The production function is the backbone of the Theory of the Firm. It describes the current state of technology and how input can be transformed into output.
The production function can be displayed in a variety of ways, including product curves and isoquants. In every optimization problem faced by the firm, the production function is included.
Key Definitions and Assumptions
Inputs, also known as factors of production, are used to make output, sometimes called product. As shown in Figure 10.1, the firm is a highly abstract entitya black boxthat transforms inputs into output.
Figure 10.1: The black box nature of the firm.
The specific details of how the firm is organized and how it actually combines the inputs to make goods and services is ignored by the theory, hidden in the black box.
Inputs are often broken down into large categories, such as land, labor, raw materials, and capital. We will simplify even further by collapsing everything that is not labor into the capital category.
Labor, L, is human toil and effort. It is measured in units of time, usually hours.
Capital has a confusing history in economics. As a factor of production, capital, K, means things that produce other things, such as machinery, tools, or equipment. That is different from financial or venture capital that is a fund of money. The title of Karl Marx’s famous book, Das Kapital, uses capital in the sense of wealth, denominated in money. The Theory of the Firm’s K is measured in numbers of machines.
Like labor, capital is rented. The firm does not own any of its machines or buildings. This is extremely unrealistic, but allows us to avoid complicated issues involving depreciation, financing of machinery purchases (debt versus equity, for example), and so on.
Another extreme simplifying assumption is that there is no time involved. Like the consumer maximizing utility subject to a budget constraint, the firm exists only for a nanosecond. It makes decisions about how much to produce to maximize profits with no worries about inventories or the trajectory of future sales. It produces the output in an instant.
We avoid complications arising from the production of more than one good or service by assuming that the firm produces only one product. That makes revenues simply price times quantity sold of the one product.
Without going into detail again about unrealistic assumptions, it seems helpful to point out that we are not trying to build an accurate model of a real-world firm. Our primary goal is to derive a supply curve. We want to know how a firm responds to a change in price, ceteris paribus. By assuming away many real-world complications, we can model the firm’s maximization problem, solve it, and do comparative statics to get the supply curve.
Mathematical Representation
Just like the Theory of Consumer Behavior, which uses a utility function to model tastes and preferences, the Theory of the Firm uses a production function to capture the ability of firm’s to transform inputs into outputs. Unlike utility, production is objective and observable. We can count how much output is made from a given number of hours of labor and machines.
The production set describes all of the technologically feasible outputs from a given amount of inputs. The production function describes the maximum output possible from a given amount of inputs. Notice how the production function assumes the inputs are being used in the best way possible.
The most abstract, general notation for a production function is \(y = f(L, K)\). The \(f()\) represents the technology available to the firm. A specific, concrete example of a production function is the Cobb-Douglas functional form: \(y=AL^\alpha K^\beta\). Let’s see what it looks like in Excel.
STEP Open the Excel workbook ProductionFunction.xls, read the Intro sheet, then go to the Technology sheet to see an example of the production function.
In Figure 10.2, the production set is the surface of the 3D object and everything inside; the production function is just the surface.
Figure 10.2: The production function.
Source: ProductionFunction.xls!Technology.
The production function implicitly includes an already solved engineering optimization problemit gives the maximum output from any given combination of inputs. In other words, we are assuming that the inputs are organized in their most productive configuration and nothing is wasted.
Notice that the Cobb-Douglas function on the Technology sheet has been set up so it can be controlled by a single parameter, \(\alpha\) (alpha), by making the exponents \(\alpha\) and (\(1 - \alpha\)). Use the scroll bar to change alpha and notice how the shape of the production function surface changes. Alpha is a parameter that takes values between zero and one.
STEP Click the button to return the sheet to its default, initial position.
Product Curves
In addition to the 3D view, the production function can be displayed in other ways. To graph the production function in two dimensions, we need to suppress an axis. If we keep output and suppress one of the input axes we get a total product curve. If we suppress output and keep the two inputs, we get an isoquant.
Product and output mean the same thing. The total product curve is the number of units of output produced as one input is varied, holding the other constant.
STEP Click the and buttons to see the product curves for labor and capital.
In addition to the total product curves, there are average and marginal product curves. The average product is simply output per unit of input. Thus, the average product of labor is Y/L and the average product of capital is Y/K.
The marginal product curves tell us the additional output that is produced as input is increased, holding the other input constant. Marginal product can be computed based on finite-size changes in an input or via the derivative.
