Teaching production theory through simulation

Abstract

The authors present a Web application they designed in the R programming language as an experiential learning tool for teaching production theory. The app simulates production decisions where a manager is tasked to find the optimal mixture of inputs through experimentation. Users of the application are instructed to use calculations and intuitions from production theory to improve profitability quickly. The underlying parameters and starting points are randomized to allow for additional practice. Variations of the exercise correspond to cost-minimization and profit-maximization problems with and without market power in the output market. Also, the app provides dynamic, interactive visualizations in 3D in order to assist in a better understanding of standard production theory graphs. The app is located at https://bit.ly/387HsHL.

Keywords:

• Experiential learning
• production theory
• simulation

JEL codes:

• A22
• A23
• D24

Managers often need to cut costs without sacrificing output quality and quantity. Because production theory examines these objectives, it is an important topic in economics, especially managerial and intermediate microeconomics economics. Despite its usefulness, production theory is very challenging to teach. The standard approach relies heavily on graphical and mathematical analysis using tangency conditions between isocost and isoquant curves. Many students feel confused and frustrated by the apparent abstract nature of this approach. Students with good math or reasoning skills may conceptually understand the graphs, but even they may not know how to apply the theory in any real-world setting.

This article is a response to our students’ oft-repeated question, “When or how would we ever use this?” We provide a publicly available Web application to demonstrate how a manager could use production theory to improve hiring and investment decisions. The app includes simplified, yet rich, simulations where users have a minimal amount of information. It illustrates that an understanding of production theory can improve profitability even with limited information. The application, located at https://bit.ly/387HsHL, also includes various interactive 3D graphs (visualizations) from production theory. Because of the amount of information on the application, using a standard Web browser on a computer or large tablet rather than on a smartphone is highly recommended.

We simulate a simple firm where, over a number of periods, users search out more profitable levels of inputs knowing only prices, input levels, and output levels. Users start with randomly determined levels of each of the inputs. They can change any or all of the inputs each period. As in many real-life situations, the underlying production function is unknown.

As a part of the experience, the application explains how to calculate key measures based on input prices, output prices, and production levels. But, because the objective of the exercise is to learn how to use these numbers, the application automatically calculates and reports these measures whenever possible.¹ To quickly find optima, users interpret and use these numbers as they adjust input levels. Without these measures, users find that maximizing profits can be very challenging. In short, the application illustrates the importance of production theory in a simplified and controlled real-world-like setting.

We present three distinct exercises corresponding to maximization problems in production theory. In the profit maximization with no market power problem, users are given static input prices, output prices, and initial input levels. The inputs to production are unskilled labor, skilled labor, and robots. Over 30 periods, users adjust input levels to maximize profits. The underlying production function, which maps input levels to output, is unknown to the user. For each period’s inputs, the program calculates and reveals output levels, along with costs and profits.

At period zero, the inputs are not optimal. The challenge for the user is to make improvements without knowing the production function. Without a good strategy, the user finds it difficult to solve this exercise. They quickly realize why an understanding of production theory is so valuable.

To effectively solve the problem, users learn how to use the main concept of marginal product and other related measures. The app explains how to approximate marginal product and/or value of marginal product by changing one input at a time. These measures inform them how to change the input mix and improve profit. In particular, they can compare the value of marginal product (VMP) to input price (wage): they increase the input when VMP > wage and decrease the input when VMP < wage. Over multiple iterations, users can systematically improve profits until they approach optimal input levels.

The second task, profit maximization (with market power), is similar to the first. The only difference is that output price decreases as more product is sold. As before, using the value of marginal product (VMP) is the key to solving this quickly. As in the previous exercise, this measure is explained and calculated in the app. This task is a bit more challenging as the Law of Demand will affect how VMP changes with total output. For example, increasing robots will increase the marginal product of unskilled workers; but it will also decrease the output price through the Law of Demand. Therefore, the effect of an increase of one input has an ambiguous effect on the VMP of the complementary inputs. These types of complications make this exercise different, and a bit more challenging, from the first.

The last task is a cost-minimization problem. This problem corresponds to the classic isoquant/isocost framework in textbooks. A real-world application is a division manager who must find the most cost-effective way to reach assigned production targets. In the app, the user is given a specific quantity to produce each period and is tasked with minimizing costs. Because they don’t know the specific production function, users produce too much or too little output as they search for optimum inputs. The users experience a financial penalty proportionate to the quantity they are off-target. As with the other examples, the user’s best option is to use the tools and intuitions from production theory to instruct their decisions.

