26 Sept 2000

Production

Production is the entire process of making goods and services. In theoretical terms production involves the transformation of inputs into outputs, and in practice involves input sourcing, capital equipment acquisition, industrial engineering, adaptation of new technology, personnel training and management, and a great deal of managerial coordination. Production is a firm's response not only to the consumer market signals of 'what' type of good or service is demanded, but also to the input market signals of 'how' to produce, given the costs associated with various labor and capital/technological inputs.

Lecture Outline

• Production Functions
• Returns to Scale and Returns to a Factor of Production
• Total, Average, and Marginal Product
• Production Planning When One Factor is Fixed: Diminishing Marginal Returns
• Production Planning When Factors are Variable: Isoquants and the Marginal Rate of Technical Substitution
• The Role of Revenue and Cost in Production
  • Marginal Revenue Product
  • Deriving Input Demand
  • Isocosts
  • Isocosts and Isoquants: Deriving the Optimal Input Combination
• A Brief Discussion of the Empirical Estimation of Production Functions

Lecture Notes

Production Functions

A simple production function describes the process of transforming a set of inputs I1, I2, I3,..., into a quantity Q of output (goods or services):

Q = f(I1,I2,I3,...,In)

In the actual world of production, often times inputs (or outputs) cannot easily be divided into very small units. In this case we describe the production process through a discrete production function. In other instances in the field, and in conceptual illustrations (textbooks, lecture notes), inputs and outputs can easily be divided into very small, marginal units. In this latter case we can describe the production process through a continuous production function.

Where do production functions come from, and why might they be helpful? First, firms and other organizations can empirically estimate a production function using regression techniques, as briefly described at the end of the unit. In other cases the technology of production calls for a very specific 'recipe' in which we know rather exactly the input combinations that yield a particular quantity of output, and we have a fairly clear idea of the extent to which one input can substitute for another. In particular, inputs may be nearly perfect substitutes for each other, as for example are coal and natural gas in some dual-fuel electricity generating plants, or nearly perfect complements, as for example are paper stock and ink in the printing process. We also think that there are situations in which inputs are substitutes, but only very imperfectly so, as for example with labor and capital equipment (ex: laborers and backhoes in ditch digging).

Returns to Scale and Returns to a Factor

First, suppose the firm is considing expansion of its facilities, and the question is how much additional output will be generated by, say, a fully proportionate doubling of all inputs. This question addresses the issue of returns to scale. Production processes may feature increasing, constant, or decreasing returns to scale over a particular range of expansion, and moreover may experience two or more of these over the entire range of production.

• Increasing Returns to Scale: Output increases more than proportionately with an increase in all inputs. For example, a doubling of all inputs will result in a more than doubling of output.
• Constant Returns to Scale: Output increases proportionately with an increase in all inputs. For example, a doubling of all inputs will result in a doubling of output.
• Decreasing Returns to Scale: Output increases less than proportionately with an increase in all inputs. For example, a doubling of all inputs will result in a less than doubling of output.

Returns to scale help the firm determine its optimal size, and thus is naturally a long-run question in microtheory.

Second, managers may need to know how total output will increase when one particular input (or subset of all inputs) is increased Returns to a factor, or factor productivity, refers to the relationship between increases in one paricular factor of production and output. Factor productivity is important because it helps firms determine the optimal production technology, or more specificially the optimal combination of various inputs in a situation in which inputs are neither perfect substitutes nor perfect complements, but instead can at least imperfectly substitute for one another.

Total, Average, and Marginal Product

We can use tabular data on Q and variation in input usage to plot a total product curve. If only one input varies, as for example with labor in short-run analysis in which the a professional services firm has committed to a particular size of capital (office space and number of computers and other equipment), then we can draw a simple two-dimensional total product curve as follows:

We can be a bit more general, however, and draw a total product surface in which we allow two (or more) inputs to vary (see Figure 7.1 in the textbook on pg. 264 for a three-dimensional discrete production function).

Marginal product (MP) of a particular factor of production, for example labor, is defined to be the change in output (Q) resulting from a one-unit change in the input (ex: one more hour worked, or one more worker employed). If again we use the symbol '@' to mean 'change in', then we have:

MP(X) = @Q/@X

where X is a particular input, such as labor, and Q is output of the final good or service produced. Marginal product is the slope of total product. If we have a nice continuous total product curve, then marginal product at a particular point on the total product curve is simply the slope of the tangent line at that point on the total product curve.

Finally, average product is the average amount of output produced with each unit of input:

AP(X) = Q/X

When we have a nice, continuous total product curve, average product at a particular point on the total product curve is the slope of a ray that comes out of the origin of the diagram and passes through the point in question on the total product curve.

Production Planning When One Factor is Fixed: Diminishing Marginal Returns

• Gains in marginal productivity as we add more of a variable factor of production (ex: workers) due to specialization
• Declines in marginal productivity as we add more of a variable factor of production due to congestion in the fixed factor (ex: a kitchen).

In the general case we do not think that diminishing marginal returns sets in immediately because there frequently are economies of specialization, especially for labor as a variable factor. For example, as we initially hire a second kitchen worker, the two kitchen workers can now specialize -- one preps food ingredients, the other combines ingredients (cooks them) to make final meals.

Thus we might make a generalization for the purposes of illustration of the concept of marginal productivity that there are three stages of production:

• Stage 1: Increasing marginal returns
• Stage 2: Diminishing marginal returns
• Stage 3: Negative marginal returns

These three stages are demarked by slim black vertical lines in the diagram below:

The first vertical line corresponds to the level of variable factor employed at which economies of specialization are exceeded (in output space) by diseconomies of congestion, leading to the boundary between increasing and decreasing marginal returns to the variable factor of production. The second vertial line corresponds with the level of variable input employed at which marginal product becomes negative, implying the boundary between diminishing and negative marginal returns.

