The impact of a subsidy on an arbitrage involving goods with perfect cross elasticity

Abstract:

If two securities are sold in two different markets, applying a subsidy in one market will result in a divergence between the price paid by consumers and what is received by producers. The subsidy will result in an arbitrage opportunity in which the purchase and sale of the same security in two different markets will result in a risk free profit (Sharpe and Alexander 1990). If the subsidy in one market is provided by the producer in the other market, the value of the securities in both markets approaches infinity.

Introduction:

Gans (2020, pp 129) defines a subsidy is a payment from a government to consumers or producers for each unit of a good that is bought or sold. It creates a divergence between the price paid by consumers and the price received by producers. The price producers receive is higher than the price consumers pay by the value of the subsidy. An arbitrage is defined as “the simultaneous purchase and sale of the same, or essentially similar, security in two different markets for advantageously different prices” (Sharpe and Alexander 1990).

Gans (2020, pp 72) defines substitutes goods are two goods for which a decrease in the price of one good will result in a decrease in the demand for the other good. In addition, substitutes are pairs of goods that satisfy similar desires as they can be used in place of each other. As a result a decrease in price of one good will lead to a decrease in demand for the other good.

According to Gans (2020, pp 94 ), elasticity is a measure of how responsive the quantity demanded or supplied is to one of its determinants such as availability of substitutes and whether the good is a necessity or luxury. Price elasticity of demand is a measure of the degree to which quantity demanded of a good changes in responses to a change in the price of the good (percentage change in quantity demanded divided by percentage change in price).

Cross-price elasticity of demand is a measure of how much the quantity demanded of one good responds to a change in the price of another good calculated as the percentage change in quantity demand of good one divided by the percentage change in price of good two as described by Gans (2020, pp 101).

• Cross price elasticity of demand = percentage change in quantity of good 1/percentage change in price of good 2

The sign of cross price elasticity of demand is determined by whether two goods are substitutes or complements. Substitutes can be used in place of one another thus an increase in the price of one good will result in an increase in demand for the other; price and quantity demanded move in the same direction thus cross price elasticity of demand will be positive. For complementary goods, cross price elasticity of demand will be negative as shown by Gans (2020, pp 101). The magnitude of the elasticity indicates the extent to which two goods are substitutes or complementary. Perfect substitutes have a high and positive cross price elasticity of demand (Khan Academy, 2022).

An arbitrage involves two identical securities and thus the securities are to perfect substitutes and will have a high and positive cross price elasticity of demand. If a subsidy is applied to one market involved in the arbitrage, the consumers will pay a lower price than that received by producers. This paper aims to explore the consequences of a producer in one market providing a subsidy to consumers in another market where an arbitrage exists between the two markets.

Methods:

An arbitrage involves identical goods traded in two different markets. Each market contains a producer and a consumer. To explore the impact of a subsidy on an arbitrage, an arbitrage will be created between two markets by the producer in the first market providing a subsidy to the consumer in the second market. An arbitrage involves the simultaneous buying and selling of identical or close to identical securities/goods. As the goods in the first and second market are identical, they are perfect substitutes as they would have identical functions. In addition, given they are perfect substitutes, an increase in the price of the good in one market should result in an increase in demand for the good in the second market.

Diagram 1: An arbitrage is created between Market 1 and Market 2

Diagram 1 shows two markets for Good j. The cost of production of good j is x. In market 1, the equilibrium price for Good j is y and quantity Q1 is produced. When producers in market 1 sell good j they make a profit of y - x. In market 2, the price received by producers is z as producers in market 1 provide a subsidy equal to z - y to producers in Market 2. Consumers in Market 2 pay z and will receive z - y from the producers in market 1 at some point in the future. As z > y for good j, an arbitrage exists in which one can buy good j in Market 1 and sell it in Market 2.

Without the subsidy, the equilibrium in market 2 would result in a lower equilibrium quantity (Q2) with a higher price (P2). However, in reality, without the subsidy, no arbitrage opportunity would exist as the price in market 1 would equal the price in market 2. When the subsidy is introduced, their is an increase in supply in market 2 resulting in a rightward shift in the supply curve from S to Ss. The quantity supplied in market 2 is limited by the quantity produced in market 1 thus the maximum quantity in market 2 will be equal to the quantity producer in market 1. The profit made in market 1 (y - x) by selling good j at the equilibrium price of y is used to provide a subsidy to consumers in market 2 as y - x = z - y thus consumers in market 2 will receive z while consumers will pay y. This creates an arbitrage opportunity between market 1 and 2 for good j.

Diagram 2:

Diagram 2 shows the interaction between Market 1 and Market 2. The producer A produces good j for a cost of x. Producer A then sells good j in Market 1 at price y to Consumer B thus making a profit of y - x. Consumer B becomes a producer in Market 2 as they sell good j in Market 2 at price z (z = y + y - x) to Consumer C. Consumer C then receives z - y = y - x (a subsidy) from Producer A at a given point in the future thus Consumer C purchases good j at y while Consumer B receives z for good j in Market 2.