For all exercises, the underlying parameters and starting points are randomized but uniquely determined by a seed number that the user chooses. By using the same seed number, users can fairly compete with each other on how quickly and profitably they improve production. But for additional practice, choosing a different seed number generates a completely different set of parameters and starting values.

In addition to the simulation exercises, there is a tab where students can manipulate three-dimensional renderings of production theory functions. These interactive graphs generate additional connections with the visual representations of these concepts. They see how isoquant and isocost curves relate to the three-dimensional quantity and cost functions of capital and labor. Students can pan, scroll, and rotate these functions as well as choose the underlying function parameters.

The innovative tools presented in this article are among a number of new, experiential techniques aimed at improving student engagement, understanding, and retention.² There have been many of these ranging from innovative writing assignments (Frank 2006; Dalton 2010; Dolan and Stevens 2006), to classroom games (Balkenborg, Kaplan, and Miller 2011; Blackwell 2011; Brouhle 2011; Meister 2011; Moinas and Pouget 2016), to using media (Simkins 1999; Al-Bahrani and Patel 2015; Miller and Watts 2011), to clickers (Brouhle 2011), to classroom discussions (Buchs and Blanchard 2011). Our Web app is best characterized as a simulation. Simulations have been a popular teaching tool in various business courses (Larreche 1987). One very popular simulation game is Glo-Bus, which covers many aspects of product design and marketing (Karriker and Aaron 2014; Bright, Bateh, and Babb 2019). As of yet, simulations are not widely used in economics (Reiss 2011). There is evidence that these new approaches improve student engagement, retention, and understanding (Emerson and Taylor 2004). Simulations have been used to teach topics in econometrics (Craft 2003; Byker, Gregg, and Mortimer 2022), macroeconomics (Santos 2002; Rogmans 2018), and microeconomics (Emerson and Taylor 2004).

While there have been innovations in teaching a variety of economic subjects, we are unaware of any in production theory. This exercise gives the user experience making these types of decisions using limited information. They get practice in making these objective decisions in a safe environment without emotional attachments to individual workers.

As previously mentioned, learning production theory can be difficult. Without a compelling reason, students struggle to find the motivation to work through the math. However, as students experience how using production theory can improve profitability, they begin to appreciate its usefulness. The simulation section of the app gives students practice using the scale condition (value marginal product = wage), the substitution condition (marginal rate of transformation = wage ratio), and marginal products. The visualization section provides rich interactive graphs that help students better intuitively understand cost functions, various production functions, isoquant curves, isocost curves, returns to scale, marginal rate of transformation, share to capital/labor, and substitutability and complementarity between inputs. Beyond production theory, these skills transfer to similar concepts in economics, such as isoprofit curves, indifference curves, and budget lines. The only major skill remaining is mathematically solving optimization problems. The app does not directly teach how to solve the algebra and calculus needed to find an optimum solution. But, after using the application, students better understand which expressions to equate and why.³

This application is appropriate for intermediate microeconomics and managerial economics courses at the junior or senior undergrad or the MBA level. The entire application may not be suitable for every class. Similar to selecting chapters in an economics textbook, instructors can direct the students to the simulations and visualizations that correspond to their course curriculum. To make the application useful for a variety of courses adopting different textbooks and to not overly clutter the application, the instructional material inside the application is minimal. Therefore, the application is most useful after students learn about production theory concepts through textbook reading and/or lecture. Students use the application to test and solidify their understanding of the concepts. Later, they focus on solving problems mathematically.

The exercises

In this section, we address potential instructors in describing these exercises, whereas the instructions in the application itself are directed to users/students. We provide a user manual for students at the end of the online appendix.⁴ Each of these three exercises is self-contained and can be assigned individually as decided by the instructor.

The default setting of the simulation is that the user doesn’t have information about the marginal products (and related measures) of the production function. This requires the users to gather such information before proceeding to making decisions to improve profits. In the remainder of the article, we discuss the simulations under this default setting. The user can elect to have the marginal products automatically calculated on the Welcome Screen. Doing this will allow the user to focus solely on the decision-making aspect of the simulation.⁵

One challenge with using the Web application is that it may not display the same on all computers. Some users report issues regarding the size of input boxes and other display items. Users experiencing these issues on a computer could zoom in/out using their Internet browser’s controls (which is often Ctrl + for zooming in and Ctrl − for zooming out).

