Definition

The price elasticity of demand, E_d is defined as the magnitude of:

 

proportionate change in quantity demanded
------------------------------------------------------------------------
proportionate change in price

 

Since the quantity demanded decreases when the price increases, this ratio is negative; however, the absolute value usually is taken and E_d is reported as a positive number.

Because the calculation uses proportionate changes, the result is a unitless number and does not depend on the units in which the price and quantity are expressed.

As an example calculation, take the case in which a product's E_d is reported to be 0.5. Then, if the price were to increase by 10%, one would observe a decrease of approximately 5% in quantity demanded.

In the above example, we used the word "approximately" because the exact result depends on whether the initial point or the final point is used in the calculation. This matters because for a linear demand curve the price elasticity varies as one moves along the curve. For small changes in price and quantity the difference between the two results often is negligible, but for large changes the difference may be more significant. To deal with this issue, one can define the arc price elasticity of demand. The arc elasticity uses the average of the initial and final quantities and the average of the initial and final prices when calculating the proportionate change in each. Mathematically, the arc price elasticity of demand is defined as:

Q2 - Q1
-----------------------
(Q1 + Q2) / 2
-------------------------------
P2 - P1
-----------------------
(P1 + P2) / 2

where

Q1 = Initial quantity
Q2 = Final quantity
P1 = Initial price
P2 = Final price

Elastic versus Inelastic

E > 1
In this case, the quantity demanded is relatively elastic, meaning that a price change will cause an even larger change in quantity demanded. The case of E_d = infinity is referred to as perfectly elastic. In this theoretical case, the demand curve would be horizontal. For products having a high price elasticity of demand, a price increase will result in a revenue decrease since the revenue lost from the resulting decrease in quantity sold is more than the revenue gained from the price increase.

E < 1
In this case, the quantity demanded is relatively inelastic, meaning that a price change will cause less of a change in quantity demanded. The case of E_d = 0 is referred to as perfectly inelastic. In this theoretical case, the demand curve would be vertical. For products whose quantity demanded is inelastic, a price increase will result in a revenue increase since the revenue lost by the relatively small decrease in quantity is less than the revenue gained from the higher price.

E = 1
In this case, the product is said to have unitary elasticity; small changes in price do not affect the total revenue.

Factors Affecting the Price Elasticity of Demand

• Availability of substitutes: the more possible substitutes, the greater the elasticity. Note that the number of substitutes depends on how broadly one defines the product.
• Degree of necessity or luxury: luxury products tend to have greater elasticity. Some products that initially have a low degree of necessity are habit forming and can become "necessities" to some consumers.
• Proportion of the purchaser's budget consumed by the item: products that consume a large portion of the purchaser's budget tend to have greater elasticity.
• Time period considered: elasticity tends to be greater over the long run because consumers have more time to adjust their behavoir.
• Permanent or temporary price change: a one-day sale will elicit a different response than a permanent price decrease.
• Price points: decreasing the price from $2.00 to $1.99 may elicit a greater response than decreasing it from $1.99 to $1.98.

 

 

Industry Concentration

 

The concentration of firms in an industry is of interest to economists, business strategists, and government agencies. Here, we discuss two commonly-used methods of measuring industry concentration: the Concentration Ratio and the Herfindahl-Hirschman Index.

Concentration Ratio (CR)

The concentration ratio is the percentage of market share owned by the largest m firms in an industry, where m is a specified number of firms, often 4, but sometimes a larger or smaller number. The concentration ratio often is expressed as CR _m, for example, CR₄.

The concentration ratio can be expressed as:

CR _m = s₁ + s₂ + s₃ +...... + s _m

where s _i = market share of the i^th firm.

If the CR₄ were close to zero, this value would indicate an extremely competitive industry since the four largest firms would not have any significant market share.

In general, if the CR₄ measure is less than about 40 (indicating that the four largest firms own less than 40% of the market), then the industry is considered to be very competitive, with a number of other firms competing, but none owning a very large chunk of the market. On the other extreme, if the CR₁ measure is more than about 90, that one firm that controls more than 90% of the market is effectively a monopoly.

While useful, the concentration ratio presents an incomplete picture of the concentration of firms in an industy because by definition it does not use the market shares of all the firms in the industry. It also does not provide information about the distribution of firm size. For example, if there were a significant change in the market shares among the firms included in the ratio, the value of the concentration ratio would not change.