Via calculus, the marginal product is simply the derivative of the production function with respect to the input. For the Cobb-Douglas function in the Technology sheet, the marginal products are found by taking the partial derivatives with respect to L and K: \[MP_L = \frac{\partial Y}{\partial L}=(1 - \alpha)AK^\alpha L^{(1-\alpha)-1}=(1 - \alpha)AK^\alpha L^{-\alpha}\] \[MP_K = \frac{\partial Y}{\partial K}=\alpha AK^{\alpha -1}L^{1-\alpha}\]
STEP Scroll down and click on cell C52 to see that the marginal product is computed via the change in output from an increase of 2 hours of labor, with \(K=4\).
This computes the marginal product of labor as the rise over the run from \(L=0\) to \(L=2\) on the total product curve.
STEP Click the button and then click on cell C58 to reveal the marginal product computed via the derivative.
Since the total product is a curve, the slope of the tangent line at \(L=2\) is not the same as the rise over the run from one point to another.
STEP Now look at the total, marginal, and average product curves.
Notice how the product curves are drawn based on a given amount of capital. If the amount of capital changes, then the product curves shift.
Marginal and average product can be graphed together because they share a common y axis scale, output per unit of input. The total product curve can never be graphed with the marginal and average product curves because the total product curve uses output as its y axis scale.
The graphs demonstrate that when total product increases at a decreasing rate, marginal product is decreasing. When total output increases at a decreasing rate as more input is applied, ceteris paribus, we are obeying the Law of Diminishing Returns. As long as alpha is between zero and one, our Cobb-Douglas production function exhibits diminishing returns.
The Law of Diminishing Returns does not deny that there can be ranges of input use where output increases at an increasing rate. It says that, eventually, continued application of more input along with a fixed factor of production must lead to diminishing returns in the sense that output will increase, but not as fast as before. Thus, the Law of Diminishing Returns is simply a statement that marginal productivity must, eventually, be falling.
As with utility, the Cobb-Douglas functional form is convenient, but there are many, many other functional forms available.
STEP Proceed to the Polynomial sheet to see a different functional form. The charts are strikingly different than before.
Unlike the Cobb-Douglas functional form, which always shows diminishing returns, the polynomial production function exhibits all three different phases of returns: increasing, diminishing, and negative returns.
At low levels of labor use, output is increasing at an increasing rate so the total product curve is curved upward and marginal product is increasing. In this range, as long as marginal product is rising and output is increasing at an increasing rate, output rockets upward, growing faster and faster.
When the marginal product curve reaches its peak, the total product curve is at an inflection point. From here, additional labor leads to increases in output, but at a decreasing rate, leveling off as L increases. We say that diminishing returns have set in.
The Polynomial sheet is color coded so it is easy to see where the total product curve changes character. Cells with yellow backgrounds signal the range of labor use where diminishing returns apply.
As more and more labor is used, total product reaches its maximum point (where marginal product is zero). Beyond this point, we are in a range of negative returns. This is a theoretical possibility, but not a practical one. No profit-maximizing firm would ever operate in this region because you can get the same amount of output with fewer workers.
It is worth remembering that the Law of Diminishing Returns does not say that we always have diminishing returns for every level of labor use. Instead, the law says that, eventually, diminishing returns will set in. It is also important to understand the difference between diminishing and negative returns. The former says output is rising, but slower and slower, while the latter says output is actually falling.
Notice the relationship between the marginal and average product curves. It is no coincidence that the marginal product curve intersects the average product curve at the maximum value of the average product. There is a guaranteed relationship between marginal and average curves: Whenever the marginal is greater than the average, the average must be rising and whenever the marginal is less than the average, the average must be falling. Thus, the only time the two curves meet is when the marginal and average are equal.
STEP Change the parameter for the b coefficient from 30 to 40.
Notice that the S shape becomes much more linear. The range of increasing returns is larger and we do not hit negative returns over the observed range of L from 0 to 25.
STEP Set the parameter for the b coefficient to 80.
Over the observed range of L from 0 to 25, we see only increasing returns.
STEP Change the \(\delta L\) parameter from 1 to 2. This makes L go up by two and the range goes from 0 to 50.
Diminishing returns do kick in; it just takes more labor for the Law of Diminishing Returns to be observed when the b coefficient is set to 80.
Diminishing versus Decreasing Returns
One extremely confusing thing about the Law of Diminishing Returns has to do with another concept called returns to scale. Unlike the Law of Diminishing Returnswhich is based on applying more and more of a particular input while holding other inputs constantreturns to scale focuses on the effect on output of changing all of the inputs by the same proportion.
There is no law for returns to scale. A production process may exhibit increasing, decreasing, or constant returns to scale, across all values of input use. For example, the Cobb-Douglas function in the Technology sheet has constant returns to scale because if you double L and K, you are guaranteed to double output.
You can see this is true by comparing the points 2,2 and 4,4 in the table in the Technology sheet. A more complete demonstration uses a little algebra.