Profit maximization with no market power

The objective of the profit maximization with no market power exercise is to find the levels of inputs that maximize total profits. The inputs are unskilled labor, $u$, skilled labor, $s$, and robots, $r$.⁶ The time period is denoted with a subscript $t$. The output function is generated using a standard Cobb-Douglas production function, with decreasing returns to scale (${\alpha }_u,{\alpha }_s,{\alpha }_r>0\mathit{\text{and\hspace{0.33em}}}{\alpha }_u+{\alpha }_s+{\alpha }_r<1$). (1) $$Q_t=F\left(u_t,s_t,r_t\right)=A{u}_t^{{\alpha }_u}{s}_t^{{\alpha }_s}{r}_t^{{\alpha }_r}.$$(1)

At the start of the simulation, users are given some initial level of the inputs $,(u_0,s_0,r_0)$, and the resulting output, $Q$, is determined by the production function (1). Users are also told the output price, $P,$ and input prices, $(w_u,w_s,w_r)$. Profits for each period, ${\Pi }_t$, are calculated using prices and input and output levels. (2) $${\Pi }_t=p{Q}_t-w_u{u}_t-w_s{s}_t-w_r{r}_t.$$(2)

An overview of how these parameters and initial values are randomly generated is provided in the online appendix.

The goal of the exercise is to achieve the highest profit possible within 30 periods. For short-run versions of this exercise, one or two of the input types are fixed at the initial level.⁷ Each period, users change the level of the variable input. The app then reveals the resulting output levels and profit for that period.

Users may attempt to solve this problem in any way they choose. In all three exercises, there are instructions to help the users solve the problem more quickly and profitably. For this exercise, the value of marginal product is most useful. When a user changes only one input type from one period to the next, the simulation estimates the value of marginal product of that input at that level, ${\mathit{\text{VMP}}}_{i_t}$, as the difference in output, $Q_t-Q_{t-1}$, divided by the difference in input levels multiplied by output price, $P$. For example, the value marginal product of unskilled workers would be (3) $${\mathit{\text{VMP}}}_{u_t}\approx P\left(\frac{Q_t-Q_{t-1}}{u_t-u_{t-1}}\right).$$(3)

If more (or less) than one input is changed at a time from one period to the next, the rows in this table are blank.

Based on this information, production theory informs us how to increase profits using the scale condition. If ${\mathit{\text{VMP}}}_{i_t}>w_i$, the input should be increased. If ${\mathit{\text{VMP}}}_{i_t}<w_i$, the input should be reduced. In the single variable input version of this exercise, i.e., the short run, it is fairly simple: users simply increase, or decrease, the variable input until ${\mathit{\text{VMP}}}_i=w_i$. However, in the multiple input cases, complementarities between inputs make the exercise more interesting. Changing one input affects the marginal productivity of the other inputs. Therefore, it is more effective for users to make simultaneous changes to all inputs after finding the $\mathit{\text{VMP}}$ for all inputs. When ${\mathit{\text{VMP}}}_i$ is very different from $w_i$, they should change the input a lot; when ${\mathit{\text{VMP}}}_i$ and $w_i$ are close, they should change the input a little. Additionally, ${\mathit{\text{VMP}}}_i$ changes as the complimentary input changes. If done well, the user will move up the gradient of the profit function as they find the profit-maximizing mix of inputs, even without knowing multivariable calculus.⁸

Profit maximization with market power

The next exercise is similar except with market power in the output market. The inverse demand curve for the output is determined by the output produced. (4) $$P_t=H\left(Q_t\right)=B{Q}_t^{-\gamma },$$(4) where $\gamma \in (0,1)$ is 1 divided by the price elasticity of demand and $B$ is a positive scaler. Therefore, profit each period is (5) $$\begin{array}{c}{\Pi }_t=H\left(Q_t\right)Q_t-w_u{u}_t-w_s{s}_t-w_r{r}_t\\ =B{\left(A{u}_t^{{\alpha }_u}{s}_t^{{\alpha }_s}{r}_t^{{\alpha }_r}\right)}^{1-\gamma }-w_u{u}_t-w_s{s}_t-w_r{r}_t.\end{array}$$(5)

The goal of this exercise is the same as in the previous exercise, to maximize total profit. The most natural way to solve this is by comparing the value of marginal products (VMP) to input prices (or wages). As noted before, the additional complication of this exercise is that price is inversely related to output.