The arrow points to the point on the total product curve where average product (AP) reaches its peak -- this point on the TP curve corresponds to the steepest ray out of the origin that touches the TP curve.

Production Planning When Factors are Variable: Isoquants and the Marginal Rate of Technical Substitution

Isoquants are to production what indifference curves are to utility.

Movement along an isoquant illustrates the concept of input factor substitution, meaning the extent to which one input (say, for example, labor) can substitute for another input (say, for example, a backhoe) in producing a given output (say, for example, a ditch):

Note that higher isoquants imply higher levels of output. Isoquants are similar in many ways to indifference curves, and serve a role in optimal input combination selection as indifference curves do in optimal consumption bundle selection.

Movement along the isoquant involves:

• A change in the input combination
• The same level of total output

Thus movement along the isoquant involves input substitution -- as more of one input is used, less of the other must occur in order to keep total output constant.

The Marginal Rate of Technical Substitution (MRTS) is the slope of the isoquant, and tells us how much of the input on the 'y' axis we must give up in order to use more of the input along the 'x' axis. Thus in the isoquant diagram above, the MRTS relates the number of backhoes that must be reduced as one utilizes more laborers in order to keep the output (yards of ditch) constant. Using the symbol '@' to denote 'change in', we have:

MRTS = @Y/@X = slope of the isoquant

Where Y and X are inputs in the production process. Since movement along the isoquant holds Q constant, we also can show that:

@Q = 0 = MPx@X + MPy@Y

rearranging the equation above gives us:

-MPx/MPy = @Y/@X

Thus the slope of the isoquant is equal to -(ratio of marginal products). Recall the law of diminishing marginal returns. As we use more and more of the 'x' input, and less and less of the 'y' input, the marginal product of the 'y' input rises while the marginal product of the 'x' input falls, meaning that the isoquant for imperfect substitutes becomes flatter as one moves down along the isoquant using more 'x' and less 'y'. When inputs are perfect complements, such as wheels and chassis for automobiles, the isoquant has a 'square L' shape (why?). When inputs are perfect substitutes, such as farmer Brown's wheat vs farmer Jones' wheat, the isoquant is a straight line (why?).

The Role of Revenue and Cost in Production

Marginal Revenue Product

MRP(X) = [@TR/@Q]@Q/@X = MR(Q)xMP(X)

MRP is a measure of the economic value of employing an additional unit of input in the production process, and derives from the value of the final good or service being produced. Recall that in general we assume that as more and more of a variable input is added to a fixed production facility, the marginal product of that input will eventually decline. Thus, for example, if the firm is selling in a perfectly competitive market, so that MR(Q) = P, then MRP(X) declines as more and more of input 'X' is employed because MP(X) declines.

Deriving Input Demand

To see this, consider the following example:

If we plot units of input X on the 'x' axis, and $MRP on the 'y' axis, we get the firm's demand curve for input X (which is derived from the demand for the final good produced and sold to consumers):

Thus we can see graphically that the profit-maximizing firm will hire 6 units of input X, where the MFC ($20) equals the MRP (crosses the factor demand curve).

Isocosts

B = P(X)X + P(Y)Y

Where P(X) and P(Y) represent the MFC of X and Y (assuming a competitive factor market) we can derive an equation of the line:

Y = B/P(Y) - [P(X)/P(Y)]X

As we move down along an isocost, two things happen:

• We increase the 'X' input and decease the 'Y' input
• We hold total costs on inputs constant

Thus we have:

@B = 0 = P(Y)@Y + P(X)@X)

@Y/@X = -P(X)/P(Y)

which says that the price ratio is equal to the slope of the isocost line.

Isocosts and Isoquants: Deriving the Optimal Input Combination

MRTS = -(price ratio) = -[P(X)/P(Y)]

And since the MRTS = - (MP(X)/MP(Y)), then the cost-minimizing input combination occurs where:

[MP(X)/MP(Y)] = [P(X)/P(Y)].

Profit maximization requires that each and every input is employed up to the point at which the MRP = MFC. While cost minimization is necessary for profit maximization, it is not sufficient, because it does not also take into account the revenue side. The MRP = MFC equation includes both cost and revenue information, however, and thus indicates the optimal amount of each input to employ:

Profit-maximization requires that firms employ inputs so that

MR = MC

Recall that

MC = MFC(X)/MP(X) = P(X)/MP(X)

and

MC = MFC(Y)/MP(Y) = P(Y)/MP(Y)

when the factor market is competitive, we can set MC = MR and get:

MR = P(X)/MP(X) and

MR = P(Y)/MP(Y)

and rearranging yields

MFC(X) = P(X) = MP(X)xMR = MRP(X)

and MFC(Y) = P(Y) = MP(Y)xMR = MRP(Y)

and which implies (since MR cancels out)

P(X)/P(Y) = MP(X)/MP(Y)

which is also the condition for the tangency of isoquant and isocost.

A Brief Discussion of the Empirical Estimation of Production Functions

Q = a + bXY + cYX^2 + dXY^2 - eYX^3 - fXY^3

which gives the general structure with increasing, following by decreasing, marginal returns.

When the data do not feature enough dispersion to provide observations over both increasing and decreasing marginal returns regions, economists can still get reasonably good results by estimating the following production function:

Q = cX^aY^b

which can be estimated using linear regression techniques by way of log transformation:

log Q = log c + alogX + blogY

Note that if (a+b) is found through estimation to be less than 1, we have diminishing marginal returns, while if (a+b) is found to be greater than 1 we have increasing marginal returns (if they are equal to 1 we have constant marginal returns). Power function forms of production functions were pioneered by Cobb and Douglas.