Herfindahl-Hirschman Index (HHI)

The Herfindahl-Hirschman Index provides a more complete picture of industry concentration than does the concentration ratio. The HHI uses the market shares of all the firms in the industry, and these market shares are squared in the calculation to place more weight on the larger firms. If there are n firms in the industry, the HHI can be expressed as:

HHI = s₁² + s₂² + s₃² +...... + s _n ²

where s _i is the market share of the i^th firm.

Unlike the concentration ratio, the HHI will change if there is a shift in market share among the larger firms.

The Herfindahl-Hirschman Index is calculated by taking the sum of the squares of the market shares of every firm in the industry. For example, if there were only one firm in the industry, that firm would have 100% market share and the HHI would be equal to 10,000 -- the maximum possible value of the Herfindahl-Hirschman Index. On the other extreme, if there were a very large number of firms competing, each of which having nearly zero market share, then the HHI would be close to zero, indicating nearly perfect competition.

The U.S. Department of Justice uses the HHI in guidelines for evaluating mergers. An HHI of less than 1000 represents a relatively unconcentrated market, and the DOJ likely would not challenge a merger that would leave the industry with an HHI in that range. An HHI between 1000 and 1800 represents a moderately concentrated market, and the DOJ likely would closely evaluate the competitive impact of a merger that would result in an HHI in that range. Markets having an HHI greater than 1800 are considered to be highly concentrated; there would be serious anti-trust concerns over a proposed transaction that would increase the HHI by more than 100 or 200 points in a highly concentrated market.

Other Considerations in Using Industry Concentration Measures

One should be aware that these measures are influenced by the definition of the relevant market. For example, the automotive industry is not the same as the market for sport utility vehicles. One also must consider the geographic scope of the market, for example, national markets versus local markets.

 

Game Theory

 

Game theory analyzes strategic interactions in which the outcome of one's choices depends upon the choices of others. For a situation to be considered a game, there must be at least two rational players who take into account one another's actions when formulating their own strategies.

If one does not consider the actions of other players, then the problem becomes one of standard decision analysis, and one is likely to arrive at a strategy that is not optimal. For example, a company that reduces prices to increase sales and therefore increase profit may lose money if other players respond with price cuts. As another example, consider a risk averse company that makes its decisions by maximizing its minimum payoff (maxmin strategy) without considering the reactions of its opponents. In such a case, the minimum payoff might be one that would not have occurred anyway because the opponent might never find it optimal to implement a strategy that would make it come about. In many situations, it is crucial to consider the moves of one's opponent(s).

Game theory assumes that one has opponents who are adjusting their strategies according to what they believe everybody else is doing. The exact level of sophistication of the opponents should be part of one's strategy. If the opponent makes his/her decisions randomly, then one's strategy might be very different than it would be if the opponent is considering other's moves. To analyze such a game, one puts oneself in the other player's shoes, recognizing that the opponent, being clever, is doing the same. When this consideration of the other player's moves continues indefinitely, the result is an infinite regress. Game theory provides the tools to analyze such problems.

Game theory can be used to analyze a wide range of strategic interaction environments including oligopolies, sports, and politics. Many product failures can be attributed to the failure to consider adequately the responses of competitors. Game theory forces one to consider the range of a rival's responses.

Elements of a Game

• Players: The decision makers in the game.
• Actions: Choices available to a player.
• Information: Knowledge that a player has when making a decision.
• Strategies: Rules that tell a player which action to take at each point of the game.
• Outcomes: The results that unfold, such as a price war, world peace, etc.
• Payoffs: The utilities (or happiness) that each player realizes for a particular outcome.
• Equilibria: An equilibrium is a stable result. Equilibria are not necessarily good outcomes, a fact that is illustrated by the prisoner's dilemma.

Game Theory Framework

When evaluating a situation in which game theory is applicable, the following framework is useful.

1. Define the problem
2. Identify the critical factors. Examples of critical factors include differentiated products, first-mover advantage, entry and exit costs, variable costs, etc.
3. Build a model, such as a bimatrix game or an extensive form game.
4. Develop intuition by using the model
5. Formulate a strategy - cover all possible scenarios.

A good strategy could be used as a set of instructions for someone who knows nothing about the problem. It specifies the best action for each possible observation. The best strategy may be formulated by first evaluating the complete set of strategies. The complete set of strategies is a list of all possible actions for each possible observation.