Cost minimization

The third exercise is cost minimization. In this exercise, users are given initial inputs and prices, as before. But now they are also given a production target quantity that stays the same for all periods. The exact target quantity is sold to another division of the company at a specified transfer price. The initial inputs might produce too many or too few of the products compared to the target. In each period, there are financial penalties for each unit above or below the target. When production is below target, the manager must purchase the difference at a cost significantly above the optimal marginal cost. When production is too high, they sell off the surplus at a price below the transfer price.⁹ Because of these prices, profits are higher when output is very close to the target. The input prices, the transfer price, the outside purchase, and the sale price are given at the top of the screen.

As before, the app contains hints on how to apply production theory to solve this problem. But now the most important measures are somewhat different from the profit-maximization exercises. These measures are still related to marginal products (MP) of input $i\in \left\{u,s,r\right\}$ and are approximated using experimentation. (6) $$M{P}_{i_t}\approx \left(\frac{Q_t-Q_{t-1}}{i_t-i_{t-1}}\right).$$(6)

If more than one input is changed at a time, from one period to the next, the rows in this table are blank.

These marginal product measures are useful when getting production levels closer to the target. For example, if a user needs to increase quantity by 20 and the marginal product of unskilled workers is 5, they can do this by increasing unskilled workers by 20/5 = 4.

Once production is close to the target, the next issue is finding the right mix of inputs. In any cost-minimizing solution, we know that the ratio of two input prices equals the marginal rate of technical substitution. The app calculates these MRTS between two inputs whenever the marginal products have been approximated within the last few periods. (7) $$\frac{M{P}_i}{M{P}_j}=\frac{w_i}{w_j}\hspace{0.17em}\mathit{\text{for all\hspace{0.33em}}}i,j.$$(7)

Alternatively, we can write this same condition as (8) $$\frac{M{P}_i}{w_i}=\frac{M{P}_j}{w_j}\hspace{0.17em}\mathit{\text{for all\hspace{0.33em}}}i,j\in \left\{u,s,r\right\}.$$(8)

Choosing which inputs to increase and decrease based on the ratio of marginal product over price, $\frac{M{P}_i}{w_i}$, is easier than MRTS and wage ratios. Students can easily compare these ratios across three or more inputs. It also has a very natural and intuitive interpretation: $\frac{M{P}_i}{w_i}$ is the additional output produced by an additional dollar spent on that input. Colloquially, it is a “bang-for-your-buck” measurement. When making changes, users should use more of the variable input that gives a larger $\frac{M{P}_i}{w_i}$ and less of the input with a smaller $\frac{M{P}_i}{w_i}$. These measures are also useful when deciding which inputs to change in order to increase, or decrease, production. Increase the variable input with the highest $\frac{M{P}_i}{w_i}$ and decrease the input with the lowest $\frac{M{P}_i}{w_i}$.

While $\frac{M{P}_i}{w_i}$ informs the direction each input needs to change, the MRTS tells us how much of one input is needed to compensate for changes in the other while holding production constant. For example, suppose the MRTS between skilled and unskilled workers is $\frac{M{P}_{\mathit{\text{unskilled}}}}{M{P}_{\mathit{\text{skilled}}}}=.8$ and we have decided to increase the unskilled labor force by 10 units. If skilled workers are decreased by (.8)(10) = 8 simultaneously, output would remain the same. In summary, to solve this problem, users need to get close to the target quantity, figure out which inputs to increase and decrease, and then adjust the inputs using MRTS to stay close to the target quantity.

Solving an exercise: Example

In order to illustrate how insights from production theory can solve these exercises quickly and efficiently, we walk through an exercise of profit maximization with no market power. Without seeing this simulation beforehand, we recorded how we solved it step-by-step using logic and the tools available to students. Because it was our first try with the simulation, neither the choices along the way nor our final solution were perfect. But, we explain our thought process at each step and how production theory informs those decisions. We present only the profit maximization with no market power walk-through in the article. Because the other exercise types have their own unique challenges, similar walk-throughs for profit maximization with market power and cost minimization are available in the online appendix.

In the introduction screen, the user first chooses one of the optimization problems and the number of variable inputs. They then decide whether they want to display marginal values. Finally, they enter a simulation ID number or click “Randomize” for the app to fill in a random number. The default number of variable inputs is two. If users choose two variable inputs, then graphs relating to the specific simulation are displayed at the end of the exercise in the solution screen.

The simulation ID is a number that determines the set of parameter and initial values for the exercise. In case the user wants to follow along or to try the problem themselves, we provide the simulation ID = 51. See figure 1 for a screenshot of the partially completed exercise. Because of limited space, not all of the information is provided in this screenshot. Instead, we provide the important information to solve this simulation in table 1.

Figure 1. Screenshot of the profit maximization with no market simulation.

