Visualizing the Production Function and Cost Curves

Single, static images of data trends aren’t the most effective way to communicate the ways the different elements of an equation or formula contribute to a trend.  This is especially true for introductory economics concepts such as cost curves or the production function. Dynamic, interactive visualizations that allow users to manipulate the variables contributing to a relationship which enables the audience to better understand how equations express trends.

Krit Petrachaianan ‘17 of DASIL programmed a visualization using R that illustrates cost curves and the production function, two core concepts of introductory economics.  DASIL’s visualization allows users to manipulate the different parts of the equations that define cost curves and the production function. For instance, users can manipulate the costs per input (denoted r and w) and the amount of a particular input (denoted K for capital and L for labor). Users can also define the productivity of the firm’s inputs.

Cost curves visualize the costs of producing different levels of output. The total cost of production for a business can be subdivided into fixed and variable costs.  Some costs, such as raw materials and production supplies, change proportionally as more or less of the good or service is produced and are known as variable costs.  Other costs, such as the annual rent or salary of workers, are independent of the level of goods or services a business produces and are known as fixed costs.

The production function shows the relationship between the output produced by a firm from a given amount of inputs (i.e. labor and capital). The productivity of inputs in producing output can vary in three ways: 1) with constant productivity, the additional output produced by a given amount of input is constant as more of the input is used, 2) with diminishing productivity, the additional output produced by a given amount of input declines as more of the input is used, and 3) with increasing productivity, the additional output produced by a given amount of input increases as more of the input is used.

Explore DASIL’s latest R visualization below, as well as in the Graphs section of the Data Visualizations page and in the Economics tab of the DASIL website.

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Modeling Population Growth in Excel

The Malthus and Condorcet Equations, simple formulas that model relatively complex trends in population growth, are now accessible with an Excel calculator that allows the user full control over every component of the equations. Students can use the Excel file to model human population growth under the assumption that a human carrying capacity exists.

The Malthus Equation expresses the growth rate of a population as a function of the current population size and current carrying capacity. Specifically, the growth rate of a population is equal to a Malthusian parameter multiplied by the current population size multiplied by the difference between the current carrying capacity and the current population size. This relationship creates a high growth rate once a population is large enough to reproduce at its full potential, but remains a low growth rate when the population is very small or when a population is nearing its carrying capacity and feeling the effect of constrained resources. The Malthusian parameter is almost invariably between zero and one because a negative Malthusian parameter would lead to a population’s gradual extinction while a Malthusian parameter greater than one would lead to explosive population growth that would greatly exceed the carrying capacity. In the latter situation, unrealistically rapid and extreme periods of growth and contraction would ensue.

The Condorcet Equation expresses the growth rate of the carrying capacity of a population as equal to the growth rate of the population multiplied by a constant termed the Condorcet parameter. The logic behind this mathematical relationship is that the carrying capacity of a population increases or decreases proportionally with the growth rate of a population because an additional person in a population can have a positive or negative effect on the carrying capacity. This implies that a Condorcet parameter greater than one results from a society where an additional individual somehow increases the number of people that can be supported even when taking into account the resources that additional individual consumes; this could result from a situation where there are increasing returns to labor. If doctors cure diseases better when more of them work together, this is reflected by a Condorcet parameter greater than one. A Condorcet parameter between zero and one is most realistic for human populations because the contribution of another person will probably grow the carrying capacity but not by more than one. A negative parameter implies that an additional person would actually lower the carrying capacity; perhaps every additional person would consume natural resources at a rate greater than the previous individual’s rate.

As Cohen (1995 Science 269: 341-346) points out, the equations are not necessarily realistic models of human population growth. There is no consensus about whether or not a human carrying capacity exists. In theory, we as a species might be able to continually develop technology at such a rate that we are unable to approach a carrying capacity. A slowdown in overall human population growth is more likely due to a global increase in income per capita that leads to altered reproductive strategies.