Bimatrix Games

In a bimatrix game, there are two players who effectively make their moves simultaneously without knowing the other player's action. A bimatrix game can be represented by a matrix of rows and columns. Each cell in the matrix has a pair of numbers representing the payoff for the row player (the first number) and the column player (the second number). The game has the following form:

 

 

Row Player (RP)	Column Player (CP)
CP Option 1	CP Option 2
RP Option 1	(Payout to RP, Payout to CP)	(Payout to RP, Payout to CP)
RP Option 2	(Payout to RP, Payout to CP)	(Payout to RP, Payout to CP)

 

The general form of equilibrium in a bimatrix game is called a Nash Equilibrium. If both rivals have dominant strategies that coincide, then the equilibrium is called a dominant strategy equilibrium, a special case of a Nash equilibrium.

A dominant strategy, if it exists, is for one of the players the strategy that is always the best strategy regardless of what one's rival plays. A dominated strategy is one that is always the worst regardless of what one's rival plays. In games having more than two rows or problems, one may find it useful to identify one option that is always better or worse than another option, in other words, that dominates or is dominated by another option. In this case, the inferior strategy can be eliminated and the game simplified such that more options can be eliminated based on the smaller matrix.

If no options dominate any others, a Nash equilibrium might still be found by evaluating each player's best option for each option of the opponent. If a cell coincides for both players, then that cell is a Nash equilibrium. A game can have more than one Nash equilibrium, but one of them may be the more likely outcome if it is better for both players.

Extensive Form Games

Extensive form games are modeled with dots with arrows that point to other dots. A node is a decision point. The beginning point is depicted by an open dot, which usually represents a state from which a situation will arise by chance. Decision points are labeled with the name of the player making the decision. The following diagram shows the structure of an extensive-form game representation.

 

Structure of an Extensive Form Game

 

Chance nodes can appear anywhere in the extensive form tree.

An information set is a collection of nodes that are controlled by the same player, but which are indistinguishable for that player. In other words, for nodes in the same information set, the player does not know which one he/she is at, but does know that these nodes are different. In the preceding diagram, the drawn information sets might arise if Decision A and Decision C were indistinguishable to Player 2, as well as Decision B and Decision D. If a single dotted line encompassed all the Player 2 decision nodes (or 4 dotted circles all connected), then Player 2 would not be able to distinguish between any of the four decisions.

An extensive form game without information sets designated is one in which the players know exactly where they are in the tree. This situation is equivalent to one of dotted circles drawn around each decision point in the tree but not connected to one another. If neither player can observe anything about the other player's action, the sequential extensive form game can be reduced to the simultaneous-action bimatrix game.

Normal-Form (Strategic Form) Game Representation

The extensive form of representing a game can become difficult to manage as the game gets larger, and the Nash equilibria may become difficult to find. The extensive form representation can be collapsed into the normal form, which encodes the game into a strategy that describes the action to take for each conceivable situation (for example, for each information set). The normal form is a complete listing of all the possible strategies along with their payoffs. For a generic case in which there are three situations (information sets) based on Player 1's move, and two possible responses by Player 2, the normal form takes the following structure:

		Player 1
		If Player 1 does A, then	If Player 1 does B, then	If Player 1 does C, then	A	B	C
Player 2	1)	d	d	d	(__,__)	(__,__)	(__,__)
	2)	d	d	e	(__,__)	(__,__)	(__,__)
	3)	d	e	d	(__,__)	(__,__)	(__,__)
	4)	d	e	e	(__,__)	(__,__)	(__,__)
	5)	e	d	d	(__,__)	(__,__)	(__,__)
	6)	e	d	e	(__,__)	(__,__)	(__,__)
	7)	e	e	d	(__,__)	(__,__)	(__,__)
	8)	e	e	e	(__,__)	(__,__)	(__,__)
	Payoffs

 

Signaling and Threats

A signal is a commitment that changes the strategic situation. For example, if there exists danger of a price war, a smaller firm may sell a plant in order to reduce capacity and limit the amount of a larger firm's demand that it can steal, signaling that it does not intend to build substantial market share. As another example, a sports player who for some reason refuses to undergo an medical exam before negotiating his salary will raise doubts as to whether he has a medical condition that might affect his performance. This player might wish to signal that he does not have such a condition by proposing that his pay be tied to his performance. For a signal to be effective, its cost to a bluffer must exceed the benefits.

A threat is credible if it is believable. A threat is believable if it is in the best interest of the one making the threat to carry it out.