Table 1. Profit maximization with no market power (simulation ID: 51).

Period	U	S	R	Profit	Price=$38	$w_u$=120	$w_s$=149	$w_r$=194
					Change in profit	$\mathit{\text{VMPu}}$	$\mathit{\text{VMPs}}$	$\mathit{\text{VMPr}}$
1	470.5	751.5	136.5	$50,401.87
2	471.5	751.5	136.5	$50,468.23	$66.36	186.36
3	471.5	752.5	136.5	$50,434.94	−$33.29		115.71
4	471.5	752.5	137.5	$50,354.25	−$80.68			113.32
5	542	700	117	$57,160.98	$6,806.73
6	543	700	117	$57,205.33	$44.35	164.28
7	543	701	117	$57,182.50	−$22.83		126.20
8	543	701	118	$57,122.71	−$59.79			134.35
9	678	595	82	$63,282.48	$6,159.77
10	679	595	82	$63,293.88	$11.40	131.40
11	679	594	82	$63,294.31	$0.43		148.57
12	679	594	81	$63,295.17	$0.87			193.13
13	725	594	81	$63,681.85	$386.67	128.41
14	726	594	81	$63,687.54	$5.69	125.69
15	726	595	81	$63,690.59	$3.05		152.05
16	726	595	82	$63,694.52	$3.93			197.93
17	776	636	87	$64,024.29	$329.76
18	792	649	89	$64,088.64	$64.36
19	793	649	89	$64,091.92	$3.28	123.28
20	793	650	89	$64,092.02	$0.10		149.10
21	793	650	90	$64,091.10	−$0.92			193.08
22	825	657	90	$64,161.41	$70.31
23	826	657	90	$64,162.11	$0.70	120.70
24	826	658	90	$64,163.31	$1.20		150.20
25	826	658	90.1	$64,163.47	$0.16			195.64
26	834	671	92	$64,178.61	$15.14
27	835	671	92	$64,179.53	$0.93	120.93
28	835	672	92	$64,179.49	−$0.04		148.96
29	835	672	93	$64,178.44	−$1.05			192.95
30	842	672	91	$64,182.32	$3.87

Note: The rows in boldface are periods where we make significant changes in the input mix and dramatically improve profits. The direction and magnitude of these changes are suggested by information on the VMPs gathered in the most recent periods before each of the changes.

The exercise starts out in period 1 with given inputs of 470.5 unskilled, 751.5 skilled, 136.5 robots, and production at 6455.96. The price of unskilled is $120, skilled is $149, and robots are $194 and an output price of $38. This gives us a revenue of $245,316.37, a cost of $194,914.50, and a profit of $50,401.87. Because we don’t know which inputs are too low or too high, in periods 2–4, we one-by-one increase a different input type by one unit and consult the Value of Marginal Product ($\mathit{\text{VMP}}$) tab to get information about these near the starting level.¹⁰ Recall that the application calculates marginals when only one input is changed from one period to another, as the change in revenue divided by change in input.

At the initial inputs, the value of marginal product is greater than its input price for only the unskilled worker (${\mathit{\text{VMP}}}_u$ = $186.357$> $120$= $w_u$); therefore, we need to increase unskilled workers. The value of marginal product is less than the input price for both skilled workers (${\mathit{\text{VMP}}}_s=$ 115.7074$<$ 120 $=w_s$) and robots (${\mathit{\text{VMP}}}_r=$ 113.316 < 194 $=w_r$). This indicates that we need to decrease both of these inputs, especially the robots.

In period 5, we make a significant change to the inputs by increasing the unskilled workers by 15 percent (from 471.5 to 542), decreasing skilled workers by 7 percent (from 752.5 to 700), and decreasing robots by 15 percent (from 137.5 to 117). We didn’t know if these changes were too big or too small; we just made a guess. Doing this both increased revenue (from $245,731 to $249,199) and reduced cost (from $195,377.50 to $192,038.00) with a net increase in profits of $6807.

Without further investigation, we don’t know which adjustments are needed to further improve profits. Therefore, in periods 6–8, we repeat the steps we did in periods 2–4 to get the value of marginal products. The application gives us $\mathit{\text{VMP}}$ for the three inputs: 164.35 for unskilled, 126.17 for skilled, and 134.21 for robots. Our changes in period 5 did move the $\mathit{\text{VPMs}}$ closer to input prices, but we need to make additional improvements. The changes we made in period 5 did move $\mathit{\text{VMPs}}$ in the right direction, but not nearly enough. So, we make bigger changes in period 9. We increase unskilled workers by 25 percent (from 543 to 678), decrease skilled workers by 15 percent (from 701 to 595), and decrease robots by 30 percent (from 118 to 82). Doing this didn’t change revenue by very much (from $249,623.71 to $249,205.48), but it lowered costs by an additional $6,558.00 (from $192,501.00 to $185,923.00).

