How can game theory improve the agronomic resource management decisions made by small-scale farmers?

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Background: 

The world is currently experiencing a food crisis. Because of inequitable distributions in food, billions live in hunger, unable to sustain basic levels of nutritional health. Moreover, international trade wars, climate change-induced weather fluctuations, and corporate farming (employing scalable technologies that dominate the market) have brought small-scale rural farming to the brink of extinction (Semuels). Though inciting a change of these large-scale factors may be difficult, improvements in agronomic resource management for individual farms would go a long way in ensuring the protection of these farmers’ livelihoods. 

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Importance of Topic: 

Farmer defaults in the U.S. are reported to be up 12% in the Midwest and 50% in the Northwest, with 100,000 farms lost between 2011 and 2018 (Semuels). Despite being well below historical highs, Chapter 12 family farm bankruptcies (the small-scale cases discussed in this project) in 2019 increased by almost a fifth from 2018 (FB). Unlike other jobs, sustained losses in farming and eventual bankruptcy can eject these farmers from their homes and land—leaving them both homeless and jobless, often without skills transferable in this digital revolution. 

In this project, I’d like to model a sequential game that reflects the agronomic resource management  decisions that farmers make on a seasonal basis. Through these models, I hope to provide some insight on the actions that farmers ought to take in these difficult economic and environmental times. 

Guiding Question: 

How can I employ game theory techniques to analyze the agronomic resource management decisions made by small-scale farmers?

Economic Foundations of Analysis:

On the left is a simple graphical model for the market of a good G. The horizontal axis represents the quantity q of the good, and the vertical axis represents the price p at which it is offered. The S and D curves are the supply and demand schedules for the good G. The demand schedule of a good represents how much of a good (quantity) is demanded at a certain price level. The supply schedule reflects a similar concept for suppliers. 

Notice that if more of a good is supplied than demanded, then there is a surplus of the good, and if more is demanded than supplied then there is a shortage of the good. 

Various supply and demand models lie at the heart of economics. Elementary models will inform our game-theoretic agronomic analysis. There are several important assumptions that must be made in both game theory and economics (rationality, maximization of utility, etc.) but for the purposes of these analyses, we will assume all to be true.

An example of a demand schedule is shown below:

Price of rice ($/kg)

Quantity of rice demanded (kg)

1

4000

2

3000

3

2000

This is a numerical example of the linear demand schedules shown in the graphs above. 

Connection to Game Theory:

We organize our game as follows:

  1. Suppose a small-scale farmer coalition, F, and a large corporation, C, each decides to harvest crops A and B such that their total does not exceed each’s capacity for production. 
  2. After observing the quantity supplied by both players for both crops, each player will sell their crop on the open market. 

Each player’s payoff will correspond to their net profit, and we want to gain conclusions about the small-scale farmer coalition’s strategy. 

Assumptions: 

  1. Both parties seek to maximize profit (rationality + maximization of utility)
  2. The two parties are the only parties that produce these specialized crops
  3. The product produced by both players is of equal quality, meaning that they can be sold at the same price on the market and are indistinguishable to consumers. 
  4. Both players must unload all their crops during the season, as surpluses are impossible to maintain. Thus, they must agree on a market price (shown in the demand schedule) at which they can both sell all of their crop. 
  5. The farmer coalition’s capacity is 15 acres and the corporation’s capacity is 40 acres. However, because the farmer coalition is of smaller individual scale, they can plant crops in increments of 5 acres, whereas the corporation must in increments of 10. 
  6. 1 acre produces 1 ton of crop. 

Evaluation of Payoffs: 

To calculate the profit per sale of each of the goods, we consider the following values: 

  • Crop A costs $2000 to maintain per 5 acres but can be sold at a higher price than Crop B. 
  • Crop B costs $1000 to maintain per 5 acres. 

We also use the following demand schedules: 

Price of Crop A ($/ 5 tons)

Prices of Crop B ($ / 5 tons)

Tons Demanded

7000

4000

5

6500

3750

10

6000

3500

15

5500

3250

20

5000

3000

25

4500

2750

30

4000

2500

35

3500

2250

40

3000

2000

45

2500

1750

50

2000

1500

55

Thus, if the coalition and corporation decide to put a combined 30 tons of crop A on the market, they must set the price at $4500. 

Also, note that even if a farm produces 30 tons of a crop, it may be advantageous to only put 25 tons of it on the market, as the price can be set higher with less crop. 

Suppose (A, B) represents how much a player allocates land to the cultivation of crops A and B, respectively. (F, C) represents the profit for the farm and corporation, respectively. Thus, the strategies of both players can be represented in the graph below.

We can find this using the data above.

