def resid(c):
F.c = c # update basis coefficients
f = F(x) # interpolate at basis nodes x
return f ** -2 + f ** -5 - 2 * x
# ### Compute function inverse
# In[4]:
c0 = np.zeros(n) # set initial guess for coeffs
c0[0] = 0.2
problem = NLP(resid)
F.c = problem.broyden(c0) # compute coeff by Broyden's method
# ### Plot setup
# In[5]:
n = 1000
x = np.linspace(a, b, n)
r = resid(F.c)
# ### Plot function inverse
# In[6]:
tnodes = BasisChebyshev(n - 1, 0, T).nodes
F = BasisChebyshev(n, 0, T, y=np.ones((2, n)))
def resid(c, tnodes, T, n, F, r, k, eta, s0):
F.c = np.reshape(c[:], (2, n))
(p, s), d = F(tnodes, [[0, 1]])
d[0] -= (r * p + k)
d[1] += p ** -eta
(p_0, p_T), (s_0, s_T) = F([0, T])
return np.r_[d.flatten(), s_0 - s0, s_T]
storage = NLP(resid, F.c.flatten(), tnodes, T, n, F, r, k, eta, s0)
c = storage.broyden(print=True)
F.c = np.reshape(c, (2, n))
nplot = 501
t = np.linspace(0, T, nplot)
(p, s), (dp, ds) = F(t, [[0, 1]])
res_p = dp - r * p - k
res_s = ds + p ** -eta
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, res_p)
plt.title('Residuals')
plt.ylabel('d(price) residual')
plt.subplot(2, 1, 2)
plt.plot(t, res_s)
plt.xlabel('time')
# ### Approximation structure
# In[3]:
n, a, b = 21, 0.5, 2.5
Q = BasisChebyshev(n, a, b)
c0 = np.zeros(n)
c0[0] = 2
p = Q.nodes
# ### Solve for effective supply function
# In[4]:
monopoly = NLP(resid)
Q.c = monopoly.broyden(c0)
# ### Setup plot
# In[5]:
nplot = 1000
p = np.linspace(a, b, nplot)
rplot = resid(Q.c)
# ### Plot effective supply
# In[6]:
demo.figure("Monopolist's Effective Supply Curve", 'Quantity', 'Price')
plt.plot(Q(p), p)
S.c = c # update interpolation coefficients
q = S(p) # compute quantity supplied at price nodes
return p - q * (p ** (eta+1) / eta) - alpha * np.sqrt(q) - q ** 2
# Notice that `resid` only takes one argument. The other parameters (`Q`, `p`, `eta`, `alpha`) should be declared as such in the main script, were Python's scoping rules will find them.
# ### Solve for effective supply function
#
# Class `NLP` defines nonlinear problems. It can be used to solve `resid` by Broyden's method.
# In[7]:
cournot = NLP(resid)
S.c = cournot.broyden(S.c, tol=1e-12)
# ### Plot demand and effective supply for m=5 firms
# In[8]:
prices = np.linspace(a, b, 501)
fig1 = demo.figure('Cournot Effective Firm Supply Function',
'Quantity', 'Price', [0, 4], [0.5, 2])
plt.plot(5 * S(prices), prices, D(prices), prices)
plt.legend(('Supply','Demand'))
# ### Plot residual
tnodes = BasisChebyshev(n - 1, 0, T).nodes
F = BasisChebyshev(n, 0, T, y=np.ones((2, n)))
def resid(c, tnodes, T, n, F, r, k, eta, s0):
F.c = np.reshape(c[:], (2, n))
(p, s), d = F(tnodes, [[0, 1]])
d[0] -= (r * p + k)
d[1] += p**-eta
(p_0, p_T), (s_0, s_T) = F([0, T])
return np.r_[d.flatten(), s_0 - s0, s_T]
storage = NLP(resid, F.c.flatten(), tnodes, T, n, F, r, k, eta, s0)
c = storage.broyden(print=True)
F.c = np.reshape(c, (2, n))
nplot = 501
t = np.linspace(0, T, nplot)
(p, s), (dp, ds) = F(t, [[0, 1]])
res_p = dp - r * p - k
res_s = ds + p**-eta
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(t, res_p)
plt.title('Residuals')
plt.ylabel('d(price) residual')
plt.subplot(2, 1, 2)
plt.plot(t, res_s)
plt.xlabel('time')
n, a, b = 21, 0.5, 2.5 Q = BasisChebyshev(n, a, b) c0 = np.zeros(n) c0[0] = 2 p = Q.nodes # ### Solve for effective supply function # In[4]: monopoly = NLP(resid) Q.c = monopoly.broyden(c0) # ### Setup plot # In[5]: nplot = 1000 p = np.linspace(a, b, nplot) rplot = resid(Q.c) # ### Plot effective supply # In[6]: