janboone
commented
1 year ago A couple of things are going wrong here. First, I do not understand how you can run this code without errors. E.g. (p_A**(1-ϵ))((1/2) + ((p_B**(1-ϵ)-p_A**(1-ϵ))/(2t*(1-ϵ)))) is not valid python code.
Second, it is not a good idea to define separate functions for A and B when, in fact, they are the same functions.
Third, in the function fixed point, x[0] should be compared to reaction_function_A (not B); although, as said, it is better not to have _A and _B to start with.
Fourth, the function fixed_point returns optimal prices. So it is a bit confusing to have the function equilibrium_output.
Does this help to work on this further?
HristinaSaparevska
We have the indifferent consumer:
$\hat{x} = \frac{1}{2} + \frac{p_B^{1-ϵ}-p_A^{1-ϵ}}{2t(1-ϵ)}$
These are the profits of the firms
$\pi_A = \hat{x} p_A^{1-ϵ}$, $\pi_B = (1-\hat{x}) p_B^{1-ϵ}$
We need to take the FOC of $\pi_A$ (maximize it) with respect to $p_A$ and then do the same for $\pi_B$ and $p_B$. Then we solve the system for $p_A$ and $p_B$ and get the equilibrium prices:
$p_A = p_B = (t(1-ε))^{\frac{1}{1-ε}}$
When we tried to calculate the equilibrium prices for different values of epsilon and t=1 we got the same result of $p_A = 0.000006$ and we think that the prices should be different for the different values of the parameter. Could you tell us where we are making a mistake?
The code:
def profit_A(p_A,p_B,t,ϵ): return (p_A*(1-ϵ))((1/2) + ((p_B(1-ϵ)-p_A(1-ϵ))/(2t(1-ϵ))))
def reaction_function_A(p_B,t,ϵ): return optimize.fminbound(lambda p_A: -profit_A(p_A,p_B,t,ϵ),0,1)
def profit_B(p_A,p_B,t,ϵ): return (p_A(1-ϵ))(1-((1/2) + ((p_B(1-ϵ)-p_A(1-ϵ))/(2*t(1-ϵ)))))
def reaction_function_B(p_A,t,ϵ): return optimize.fminbound(lambda p_B: -profit_B(p_A,p_B,t,ϵ),0,1)
def fixed_point(x,t,ϵ): return [ x[0] - reaction_function_B(x[1],t,ϵ), x[1] - reaction_function_A(x[0],t,ϵ)]
optimize.fsolve(lambda x: fixed_point(x,1,0), [0.5,0.5])
x0 = [0.2, 0.2]
def equilibrium_output(t, ϵ): sol = optimize.fsolve(lambda x: fixed_point(x, t, ϵ), x0) return sol
equilibrium_output (0.5, 0.5)
t = 1 range_ϵ = np.arange (0,1,0.1) range_p_A = [equilibrium_output (t, ϵ) [0] for ϵ in range_ϵ]
df = pd.DataFrame({'ϵ': range_ϵ, 'p_A': range_p_A})