs as \(x_1\) and \(x_2\), this is shown in Figure 1. Remember that that each isoquant shows the same level of output and the slope of the isoquant shows the marginal rate of substitution (MRS) between inputs. Importantly, if we move along a line with constant slope drawn out of the origin of the graph (called an expansion path), the MRS is constant whenever it intersects the isoquant. In all three graphs the isoquants represent a proportionate increase in output. In graph one the respective proportionate increases in output is achieved by proportionate increases in inputs \(x_1\) and \(x_2\) (as indicated by the red lines) and thus represents constant returns to scale. Graph B shows a situation where less inputs is needed to achieve the next proportionate increase in output and thus represents increasing returns to scale. And Graph C therefore shows decreasing returns to scale since each proportionate increase in output requires more inputs to achieve it.

Figure 1: Isoquants and returns to scale. Source: Debertin (2012,160)

Figure 1: Isoquants and returns to scale. Source: Debertin (2012,160)

1.3 Homogeneous Production Functions

The terms economy or diseconomy can be quantified in the case of homogeneous production functions as a particular class of production functions. Homogeneous production functions has two important properties: One, the isoquants are symmetrical and two, the function exhibits the same returns to scale throughout. This is no the case with homothetic production functions which have symmetrical isoquants but who can have regions of constant, increasing or decreasing returns to scale.

A production function is said to be homogeneous of degree \(h\) when, if each input is multiplied by some number \(t\), then output increases by the factor \(t^h\). Assuming that the time period is sufficiently long such that all inputs can be treated as variable and are included in the production function, the degree of homogeneity (\(h\)) refers to the returns to scale. A function homogeneous of degree 1 is said to have constant returns to scale, or neither economies or diseconomies of scale. A function homogeneous of a degree greater than 1 is said to have increasing returns to scale or economies of scale. A function homogeneous of degree less than 1 is said to have diminishing returns to scale or diseconomies of scale.

Lets look at two examples. Suppose the production function \(y = Ax_1^{0.5}x_2^{0.5}\). If we multiply both \(x_1\) and \(x_2\) with \(t\), then we get the following:

\[ \begin{aligned} A(tx_1)^{0.5}(tx_2)^{0.5} &= At^{0.5}x_1^{0.5}t^{0.5}x_2^{0.5}\\[10pt] &= t^{0.5+0.5}Ax_1^{0.5}x_2^{0.5}\\[10pt] &= t^1Ax_1^{0.5}x_2^{0.5}\\[10pt] &= t^1y \end{aligned} \] Thus, this function exhibits constant returns to scale without any economies or diseconomies. Now suppose the production function \(y = Ax_1^{0.5}x_2^{0.8}\). If we multiply both \(x_1\) and \(x_2\) with \(t\), then we get the following:

\[ \begin{aligned} A(tx_1)^{0.5}(tx_2)^{0.5} &= At^{0.5}x_1^{0.5}t^{0.5}x_2^{0.8}\\[10pt] &= t^{0.5+0.8}Ax_1^{0.5}x_2^{0.8}\\[10pt] &= t^{1.3}Ax_1^{0.5}x_2^{0.8}\\[10pt] &= t^{1.3}y \end{aligned} \] This function is homogeneous of degree 1.3 and thus increasing returns to scale / economies of scale exist.

1.4 Euler’s Theorem

Leonhard Euler (pronounced “oiler”) was a Swiss mathematician who lived from 1707 to 1783. Euler’s theorem is a mathematical relationship that applies to any homogeneous function (symmetrical isoquants and the same returns to scale throughout). Euler’s theorem states that if a function is homogeneous of degree \(h\), the following relationship holds:

\[(\frac{\partial{y}}{\partial{x_1}})x_1+(\frac{\partial{y}}{\partial{x_2}})x_2=hy\] If the function is a production function, then we can substitute the partial derivatives.