As an arbitrage exists between Market 1 and Market 2, rational participants would purchase good j in Market 1 and sell it in Market 2. This would result in an increase in demand in Market 1 and a rightward shift in the demand curve resulting in an increase in price in Market 1. However, as the price in Market 1 rises, a second arbitrage opportunity results. As Consumer C purchased good j in Market 2 for z and will receive z - y at a given point in the future, they essentially purchased good j for price y. Thus if the price of good j in Market 1 is greater than y, Consumer C can sell good j in Market 1 then receive z - y (the subsidy) from Producer A at a given time in the future and to make a risk free profit.

Results:

Implications for stakeholders

Market 1:

• Producers - Producers in Market 1 produce good j at price x, sell it at price y thus making a profit of y - x. They then use the profit to provide a subsidy for Consumers in Market 2.

• Consumers - Consumers purchase good j at price y and have the opportunity to sell at price z in market 2 and thus make a risk free profit.

Market 2:

• Producers - Consumers in Market 1 become producers in Market 2. They purchase good j in Market 1 and sell it in Market 2 to make the risk free profit.

• Consumers - Consumers purchase good j in market 2 at price z (z = y + [y - x]) which is higher than the price y in Market 1. Consumers will receive z - y from the Producers in Market 1 at a given point in the future. Thus Consumers are paying a price closer to y for good j.

Discussion:

Market 1:

• Producers - Producers produce good j and use the profit made to provide a subsidy to consumers in Market 2. As the arbitrage is maintained, the price in Market 1 will increase however the subsidy will remain constant. Initially, y - x = z - y however, as the price in Market 1 increases, y - x > z - y and Producer A will only need to pay z - y to the Consumer in Market 2. Thus Producer A will continue to produce more of good j due to increasing marginal profits due to the increasing market price in Market 1 while the cost of the subsidy in Market 2 remains constant.

• Consumers - Consumers can make risk free profit thus they will purchase good j in Market 1 and sell it in Market 2. However, the amount they are willing to pay for good j in market 1 is dependent on the amount of time it takes to purchase the good in market 1 and sell in market 2. The faster this process can occur, the closer the price a consumer is willing to pay is to infinity. This area will be explored further in future studies.

Market 2:

• Producers - Consumers in Market 1 become producers in Market 2. They purchase good j in Market 1 and sell it in Market 2 to make the risk free profit.

• Consumers - Consumers purchase good j at price z which is higher than the price y in Market 1. Consumers will receive z - y from the Producers in Market 1 at a given point in the future. Thus Consumers are paying a price closer to y for good j.

Good j in Market 1 is a perfect substitute for good j in Market 2. As the price of good j in Market 1 rises, the demand for good J in Market 2 should rise given good j costs the consumer price y in both markets initially. An increase in demand in Market 2 should increase the price of good j in Market 2. As a result, an increase in price in Market 1 will result in an increase in price in Market 2. This seems to suggest that the arbitrage will not be eliminated. The subsidy results in consumers paying a lower price than what producers receive. The price paid by consumers in Market 2 is equal to the price paid by consumers in Market 1. As a results, unlike traditional arbitrages, the impact of perfect price elasticity of demand is observed. In traditional arbitrage situations, the consumer in Market 2 would pay a higher price than the consumer in Market 1 thus the increasing price in Market 1 will not result in an increase in demand in Market 2 as the price is Market 2 would still be greater than the price in Market 1. In the mechanism being discussed, given the price paid by consumers in Market 1 and 2 is identical, the rise in price in Market 1 which will occur due to the arbitrage will result in an increase in demand for good j in Market 2 which will result in an increase in the price in Market 2.

Conclusion:

The creation of an arbitrage using a subsidy creates an opportunity for consumers and producers in both markets to make risk free returns. In addition, the subsidy maintains the arbitrage resulting in the securities being traded being worth a value approaching infinity. Further articles will explore the calculations that illustrate that good j is worth a value approaching infinity.

References:

Gans, J., King, S., Byford, M., & Gregory M.N (2020). Principlaes of Microeconomics (8th Asia Pacific Edition). Engage Learning Australia.

Sharp, W. & Alexander, G. (1990). Investments, 4th edition, Prentice Hall, Eangle-Wood Cliffs, N. J.

Khan Academy. (24/07/2022). Lesson Overview - Cross Price Elasticity and Income Elasticity of Demand. https://www.khanacademy.org/economics-finance-domain/microeconomics/elasticitytutorial/ income-elasticity-of demand/a/lesson-overview-cross-price-elasticity-and-income-elasticity- of-demand