We begin with the production function: \[AK^\alpha L^{1-\alpha}\] Next, we double both L and K: \[A(2K)^\alpha (2L)^{1-\alpha}\] We expand the terms with exponents: \[A(2^\alpha) (K^\alpha) (2^{1-\alpha})(L^{1-\alpha})\] We collect the "2" terms: \[A(2^{\alpha + (1-\alpha)} (K^\alpha) (L^{1-\alpha})\] The alphas add to zero (\(\alpha - alpha = 0\)) so we get: \[A2K^\alpha L^{1-\alpha}\] Thus, we have shown that doubling the inputs from any input levels leads to doubling the output, and this is called constant returns to scale. If the exponents in the Cobb-Douglas function do not sum to 1, then the function does not exhibit this property.
The Cobb-Douglas function in the Technology sheet obeys the Law of Diminishing Returns for each input (with \(0 < \alpha <1\)), yet it has constant returns to scale. Do diminishing returns imply decreasing returns to scale? No, absolutely not. The two concepts are independent. They ask different questions. The Law of Diminishing Returns is about what happens to output when a single input is increased, ceteris paribus, and decreasing returns to scale says that output will less than double when all inputs are doubled.
Isoquants
In addition to product curves, another way to represent the production function uses the isoquant. The prefix iso, meaning equal or the same (as in isosceles triangle), is combined with quant (referring to the quantity of output) to convey the idea that the isoquant displays the combinations of L and K that yield the same output.
STEP Return to the top of the Technology sheet and click the button (near cell H28) to see the isoquant map, as displayed in Figure 10.3.
Figure 10.3: Isoquants for a Cobb-Douglas technology.
Source: ProductionFunction.xls!Technology.
An isoquant is simply a 2D, top down view of the 3D surface. Unlike the product curves, which give a view from the side, the isoquant shows L and K on the x and y axes, respectively, and suppresses output.
Notice that Excel cannot correctly draw the isoquant map, putting garbled characters in the bottom left-hand corner of the chart and producing a squiggly, jagged display at the bottom.
You might be thinking that it looks a lot like an indifference map. There are definitely strong parallels between isoquants and indifference curves. Both are top-down views of a 3D object and, therefore, both are level curves or contour plots. Both are used to find and display the solution to an optimization problem.
However, there is one critical difference: unlike an indifference curve, each isoquant is, in principle, directly observable and the isoquants can be compared on a cardinal scale. With indifference curves, the utility function is a convenient fiction and the numerical values merely reflect rankings. No one cares that a particular indifference curve yields 28 utils of satisfaction. This is not the case for isoquants because the suppressed axis, output, is measurable. You can certainly say that one isoquant gives twice the output as another or that one isoquant gives 17 more units of output than another.
One way in which indifference curves and isoquants are the same is that we can compute the slope between two points or the instantaneous rate of change at a point on an isoquant. To avoid confusion with MRS, we call this slope the technical rate of substitution, TRS. With labor on the x axis and capital on the y axis, the TRS tells us how much capital we can save if one more unit of labor is used to produce the same level of output.
From one point to another, the TRS can be computed as the rise over the run, \(\frac{\delta K}{\delta L}\). At a point, we compute the TRS as the ratio of the derivatives with respect to L and K from the production function: \[TRS=-\frac{MP_L}{MP_K}=-\frac{\frac{\partial f(L,K)}{\partial L}}{\frac{\partial f(L,K)}{\partial K}}\] Whereas MRS is universally used for the slope of an indifference curve, MRTS (marginal rate of technical substitution) is sometimes used for the slope of the isoquant. MRTS and TRS are perfect synonyms. We will use TRS.
The TRS (like the MRS) is a number that expresses the substitutability of labor for capital at a point on an isoquant. So, the TRS of two different L and K combinations on the same isoquant might be \(-100\) and \(-2\). The TRS = \(-100\) value says that the firm can replace 100 units of capital with 1 unit of labor and still produce the same output. The isoquant would be steep at this point. If a point has a TRS = \(-2\), 1 unit of labor can replace 2 units of capital to get the same output. The isoquant at this point would be much flatter than the point with the TRS = \(- 100\).
Just like the MRS, the TRS tells us how steep the isoquant is at a point. The steeper the isoquant, the more capital can be replaced by labor and still produce the same output.
Technological Progress
Over time, technologyour ability to transform inputs into outputimproves. Electric power and computers are examples of technological progress that enables more output to be produced from the same input.
There are two kinds of technological change. The Cobb-Douglas functional form can be used to illustrate each type.
Suppose increased education improves the productivity of labor. This would be modeled as an increase in the exponent for labor in the Cobb-Douglas production function. Small changes, say from 0.75 to 0.751, lead to large responses (e.g., in output or labor use) because we are working with an exponent. This is known as labor-augmenting technological change.