\[MPP_{x_1}x_1 + MPP_{x_2}x_2 = hy = Ey\] where \(E\) is the scale elasticity parameter. If we multiply the function by the output price \(P_y\) then we get

\[ \begin{aligned} P_y MPP_{x_1}x_1 + P_yMPP_{x_2}x_2 &= hP_yy\\[10pt] VMP_{x_1}x_1 + VMP_{x_2}x_2 &= hTR\\[10pt] \end{aligned} \] where
\(VMP\) is the return from the use of one additional unit of the input \(x_n\)
\(TR\) is the total revenue as \(P_yy\)

Now lets apply Euler’s theorem. Suppose that a farm produces a single output using two inputs as labour and capital. Also suppose that the returns to scale is constant, i.e. \(h=1\). Following Euler’s theorem, the return to each unit of capital would be the \(VMP\) of the last unit multiplied by the quantity that is used, assuming that the farmer owned the capital. The wage rate for each unit of labour on the farm would be equal to its \(VMP\) and thus labourers would receive a wage rate equal to their \(VMP\). Since returns to scale is constant (and \(h=1\)) the payments for capital and labour would just exhaust the total revenue produced by the firm, hence there would be no pure or economic profit.

Now lets suppose a farm that produces as single output using capital and labour through a production process that exhibits increasing returns to scale. If the degree of homogeneity of the function were 3, and if each factor of production were paid according to its \(VMP\), total revenue would be more than exhausted (three times to be exact). The farmer would not have sufficient revenue to do this because the \(VMP\)s paid to each input or factor would be so very large.

Studies on returns to scale in agriculture often report a return to scale of 1.3 (Kislev & Paterson, 1995) which implies that inputs receive $1.3 for each dollar of output. How can this be sustainable? One possibility is to treat land as a residual claimant, i.e. you pay all else and only thereafter land. The coefficient of land in production function estimates typically is in the range of 0.1 and 0.15. If we assume that its 0.15, then it implies that 88% of output is paid to non-land inputs (\(100-\frac{0.15}{1.3} \times100\)) which receive $1.14 for each dollar of output (\(1.3\times 0.88\)). For this to occur, assuming that the farmer pays the \(VMP\) for all the other inputs, the return on land must be negative ($\(-14\)). However, the cost of land is positive and thus there is an inconsistency in the sense that the economies of scale could be over estimated because of measurement errors and the omission of other inputs.

2 The Cobb-Douglas function

The paper describing the Cobb Douglas production function was published in the journal American Economic Review in 1928. The original article dealt with an early empirical effort to estimate the comparative productivity of capital versus labour within the United States. Since the publication of the article in 1928, the term Cobb Douglas production function has been used to refer to nearly any simple multiplicative production function. The original production function contained only two inputs, capital (K) and labour (L), and the function was assumed to be homogeneous of degree 1 in capital and labour. A generalised Cobb Douglas production function can be specified as follow:

\[ y=Ax_1^{b_1}x_2^{b_2} \] where
\(y\) represents output
\(A\) represents Total Factor Productivity (TFP)
\(x_1\) represents the first input
\(x_2\) represents the second input
\(b_1\) represents the coefficient of \(x_1\), which represents the partial elasticity thereof \(b_2\) represents the coefficient of \(x_2\), which represents the partial elasticity thereof

This can be generalised to the following:

\[ y=A\prod^N_{i=1}x_i^{b_i} \] A Cobb Douglas function can be estimated using ordinary least squares after converting both sides of the equation to logarithms:

\[ \log y = \log A + b_1\log x_1 +b_2\log x_2 + e \] where in addition to the above,
\(e\) represents the regression error term.

Again, this can be generalised to the following:

\[ \log y= b_o +\sum^N_{i=1}b_i\log x_i \] where \(b_0 =\log A\)

The sections below will expand on the properties of the Cobb Douglas function which will serve as basis for understanding the other functional forms.

