Question

Let $c_i$ be the constant marginal and average cost for firm $i$ (so that firms may have different marginal costs). Suppose demand is given by $P=1-Q$ a. Calculate the Nash equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output, market price, firm profits, industry profits, consumer surplus, and total welfare. b. Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in firm 1 's cost would change the equilibrium. Draw a representative isoprofit for firm 1

Short answer

In a Cournot market with two firms, given the demand function $P=1-Q$ and constant marginal costs $c_1$ and $c_2$ for firm 1 and firm 2 respectively, we first determine the Nash equilibrium quantities for both firms, which are $q_1^{\ast }=\frac{2-c_1-c_2}{3}$ and $q_2^{\ast }=\frac{2-c_1-c_2}{3}$. From this, we can calculate market output, market price, firm and industry profits, consumer surplus, and total welfare. After plotting the best-response functions for both firms on a diagram, we can observe the effect of a reduction in firm 1's cost on equilibrium quantity. A decrease in firm 1's cost leads to an upward shift in its best-response function, resulting in a higher Nash equilibrium quantity for firm 1 and a lower Nash equilibrium quantity for firm 2. By drawing an isoprofit curve for firm 1, we can also see that in a Cournot model, firm 1's profit increases with both a higher Nash equilibrium quantity for firm 1 and a lower Nash equilibrium quantity for firm 2.

Step-by-step solution

Determine the inverse demand function

Given the demand function is $P=1-Q$, where $Q$ is the total quantity supplied.

Define firm 1 and firm 2 quantities, cost functions, and reaction functions

Let $q_1$ be the quantity supplied by firm 1 and $q_2$ be the quantity supplied by firm 2. Total output in the market is $Q=q_1+q_2$. Now, let the constant marginal and average costs for the two firms be denoted as $c_1$ and $c_2$. The profit function for firm i can be denoted as ${\pi }_i=P\times q_i-c_i\times q_i$. To derive the reaction functions for both firms, we need to find the maximization of their profit functions with respect to their quantity. First, profit maximization for firm 1: ${\pi }_1=(1-Q)q_1-c_1{q}_1$ Now, we derive the first-order condition with respect to $q_1$: $\frac{\partial {\pi }_1}{\partial q_1}=1-2q_1-q_2-c_1=0$ This leads to the reaction function for firm 1: $q_1^{\ast }=\frac{1-c_1-q_2}{2}$ Similarly, we derive the reaction function for firm 2: $q_2^{\ast }=\frac{1-c_2-q_1}{2}$

Solve for Nash Equilibrium quantities

To find the Nash equilibrium, we need to solve the reaction functions simultaneously. Substitute the reaction function for firm 1 into the reaction function of firm 2: $q_2^{\ast }=\frac{1-c_2-(\frac{1-c_1-q_2}{2})}{2}$ By solving this quadratic equation, we obtain the Nash equilibrium quantities for both firms: $q_1^{\ast }=\frac{2-c_1-c_2}{3}$ $q_2^{\ast }=\frac{2-c_1-c_2}{3}$

Calculate market output, market price, and firm and industry profits

Market output: $Q=q_1+q_2=\frac{4-2(c_1+c_2)}{3}$ Market price: $P=1-Q=\frac{1+c_1+c_2}{3}$ Firm 1 profit: ${\pi }_1=P\times q_1-c_1\times q_1=(\frac{1+c_1+c_2}{3})(\frac{2-c_1-c_2}{3})-c_1(\frac{2-c_1-c_2}{3})$ Firm 2 profit: ${\pi }_2=P\times q_2-c_2\times q_2=(\frac{1+c_1+c_2}{3})(\frac{2-c_1-c_2}{3})-c_2(\frac{2-c_1-c_2}{3})$ Industry profit: ${\pi }_{industry}={\pi }_1+{\pi }_2$

Calculate consumer surplus and total welfare

Consumer surplus: $CS=\frac{1}{2}(1-P)(Q)$ Total welfare: $TW=CS+{\pi }_{industry}$ #Part B: Nash Equilibrium Representation on Best-Response Diagram#

Plot best-response functions

On a best-response diagram, plot the reaction functions for firm 1 and firm 2 ($q_1^{\ast }=\frac{1-c_1-q_2}{2}$ and $q_2^{\ast }=\frac{1-c_2-q_1}{2}$) with $q_1$ on the x-axis and $q_2$ on the y-axis. The intersection point of these reaction functions is the Nash equilibrium we computed earlier.

Analyze the effect of a reduction in firm 1's cost on equilibrium

If firm 1's cost $c_1$ decreases, the best-response function for firm 1 shifts upward, with the slope still at -0.5. The new equilibrium occurs at the new intersection of the best-response functions, resulting in a higher Nash equilibrium quantity for firm 1 ($q_1^{\ast }$) and a lower Nash equilibrium quantity for firm 2 ($q_2^{\ast }$).

Draw representative isoprofit for firm 1

On the same best-response diagram, draw an isoprofit curve for firm 1 (indicating the set of points where firm 1 makes the same profit). The curve should be upward-sloping, with the slope greater than 0. This is because, in a Cournot model, firm 1's profit increases with both a higher $q_1^{\ast }$ and a lower $q_2^{\ast }$.