After finding them in periods 10–12, the marginal products are much closer to wages. The $\mathit{\text{VMP}}$ for skilled worker is still $11 higher than skilled worker’s wage. The other two inputs’ $\mathit{\text{VMP}}$s are only slightly lower than their respective wages. In period 13, we leave skilled workers and robots unchanged and increase unskilled workers by only 7 percent.

The reported $\mathit{\text{VMP}}$ for unskilled workers decreased by $3 in period 13. The temptation is to again increase unskilled workers by a lot more in period 15, but we remember that the $\mathit{\text{VMP}}$ measurement is not as accurate when we make large changes in the input. We check this in periods 14–16 by a 1-unit increase in the inputs. We see that all three values of $\mathit{\text{VMP}}$ are about $3 greater than their wage; all three inputs should be increased. If we increase all inputs at the same time, we expect that $\mathit{\text{VMP}}$ will not fall rapidly because of complementarities between inputs. Instead of experiencing the law of diminishing marginal returns as we did in period 13 when we kept all other inputs constant, we are now mostly going to increase scale. This exercise has decreasing returns to scale and there is a finite maximum, and $\mathit{\text{VMP}}$ will equal input prices.

In period 17, we increase inputs by 7 percent, which improves profits by $329.76 (from $63,694.52 to $64,024.29). In period 18, we increase inputs by another 2 percent and further increase profits by another $64.36. At this point, we see that improvements in profits are becoming more and more modest in size, which means that we are probably very close to the optimum mix of inputs.

In periods 19–21, we again find all $\mathit{\text{VMP}}$ measures so that we can compare them to input prices. We find $\mathit{\text{VMPu}}$ = $123.28 is $3.28 higher than $w_U$ = $120, $\mathit{\text{VMPs}}$ = $149.10 is very close to $w_s$ = $149, and $\mathit{\text{VMPr}}$ = $193.08 is within a dollar of $w_r$ = $194.

These numbers indicate that we need a few more unskilled workers. An increase in these workers will also increase the productivity of the other two. Therefore, in period 22 we decide to increase unskilled by 4 percent (from 793 to 825), increase skilled by 1 percent (from 650 to 657) and leave robots unchanged at 90. We again see an increase in profits by an additional $70.31.

In periods 23–25, we check all $\mathit{\text{VMP}}$s and find we are still slightly higher but getting closer to input prices ($\mathit{\text{VMPu}}$ =120.70, $\mathit{\text{VMPs}}$ = 150.20, and $\mathit{\text{VMPr}}$ = 195.64). In period 26, we increase unskilled workers by 1 percent, skilled workers by 2 percent, and robots by 2 percent, and again increase profits by an additional $15.14.

In periods 27–29, we find that the main input needed is the unskilled labor and that we might have slightly too many robots. In the final round, we make some minor adjustments and increase profits by $3.87.

The final inputs were unskilled workers at 842, skilled workers at 672, and 91 robots with a total profit of $64,182.32. This is $4.25 away from the highest possible profit. We see that most of the improvements happened very quickly at the beginning when the theory indicated that certain improvements would be very profitable. With minimal knowledge of the production function, we demonstrate how to lower costs and increase revenue by simply applying production theory concepts.

Visualizations

Economics courses are full of charts and graphs that help students process complex information and improve reasoning and intuition. Most of these graphs are two-dimensional, such as supply, demand, the Keynesian cross, and the production possibilities frontier. The standard graphs in the production theory, such as isoquants, isocosts, isorevenue, and isoprofit curves, are, out of necessity, also often drawn in two dimensions. But these graphs actually represent objects in higher than two-dimensional space. They are functions of at least two inputs: capital and labor. Without establishing the connection with a three-dimensional model, the isocost, isoquant, and other “iso” curves can be challenging for students to understand fully.

In an effort to help students conceptualize the various “iso” curves, the app also presents a series of interactive two- and three-dimensional production theory graphs. When users manipulate these graphs in 3D, they see how these functions can be represented on the two-dimensional XY plane. They also can visualize what an optimal set of inputs means. Users adjust function parameters such as prices and productivity and change the shape of the curves. They can also rotate, pan, and manipulate these graphs so that they get a sense of their true 3D nature. After playing with the graphs for just a few moments, users have a better understanding of various graphs and measures in production theory.