2.1 Marginal products

To get the marginal product of \(x_1\) we need to take the partial derivative of the function with respect to \(x_1\).

\[ \begin{aligned} MPP_{x_1}=\frac{\partial y}{\partial x_1} &= b_1(Ax_1^{b_1-1}x_2^{b_2})\\[10pt] \frac{\partial y}{\partial x_1} &= b_1(\frac{Ax_1^{b_1}x_2^{b_2}}{x_1})\\[10pt] \frac{\partial y}{\partial x_1} &= b_1(\frac{y}{x_1})\\[10pt] \frac{\partial y}{\partial x_1} &= b_1(APP_{x_1})\\[10pt] \end{aligned} \] Hence the marginal physical product of an input is simply the product of its coefficient (partial elasticity) and its average product. A Cobb Douglas exhibits declining marginal products under normal circumstances and is thus only defined for Stage II of the production process. This can be proven using the second derivative which will be smaller than 0 provided that its coefficient is smaller than one. In some cases the coefficient of an input can be greater than one but this could be the result of statistical problems such as multicollinearity.

\[ \frac{\partial^2 y}{\partial^2 x_1}<0\ \ \ \ \ if \ \ \ \ b_1<1 \] ## Inputs are compliments

As you will recall from lecture 5, two inputs are compliments if the MPP of one increases if the quantity of the other is increased. Inputs of a Cobb Douglas production process are compliments as long as they are positive, i.e. production takes place in Stage II.

2.2 Substitution elasticity equal to one

The elasticity of substitution of a Cobb Douglas production function is constant and equal to one. If you recall from the previous lecture this means that the isoquants are homogeneous and the inputs are imperfect substitutes. The fact that elasticity is equal to one can be proved mathematically:

\[ \begin{aligned} \sigma&=\frac{d(\frac{x_1}{x_2})}{\frac{x_1}{x_2}}/\frac{d(\frac{MPP_{x_2}}{MPP_{x_1}})}{\frac{MPP_{x_2}}{MPP_{x_1}}}\\[10pt] &=\frac{d(\frac{x_1}{x_2})}{d(\frac{MPP_{x_2}}{MPP_{x_1}})}.\frac{\frac{x_1}{x_2}}{\frac{MPP_{x_2}}{MPP_{x_1}}}\\[10pt] since \ \ \ \ MPP_{x1}&= b_1(\frac{y}{x_1})\ \ and \ \ MPP_{x2}= b_1(\frac{y}{x_2})\ \ then\\[10pt] \frac{MPP_{x_2}}{MPP_{x_1}}&=b_2\frac{y}{x_2}/b_1\frac{y}{x_1}\\[10pt] &=b_2\frac{y}{x_2}.\frac{1}{b_1}.\frac{x_1}{y}\\[10pt] &=\frac{b_2}{b_1}\frac{x_1}{x_2}\\[10pt] \frac{b_2}{b_1}&=\frac{\frac{MPP_{x_2}}{MPP_{x_1}}}{\frac{x_2}{x_1}}\\[10pt] therefore\\[10pt] \sigma&=\frac{b_1}{b_2}.b_2.\frac{y}{x_2}.\frac{1}{b_1}.\frac{x_1}{y}.\frac{x_2}{x_1}=1 \end{aligned} \]

2.3 Elasticity of production / output elasticities

You would recall from lecture 4 that the elasticity of production is defined as the percentage change in output divided by the percentage change in input, as the level of input use is changed:

\[ \begin{aligned} E_{x_1} &= \frac{y'-y''}{y}/\frac{x_1'-x_1''}{x_1}\\[10pt] &= \frac{\Delta y}{\Delta x_1}.\frac{x_1}{y} \end{aligned} \] In the case of the Cobb-Douglas production function, the output elasticities of the inputs(\(x_i\)) are equal to their corresponding coefficients(\(b_i\)). Therefore, if the the coefficient of capital is equal to 0.65 it means that if capital is increased by 1%, then output will increase by 0.65%.

2.4 Returns to scale

As we’ve seen in Section 1, the sum of the coefficients of a Cobb Douglas production function yields an estimate of the returns to scale of the production process.
\(\sum b_i<0\) decreasing returns to scale
\(\sum b_i=0\) constant returns to scale
\(\sum b_i>0\) increasing returns to scale

It is important to note that returns to scale can only be used as indicator for inter-firm comparisons and not for industries, acres or enterprises. In addition, only inputs controlled by the firm should be included. Therefore an input like rain should not be included when summing the coefficients.