# Math

1. Suppose the demand for output is given by Q = 120 – P and the marginal (and average) cost of production is constant, at 30. Assuming two firms and Cournot competition, compute the Nash equilibrium outputs, prices, and profits. Graph the reaction curves. Calculate the Lerner index for the typical firm in this market.

2. Can you provide definitions of the following concepts in economics for me?

• Values Q and P
• Cost of production, especially average cost of production and marginal cost of production
• Cournot competition model
• Nash equilibrium
• Reaction functions curves
• Lerner index a) The demand equation is:

`Q=Q_1+Q_2=120-P`

So, `P=120-Q_1-Q_2`

Average cost=Marginal cost=30

To determine the Cournot-Nash equilibrium, the reaction function for each firm has to be calculated first.

Profit for Firm 1=total revenue-total cost

`=(120-Q_1-Q_2-30)Q_1`

`P_1=90Q_1-Q_1^2-Q_1Q_2`

Differentiating this profit function with respect to Q1, and setting the derivative equal to zero gives the reaction function for the first firm,

`R_1=gt 90-2Q_1-Q_2=0`

`Q_1=45-0.5Q_2`

By symmetry, reaction function for the second firm,

`R_2=gt Q_2=45-0.5Q_1`

Substituting the value of Q2,

`Q_1=45-0.5(45-0.5Q_1)`

So, `Q_1=22.5/0.75=\$30`

And also, `Q_2=\$30`

To determine the price at profit maximization, Q1 and Q2 has to be substituted into the demand equation:

P=120-30-30=\$60

Putting the values for price and quantity into the profit function,

`P_1=60xx30-30xx30=\$900`

`P_2 ` is also equal to \$900.

b) The price is \$60, marginal cost is \$30

So, Lerner index `=(P-MC)/P=(60-30)/60=0.5`

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Approved by eNotes Editorial Team In the Cournot competition model developed in France in 1838 by mathematician Augustin Cournot to reflect a duopoly, Q and P represent quantity of output (Q) and product price (P). On a graph, P is the vertical (up-and-down) axis (line) while Q is the horizontal (side-to-side) axis.

Costs of production are classed as total cost, average cost and marginal cost. Total cost is the sum of all variable costs, such as the variable cost of raw materials, and all fixed costs, such as the fixed cost of operating costs.

Average cost is the average cost per each unit produced: Average cost is total cost divided by the total number of units produced during a fiscal period. Consequently, average cost can be stated as the cost to produce one unit of product.

Marginal cost builds on the average cost (cost of one unit) to calculate the added cost of one unit more of the product. In other words: If production is X and average cost Y, then X+1 will equal Y+?. The formula for calculating the cost that results from a one unit change in production output is this:

Marginal Cost (MC) = Change in Total Cost = ΔTC
____________________   ____

Change in Output         Δq

Cournot competition assume a duopoly restrict, per definition, to two firms. [Research has shown the model can be expanded to multiple firms.] Cournot assumes that quantity is the variable and price is the constant.

In other words, companies fix their production on a predetermined price range so that price is stable and so that quantity of output is variable. [Research has shown that, in fact, it is the other way round: quantity is stable while price that is the variable (changeable) factor.]

The Nash equilibrium reflects conditions when prices are stable, as in the Cournot model, and is thus used instead of the price taking competitive equilibrium model that reflects variable prices. Nash equilibrium is appropriate to the Cournot duopoly model because it assumes stable prices with variable quantity output.

With stable prices for two companies--both of which operate under the same assumptions of stable price and homogenous (identical) products--the Nash equilibrium reflects the industry price set by the demand curve, thus reflects the revenues of each company in the duopoly. The Nash equilibrium show the intersection where both firms are choosing optimal production output in consideration of the output of the other firm.

In other words, with prices stable, the only variable is production output. The Nash equilibrium shows the optimal output for each company base upon, or given, the output of the single competitor in the duopoly.

Reaction function curves: With R being the reactions of firms 1 and 2 (R1 and R2), this is shown in graphed reaction functions curves where q1=R1(q2) gives one firm's optimal output in reaction to a given output (known output) for the competitor q2 firm. The vertical axis is R2(q1) while the horizontal axis is R1(q2). Equilibrium is where q1 and q2 intersect on the R2-R1 graph (see image).

The Lerner index uses exact price information, cost structure of a firm, or price elasticity of demand to measure market power. The level of market power for each firm is measured on the Lerner index by measuring price (P) to marginal cost (MC). When using price elasticity of demand, the Lerner index is the inverse of elasticity to maximized price. The formula is:

L = P-MC =  1
____   ___
P       lEl

[Image from Wikipedia Commons and provided by Twisp, Bluemoose (Own work) (Public domain).]