We have three sections of the visualization exercises in the application—profit maximization, cost minimization, and the production function. The first tab in the profit-maximization section examines the isorevenue curve (see figure 2). It presents a 3D visualization of the revenue function, $R\left(L,K\right)={\mathit{\text{PAL}}}^{\alpha }{K}^{\beta }$, and shows how isorevenue curves are a projection onto the XY plane. Users choose the parameters for the function: an output price, $P$, the productivity coefficient, $A$, share of labor, $\alpha$, and share of capital, $\beta$. Users can see how the revenue function and its isorevenue curves change as they change these parameters.

Figure 2. Screenshot of isoquant curve in the profit-maximization visualization.

Iso-cost is the second tab in the profit-maximization section (see figure 3) and is similar to the iso-revenue tab. Users select input costs, i.e., wages. The cost function with labor and capital are on the X and Y axes and costs is on the Z axis with a shade gradient. As they scroll over the function, the app displays the coordinates and projects these coordinates on all three axes. On this 3D graph, isocost curves are projected on the XY plane. To the right of this 3D graph is a set of standard isocost curves with a shade gradient showing that the darker shades correspond to higher costs.

Figure 3. Screenshot of isocost curve in the profit-maximization visualization.

The optimization tab is the final tab in the profit maximization section (see figure 4). It combines both the cost and revenue curves on a single graph, and the users get to choose all the parameters from both of these functions. A 3D graph showing how the profit is the gap between the revenue and cost functions is on the left side of the page. The graph on the right side of the page is the profit function, which is revenue minus cost as a function of the inputs. Because of the parameter restrictions we impose, this curve is concave, and maximum profit is marked with a dot. This same point is also marked on the left-hand side to illustrate the connection between the two graphs. Users can explore how profits change as they move away from the optimum.

Figure 4. Screenshot of optimization tab in the profit-maximization visualization.

The second section is about cost minimization. The first tab in this section is isoquant and is similar to the isorevenue tab except there is no output price and the user chooses an output level that is marked by the app as the isoquant curve (see figure 5). The second tab, isocost, is very similar to one in the profit-maximization section except the users select the spending level (see figure 6). The application then draws the resulting isocost curve on the two graphs. The final tab in the cost minimization tab shows a version of the classic 2D isoquant/isocost graph with a few dynamic enhancements (see figure 7). The first enhancement is that the user can change the curve by manipulating input costs, productivity parameters, and target output. Second, as the user moves the cursor over the graph, the program displays a flyover with information pertaining to the cost and revenue functions. The flyover displays the capital and labor costs and output at that point. It also displays both marginal products, the marginal rate of technical substitution, and the price ratio. The user can trace the isocost and isoquant lines and see how these measures change and how they relate to cost and output.

Figure 5. Screenshot of isoquant curve in the cost minimization visualization.

Figure 6. Screenshot of isocost curve in the cost minimization visualization.

Figure 7. Screenshot of optimization tab in the cost minimization visualization.

The third section in the visualizations portion of the app is called the production function (see figure 8). In this section, the app plots a constant elasticity of substitution (CES) production function with varying returns to scale. Changing the elasticity of substitution allows the users to explore substitutability and complementarity between inputs. If the user chooses elasticity of substitution close to 0, the production function becomes Leontief. If they choose a very large elasticity, the production function becomes linear. If the elasticity is close to 1, the function becomes Cobb-Douglas. Choosing intermediate ranges shows the transitions between the extremes. The other interesting parameter that users can adjust is returns to scale. As they go from decreasing to increasing returns to scale, they see how the production function and its isoquants change. The ability to rotate the 3D graph is particularly useful in visualizing how returns to scale affects the production function.

Figure 8. Screenshot of the isoquant tab in the production function visualization.

Discussion

As mentioned in the exercises section, the user can opt to have the marginal products calculated automatically or not. Users who choose to have the marginal products calculated for them automatically have an unfair advantage over those who do not. The simulation ID of the simulations of those with no automatic marginal products is 72 more than that with automatically generated marginal products. For example, a simulation ID of 172 for the no automatic marginal products and a simulation ID of 100 for automatic marginal products give the same exercise. If the instructor keeps this a secret, then the students can compete fairly. If the instructor reveals this information, students can do the same exercise with or without the calculated marginal products.