With r=0.1 and c=0.1 as parameters, the population experiences a positive but steadily decreasing growth rate because the carrying capacity increases at 1/10th the rate of population growth, and since population growth slows as the population size approaches the carrying capacity, we observe almost asymptotic behavior. This is a realistic pattern for human population growth if a carrying capacity exists.

Figure 1: with r=0.1 and c=0.1 as parameters, the population experiences a positive but steadily decreasing growth rate because the carrying capacity increases at 1/10th the rate of population growth, and since population growth slows as the population size approaches the carrying capacity, we observe almost asymptotic behavior. This is a realistic pattern for human population growth if a carrying capacity exists.

The calculator defines the Malthus Equation as dP(t)/dt=rP(t)[K(t)-P(t)] and the Condorcet Equation as dK(t)/dt=c dP(t)/dt (See Cohen 1995: 343). The user may enter values for the initial states of r (the “Malthusian parameter”), P(t), (population size), K(t) (carrying capacity), c (“Condorcet parameter”), t_0 (the starting time for the model) and dt (the length of one interval in time) that determine all of the future changes in population size. The rates of change of population and carrying capacity at time t, dP(t)/dt and dK(t)/dt respectively, are determined by the equations. The Malthusian and Condorcet parameters are constant in a growth model provided that there are no exogenous shocks that affect the nature of population or carrying capacity growth. Because of this, they do not vary as a function of t.

To explore the Malthus-Condorcet calculator, please follow this link to an automatic download of the Excel spreadsheet containing the calculator.

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Data Across the Curriculum: Teaching Data Skills in Sociology

Casey Oberlin, Assistant Professor of Sociology, understands the importance of using data in the classroom, especially in such a discipline as Sociology, which is commonly viewed by others outside the discipline as a field with less real-life application of hard skills (e.g. data analysis). This conception is far from the truth, and Oberlin’s approach with data in the classroom gives her students a very holistic and interactive view of data analysis in the field that shows how data is part and parcel to the discipline.
Oberlin uses both her introductory Sociology courses and Research Methods courses as opportunities for students to get deeply entrenched with the data-rich, multi-tiered research process of the field. Data in Sociology is very diverse, as it involves both quantitative and qualitative measures, so Oberlin’s approach focuses on getting students exposed to the vast array of data types, as well as the techniques, technologies, and methods used to interpreting each type.

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At the introductory level, Oberlin focuses on data consumption as a first step to data concepts. Students study infographics (see Figure 1) and other data visualizations to learn how to present data and interpret the data being presented. Oberlin’s Research Methods courses are reserved for her experiential-based approach with data that teaches students two data software programs throughout the semester, one quantitative (SPSS) and the other qualitative (Nvivo), shows students the wide range of data utilized by Sociology, and has students grapple with the entire research process for themselves. In Research Methods, students create research questions, hypotheses/expectations, clean or assess the dataset, analyze their results, and present their work in a professional manner. Her heavy guidance through the research process helps to mitigate understandable anxiety about trying new techniques and presenting their ongoing work, setting her students up to then develop their own sustained research project throughout the semester. Oberlin states this immersive method is beneficial to and enthusiastically received by students, as the practice in research opens doors to internships, jobs, and grad schools.

All in all, Casey Oberlin’s utilization of data in the class gives students exposure to the intensive research process that is integral to Sociology and teaches important data skills and concepts that are applicable both in the real-world and in a classroom setting.

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Data Across the Curriculum: Integrating Data Analysis with Narrative in Political Science

From a pedagogical standpoint, Danielle Lussier, Assistant Professor of Political Science, stresses data as a tool for helping students approach problems from multiple perspectives. Working interactively with data allows them to better compare narratives and better understand the research process in both lower-level and upper-level material.