We could also have a situation where the coefficient A in the function \(AK^\alpha L^\beta\) increased over time. As A rises, the same number of inputs can make more output. This technological progress is said to be neutral (in terms of the utilization of L and K) because TRS does not depend on A.
We can show this by walking through the steps needed to find the TRS. First, we compute the marginal products of L and K from the function, \(Y=AL^\alpha K^\beta\): \[MP_L = \frac{\partial Y}{\partial L}= \alpha A L^{\alpha-1}K^\beta\] \[MP_K = \frac{\partial Y}{\partial K}=\beta AL^\alpha K^{\beta -1}\] The TRS is minus the ratio of the marginal products: \[TRS=-\frac{MP_L}{MP_K}=-\frac{\alpha A L^{\alpha-1}K^\beta}{\beta AL^\alpha K^{\beta -1}}=-\frac{\alpha K}{\beta L}\] The A terms cancel out, which means that the ratio of the marginal productivities of each input depends only on each input’s exponent and the amount of the input used.
The Firm as a Production Function
The production function is the starting point for the Theory of the Firm. As with utility, many, many functional forms can be used to represent real-world production processes.
Economists represent the production function not as a 3D object, but in two dimensions. We get product curves (total, marginal, and average product curves) by focusing on output as a function of a single input, holding all other inputs constant. An isoquant suppresses the output and shows the different combinations of L and K that produce a given level of output.
The TRS is similar to the MRS, and it will play an important role in the understanding the firm’s cost minimizing input choice.
Remember to keep straight the difference between the Law of Diminishing Returns and idea of returns to scale. The former applies more and more of a single input, holding all other inputs constant; the latter reports what happens to output when all inputs are changed by the same proportion. Those are two different things.
Exercises
Starting from a blank workbook, with K = 100, draw total, marginal, and average product curves for L = 1 to 100 by 1 for the Cobb-Douglas production function, \(Q=L^\alpha K^\beta\), where \(\alpha = 3/4\) and \(\beta = 1/2\). Use the derivative to compute the marginal product of labor.
Hint: Label cells in a row in columns A, B, C, and D as L, Q, MPL, and APL. For L, create a list of numbers from 1 to 100. For the other three columns, enter the appropriate formula and fill down. For MPL, do not use the change in Q divided by the change in L; instead enter a formula for the derivative for the MPL at a point.
For what range of L does the Cobb-Douglas function in question 1 exhibit the Law of Diminishing Returns? Put your answer in a text box in your workbook.
Determine whether this function has increasing, decreasing, or constant returns to scale. Use the workbook for computations and include your answer in a text box.
From your work in question 3 and the comment in the text that you cannot have constant returns to scale "if the exponents in the Cobb-Douglas function do not sum to 1," provide a rule to determine the returns to scale for a Cobb-Douglas functional form.
Is it possible for a production function to exhibit the Law of Diminishing Returns and increasing returns to scale at the same time? If so, give an example. Put your answer in a text box in your workbook.
Draw an isoquant for 50 units of output for the Cobb-Douglas function in question 1.
Hint: Use algebra to find an equation that tells you the K needed to produce 50 units given L. Create a column for K that uses this equation based on L ranging from 20 to 40 by 1 and then create a chart of the L and K data.
Compute the TRS of the Cobb-Douglas function at L = 23, K = 312.5. Show your work on the spreadsheet.
References
The epigraph comes from page 152 of "A Theory of Production" by Charles W. Cobb and Paul H. Douglas, The American Economic Review, Vol. 18, No. 1, Supplement, Papers and Proceedings of the Fortieth Annual Meeting of the American Economic Association (March, 1928), pp. 139–165, www.jstor.org/stable/1811556.
Douglas, an accomplished professor and US Senator from Illinois, explained how he and Cobb used the functional form that would be named after them:
I was then temporarily lecturing at Amherst College, and consulted with my friend and colleague, Charles W. Cobb, a mathematician. At the latter’s suggestion, the formula \(P = bL_kC_{k-1}\) was adopted, a form that had also been used by Wicksteed and Wicksell.
See p. 904 in Paul H. Douglas, "The Cobb-Douglas Production Function Once Again: Its History, Its Testing, and Some New Empirical Values," The Journal of Political Economy, Vol. 84, No. 5 (October, 1976), pp. 903–916, www.jstor.org/stable/1830435
Figure 10.1: The black box nature of the firm.
button to return the sheet to its default, initial position.
and 
buttons to see the product curves for labor and capital.
button and then click on cell C58 to reveal the marginal product computed via the derivative.
button (near cell H28) to see the isoquant map, as displayed in Figure 10.3.