GAME THEORY

Game Theory

Game Theory

Question 1

(a) The definition of substitutes is that . However, we don't have Hicksian demand functions here, and there's no good way to get them. So instead, we can define substitutes based on willingness to pay: if the willingness to pay for good i is positively related to the quantity of good j, then the goods are complements. Using this definition, and the fact that the inverse demand functions represent willingness to pay, we get that the goods are substitutes if z < 0, and complements if z > 0.

Using the results from the later parts of the problem, you should find that z = -1 implies perfect substitutes, and z = 0 implies completely unrelated (or heterogenous) goods. Again notice that the two firms are identical, so we can solve the optimization problem for one firm and then use symmetry to find the solution:

Giving the FOC (interior solution):

This is firm 1's best response function. By symmetry, firm 2's BRF is:

You can now plug one BRF into the other to solve. Or, you can use this often-quicker symmetry strategy: using the fact that q1*=q2*, so you can change the q2 in firm 1's BRF to a q1* and quickly solve for the optimal quantities (note that you can only do this after taking the first-order conditions).

Now plug these quantities into the inverse demand functions to find equilibrium prices:

We are asked to convert the two inverse demand functions into demand functions. So rearrange each equation and then plug one into the other:

And of course, by symmetry:

(b) The strategy looks very similar to part (a), except now we will write profits as q(p)p instead of p(q)q, and we will maximize with respect to p. Consider firm 1:

The FOC (interior solution) is:

This gives firm 1's best response function. By symmetry, firm 2's BRF is:

Now let's use the fact that p1*=p2* in equilibrium (again by symmetry) to solve firm 1's BRF for p1*:

Now substitute in to the demand functions to find equilibrium quantities. Before doing that, let's simplify the demand functions by recognizing that p1*=p2*:

And by symmetry:

(c) First let's compare prices. Prices under quantity competition will be higher than prices under price competition if:

This will always be true for values of z between -1 and 1 (Note: if you do not restrict the values of z, you have to be careful. If z > 2 or -2 < z < -1, then when you try to cross multiply, the sign flips. (But also note, if z>1, then you will get a negative quantity under quantity competition, which doesn't make sense.) If z=0, ie the two goods are completely unrelated or heterogenous, then both sides of this equation are equal. If z = -1, we get the Cournot and Bertrand results for the case where the firms produce the same exact product.

Now let's compare quantities. Quantities under quantity competition will be lower than quantities under price competition if:

This will always be true for values of z between 0 and ...