Political science is both a quantitative and qualitative field, so students at all levels of Lussier’s political science classes delve into both data types extensively and build data analytic skills as students progress in the major. Every class taught by Lussier involves data labs that draw on both cross-national data with countries as the unit of measure and on data with individuals as the unit of measure. The labs directly relate to readings, concepts, and/or countries that students study.

At the 100-level, students gain both an introduction to fundamental data concepts such as the construction and measurement of variables and to analytical computer programs like STATA, a statistical package, and ArcGIS, which analyses spatial data. The image below is of a GIS map her introductory political science students make in a data lab.

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At the 200-level, Lussier’s students delve into applied data analysis and write in-depth data reports that compare data analyses from the course readings to data analyses that students reconstruct and update from the readings.

At the 300-level, students get the opportunity to pose questions about class readings and use lab time to test their inquiries with actual data from the readings. In addition, Lussier assigns students research modules that allow them to create their own qualitative variables from cross-national data that they then transform into quantitative data, giving students the opportunity to apply the data skills they’ve accumulated in each course level.

The positive impact of incorporating data into classroom work is not lost on students. Students in all levels of her courses are widely receptive to data in coursework and have viewed working with data in her classes as an integral stepping stone to both academic and professional pursuits. Adam Lauretig ’13, the first Post-Baccalaureate Fellow for DASIL, was inspired by Lussier’s data-driven coursework to pursue more advanced courses in spatial statistics, and subsequently created visualizations like the interactive timeline map of historical coups d’etat. Additionally, many of her students have cited the research and data skills developed in her class work as marketable to employers and graduate programs.

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Data Across the Curriculum: Helping the Local and the International with Consulting Research

Students in Monty Roper’s Anthropology and Global Development Studies classes gain practical experience in fieldwork, data analysis, and ways to deal effectively with clients when they act as consultants for both local organizations in Grinnell and internationally in an agricultural village in Costa Rica.  The clients they work with get free research which is presented to them both in the form of an oral consultation and in a written report.

 


From left: Roni Finkelstein ’15, Ellen Pinnette ’15, Liberty Britton ’14, Rosalie Curtain ’15, Emily Nucaro ’14, Ben Mothershead ’15, Zhaoyi Chen ’14, and M’tep Blount ’15, listen to Juan Carlos Bejarono explain the palm growing process.

 

For a Global Development Studies/Anthropology seminar, students prepare research plans during the first half of the semester and then travel to a rural agricultural community in Costa Rica to spend the two weeks of spring break collecting data which is then analyzed and written up during the remaining weeks of the semester.  The first year of the project, the class conducted an in-depth community development diagnostic.  Since then, they have investigated a variety of rural development issues, mainly focusing on tourism, women’s empowerment, and organizational issues and agricultural projects of the town’s two cooperatives.

 

From left: Chloe Griffin ’14 and Samanea Karrfalt ’14 present their research on “Professional Black Hair Care in Grinnell, IA”


From left: Irene Bruce ’15 and Matt Miller ’15 present their research and answer questions.

 

In Grinnell, Monty works with Susan Sanning, Director of Service and Social Innovation, to identify and explore possible collaborations with community partners who have research needs.  In the past, for example, Mid-Iowa Community Action (MICA) was interested in knowing why families dropped out of their Family Development and Self-Sufficiency Program (FaDSS) before their benefits were fully used, Drake Library was interested in what kinds of programming would best serve the town’s “tween” population, and a hair salon wanted to find out whether it was economically viable to invest in special hair care products and services for black customers.

Ideally positive change occurs because of the class’ research.  Grinnell students, Dillon Fischer ’13 and Sarah Burnell ’13, interviewed graduates of Grinnell High School who had gone on to attend college about their preparedness for college academics. According to the GHS Principal, these findings led the school to revise its minimum writing standards, making them more challenging. The local after school youth program, Galaxy, requested a study on donor perceptions and desires and subsequently used the results to write a successful grant proposal for support. This year’s class is planning to do more follow-ups on previous projects to ascertain longer term results.

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