Suppose that we have two firms that face a linear demand curve p(Y)= a-b Y and have constant mar

step 1 We want to graph the demand and marginal cost curves. So we're given P to the negative two, so w

Suppose that we have two firms that face a linear demand...

Key Concepts

Constant Marginal Costs

Constant marginal costs imply that the cost of producing an additional unit of output remains the same regardless of the level of production. This assumption is commonly used in economic models to simplify the profit maximization process, as it yields linear cost functions and straightforward equilibrium analysis.

Nash Equilibrium

Nash equilibrium is a fundamental concept in game theory, representing a situation where no player can increase their payoff by unilaterally changing their strategy given the strategies selected by others. In a Cournot duopoly, the equilibrium is reached when each firm's output choice is the best response to the other's, so neither firm has an incentive to deviate from its equilibrium quantity.

Linear Demand Curve

A linear demand curve is one in which the price is a linear function of the quantity, typically expressed as p = a - bY, where 'a' and 'b' are parameters. This simple form facilitates analytical tractability in economic models, as the effect of changes in total output on market price is constant.

Cournot Competition

Cournot competition is a model of oligopoly where firms compete by choosing outputs rather than prices. Each firm makes its decision assuming the quantity produced by its competitor is fixed, and the market price is determined by the total output produced by all firms. This framework highlights the strategic interdependence between firms in their production decisions.

Reaction Function

A reaction function in the context of Cournot competition describes how a firm's optimal output decision depends on the output level chosen by its rival. It is derived from the firm's profit maximization condition and illustrates the best response strategy that maximizes a firm's profit given the competitor's quantity.

Transcript

00:01 We want to graph the demand and marginal cost curves.

00:08 So we're given p to the negative two, so we can put p squared on the bottom.

00:19 Now let's get p in terms of q.

00:54 So if we use a graphing calculator and we input this function, basically we get a graph that looks like this.

01:10 So both ends of the function are approaching q is equal to zero and p is equal to zero.

01:26 We also want to graph the marginal cost curve.

01:38 So if we do this, we're basically going to get a line with a slope of 0 .001, so it's very close to a straight horizontal line.

01:52 You see it's steadily increasing as q increases.

02:02 Now we can determine where they cross if we want.

02:19 So set them equal.

02:32 And then we're going to determine at which value of q they cross.

02:40 Square both sides.

03:29 And they will cross the very high value of q, so it's hard to show.

03:33 So we're not going to include it.

03:37 And if you want, you can just change the scale of the axes and you could show that they cross at this value of q.

03:50 The question doesn't really require it.

03:56 Calculate marginal revenue associated with the demand curve.

04:00 So we already got the demand in terms of p.

04:28 Total revenue is equal to price times quantity.

04:31 So we can plug in this value for p and multiply by q.

04:55 So we know that if we have q on the bottom, we're going to subtract the exponent.

05:32 And we'll take the square root of the top.

05:57 Marginal revenue is the derivative of total revenue.

06:25 So then we're able to get marginal revenue.

06:37 And we also want to graph this curve.

06:46 So then using our graphing calculator, and i'm going to put it on the same graph as before so that we can compare these two.

07:00 Basically we get a graph that is going to be even closer to the axes.

07:18 So basically it looks something like this.

07:51 At what output level does marginal revenue equal marginal cost? so here we can do it algebraically.

07:59 So we're going to set marginal revenue equal to marginal cost...

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