 

All profits are in thousands of dollars

Corporation

(0, 40)

(10, 30)

(20, 20)

(30, 10)

(40, 0)

Farm

(0, 15)

(7.5, 20)

(15, 75)

(22.5, 100)

(30, 95)

(37.5, 60)

(5, 10)

(32.5, 30) 

(32.5, 77.5)

(32.5, 95)

(32.5, 82.5)

(32.5, 40)

(10, 5)

(50, 40)

(42.5, 80)

(35, 90)

(27.5, 70)

(20, 20) 

(15, 0)

(60, 50)

(45, 82.5)

(30, 85)

(15, 57.5)

(0, 0)

To find our solution to this game, we first employ the movement diagram method to find the Nash Equilibrium—the solution where both players cannot unilaterally change their strategy for a higher payoff. The movement diagram helps us do this by finding the highest payoffs for each row and column, and seeing where they intersect.

I would like to make several conclusions regarding this result: 

I would first like to determine whether the solution is Pareto Optimal. 

The convex hull of the graph is shown on the right. Suppose that the x-axis represents the farm’s profits, and the y-axis represents the corporation’s profits.

A Pareto optimal solution is one where any shift in strategy leading to an increase in payoff for one party, leads to a negative tradeoff for at least one other party. In a graphical context, a Pareto optimal solution must lie in the top right corner of the graph. Thus, it is fairly obvious that there are multiple Pareto optimal solutions, lying on the top right three edges of the convex hull of this graph.

Regarding more real-world conclusions: 

  1. We know that the farm has 15 acres and the corporation has 40 acres of arable land, meaning that they have 8/3 times the production capability. In the optimal solution with value (35, 90), the farm has disproportionate market power, as they have made more than 3/8ths of the corporation in profit. However, in most of the other cases, the difference between the two players is stark.
  2. An interesting observation to make here is whether comparative advantage is a cooperative strategy. Comparative advantage is where each entity specializes in the production of the item that they are comparatively better at producing. Because the farm and corporation both produce at the rate of 1 ton/acre, we can’t simulate this with our current model, but it would be interesting to do an extension later, adding this factor. 
  3. Though this is the optimal solution in the short run, a corporation could choose to sacrifice some of its own profits in exchange for reducing a farm’s profits. Then, after driving the farm out of business, they could acquire these lands for reduced rates and then increase the size of their monopoly. This is a morally dubious, but valid strategy that a corporation could take to increase its own profits. Governments should enforce some external controls to ensure that a corporation cannot use its disproportionate market power to destroy the profitabilities of smaller farms. 

For Now Response:

Through this project, I was able to create a basic game-theoretic model for the agronomic considerations undergone by small-scale farmers on a seasonal basis. Even without accounting for the technological and political advantages offered to large corporations, as well as the monopolistic economic practices they could undertake to destroy the profitabilities of smaller farms, I am harrowed at the prospect that those who subsist on agriculture may no longer have homes or jobs. 

I’d like to end my presentation with the following short conclusion. In a free-market economy, the largest market share often corresponds to the largest market power, which can lead to increases in the disparity of power between the big and little guy. Without government intervention, these differences can spread to unhealthy levels, potentially leading to damaging consequences for both the producers, as well as ourselves, the consumers. In Japan, diversion programs incentivize local farmers to produce other crops under better economic conditions, and government subsidies support small-scale farmers in their endeavors (Fukuda, et al.) Similar programs across the world would do wonders in democratizing the industry again. What we can do now is be conscientious consumers and not feed into the monopoly that large corporations are trying to create. 

Request for Feedback:

  • Are there any other factors you would have liked to see in my game-theoretic analysis?
  • How do you think irrationality plays a factor in agronomic resource management?
  • Is there a way to mitigated the effects of climate-changed-induced disasters for small-scale farmers?
  • Any interesting facts about agriculture would be much appreciated!

Works Cited: 

Fukuda, Hisao. and Dyck, John H. and Stout, Jim. and United States. Department of Agriculture. Economic Research Service.  Rice sector policies in Japan [electronic resource] / Hisao Fukuda, John Dyck, and Jim Stout  U.S. Dept. of Agriculture, Economic Research Service [Washington, D.C.]  2003

Semuels, Alana. American Farmers Are in Crisis. Here’s Why. 27 Nov. 2019, time.com/5736789/small-american-farmers-debt-crisis-extinction/. 

“The Verdict Is In: Farm Bankruptcies Up in 2019.” American Farm Bureau Federation – The Voice of Agriculture, 29 Jan. 2020, www.fb.org/market-intel/the-verdict-is-in-farm-bankruptcies-up-in-2019/. 

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