This exercise can be graded in a number of ways. It could be done as an in-class exercise as part of participation points. In the online appendix, there is a section labeled “Instructor Resources,” which is a list of suggested questions to facilitate classroom discussion. It could be given as homework, and then a screenshot of the final stage sent to the professor. It could be a competition where the winner receives a prize or bonus points. Of all these approaches, in-class competition seems to generate the most enthusiasm in our experience.

We solicited student feedback while developing this application to help improve the experience. These were initially intended to be excluded from a study. But afterward, we felt that such responses could be useful. Therefore, in order to share the following quotes from recent anonymous students about the application, we applied for and received an exemption and a waiver for the requirement of consent from the Human Use Committee at Louisiana Tech University (assigned number 23-141).

• “I enjoyed the production application; it was an interesting concept and a great idea to implement the things we have been taught. Playing around with different numbers also helps to get an idea of where each number is going and what it is doing.”

• “I think it helped spark interest and more class involvement in the production topic. Overall it was a fun way to learn/experiment with how changes in production can affect profit.”

• “Very intuitive to use, overall had a good time with it, came close to beating Dr. Peterson too.”

• “I enjoyed the app. I thought it was very well laid out and explained a lot of what we had already discussed in class.”

• “I thought the app was a great way to get a better grasp on this topic! I feel more comfortable in this topic going forward. The app reminded me much of the business game we played in capstone!”

Conclusion

The study of economics provides an invaluable set of intuitions, frameworks, and understandings that can vastly improve economic decision-making. However, for some students, the value of such learning is hidden behind cumbersome and intimidating abstractions. It is vital that we search out new and innovative ways for students to connect with and understand economic theory. Once students realize the value of these theories in solving exercises and simulations, they can find ways to apply these concepts in their lives and livelihoods.

The version of the application described in this article is designed to model situations where users can most directly apply production theory. In many real-life applications, there are additional complications that may require more sophisticated solutions. Such frictions include changing input and output prices, the cyclicality of input and output prices, randomness in output, the ability to inventory excess output, etc. For purposes of instruction, we have left these complexities out of the simplest versions of the application. Future work will enrich this application to give users experience in solving situations with a variety of such complications.

Acknowledgments

The authors are grateful to two anonymous referees for their invaluable comments. They also thank many former students who were unafraid to express feedback, positive and negative, during the conception and development of this application, and they also thank the grrr R Slack group for help in developing the tools to make this happen.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Marginal products and related measures are only estimated when just one input is changed from one period to the next.

2 See Hawtrey (2007) and Sheridan, Hoyt, and Imazeki (2014) for further motivation for and examples of experiential learning techniques.

3 For example, a standard problem in traditional approaches is to give students a Cobb-Douglas function with input prices and target quantity, and the students mathematically solve for the cost-minimizing set of inputs. Our app helps students intuitively understand what the marginal rate of technical substitution means and why it should equal the wage ratio in the optimum.

4 The online appendix can be found at bit.ly/41lO0fu.

5 If the user chooses to have the simulation provide the marginal products, this is akin to having a production team that provides this information. The default setting is to not provide these because our MBA students have asked where information on marginal products could be obtained in a business setting and were not satisfied with the suggestion of consulting a production team. Also, finding marginal products through input choice reinforces what marginal products mean.

6 We chose the notation in this article to be consistent with the application. There were some typesetting limitations in designing the application, so the best notation we could choose might still differ from standard.

7 The number of variable inputs is decided on the first page of the application.

8 This application is designed so that even the least mathematically sophisticated economics student can gain a working understanding of the major concepts of production theory. But, more advanced courses could use this application to demonstrate deeper connections between the mathematics and the application of production theory, for example, because the Cobb-Douglas production function (1) is log-linear. $\mathrm{\text{ln}}\hspace{0.17em}\left(Q_t\right)=\mathrm{\text{ln}}\hspace{0.17em}\left(A\right)+{\alpha }_u\mathrm{\text{ln}}\left(u_t\right)+{\alpha }_s\mathrm{\text{ln}}\left(s_t\right)+{\alpha }_r\mathrm{\text{ln}}\hspace{0.17em}\left(r_t\right)$. After at least four periods of choosing a variety of inputs, the underlying parameters can be calculated using mathematical tools such as linear algebra or regression analysis. Once these model parameters are found, the students can solve for the optimal mix of inputs and immediately reach the profit-maximization point.

9 One might think of other penalties for missing production targets, such as excessive overtime costs, delays in production, storage costs, etc. This particular mechanism seems most tractable and still be fairly realistic.

10 If the display marginal products option was selected in the Welcome Screen, the VMPs are already available. The user can go right to analyzing them.