# first we find the cutoff below which borrowers prefer direct lenders to banks
# guess a qD
qD=rH/etaB*(1+sigma/rho) # this is what the price would be if the success rate of direct lenders was the same as that of banks
# and now we look for the fixed point we need
# unlike most papers in econ, we won't use dichotomy here, because that's bound to give the wrong root
# instead we do an exhaustive search
# a bit expensive, but the only way to get the right answer safely
gridsize=200
qtry=np.linspace(rL/1*(1+sigma/rho),4*rL/1*(1+sigma/rho), gridsize) # direct lenders need at least rH so it is low enough
# whether the upper bound is high enough will need to be found out below
# recall that no solution at all may exist
gaptry=np.zeros(gridsize)
etaDtry=np.zeros(gridsize)
etaBtry=np.zeros(gridsize)
deltaDtry=np.zeros(gridsize)
deltaBtry=np.zeros(gridsize)
plowtry=np.zeros(gridsize)
pbartry=np.zeros(gridsize)
netincomeBtry=np.zeros(gridsize)
# ready for recording purposes
# we could use list comprehension below
# but writing out an explicit loop helps clarity
gap=-5
i=0
# while gap<0: # use this opening if pressed for time
while i<gridsize:
qD=qtry[i]
# find the bank threshold quality level
fB = lambda p: lam/(lam+rho+sigma)*(p *(y-rL/p*(1+sigma/rho))/(sigma+rho) +(1-p)*w/(sigma+rho)) \
-(p*(y-qD)/(sigma+rho)+(1-p)*w/(sigma+rho))
pbar=fsolve(fB, 0.5)
pbar=min(pbar,0.995) # to avoid issues, graphs will show if we have to worry about it or not
# now compute the average producer income
#fig,ax=plt.subplots(1,1)
#x = np.linspace(0.1,1, 100)
#ax.plot(x, f(x),'r-', lw=2, alpha=0.5)
#plt.show()
# we also need the threshold past which people choose not to seek funding
f = lambda p: (p*(y-qD)/(sigma+rho)+(1-p)*w/(sigma+rho))-c - w/(sigma+rho)
plow=fsolve(f, 0.5)
plow=min(plow,pbar) #if plow>pbar all agents choose banks
#now we measure the fraction of projects that are between plow and pbar
deltaD, err = quad(pdensity,plow, pbar)
#now we measure the fraction of projects funded directly that succeed
def xpdensityD(x):
return x*pdensity(x)/deltaD # note this the deltaB from
etaD, err = quad(xpdensityD,plow, pbar)
#now we measure the fraction of projects that are above pbar
deltaB, err = quad(pdensity,pbar, 1)
def incB(x):
return (y-rL*(1+sigma/rho)/x)*pdensity(x)/deltaB
netincomeBtry[i], err = quad(incB,pbar, 1) # call quad to integrate premiumB from pbar to 1
#now we measure the fraction of projects funded directly that succeed
def xpdensityB(x):
return x*pdensity(x)/deltaB
etaB, err = quad(xpdensityB,pbar, 1)
# now we check the break-even condition
# I already selected parameters so that we are almost at equilibrium
gap=qD-rH/etaD*(1+sigma/rho)
#print("at iteration " + str(i) + " the gap is:", gap)
#print("plow is:", plow)
#print("pbar is:", pbar)
#print("etaD is:", etaD)
#print("\n")
# record mapping if to be displayed
etaDtry[i]=etaD
etaBtry[i]=etaB
deltaDtry[i]=deltaD
deltaBtry[i]=deltaB
plowtry[i]=plow
pbartry[i]=pbar
gaptry[i]=gap
i +=1
if i==gridsize:
gap=1
# now we display
fig, axs=plt.subplots(2, 2,figsize=(12, 8))
axs[0,0].plot(qtry,plowtry,label='lower')
axs[0,0].plot(qtry,pbartry,label='upper')
axs[0,0].legend()
axs[0,1].plot(qtry,etaDtry,label='direct lenders')
axs[0,1].plot(qtry,etaBtry,label='banks')
axs[0,1].legend()
axs[1,0].plot(qtry,deltaDtry,label='direct lenders')
axs[1,0].plot(qtry,deltaBtry,label='banks')
axs[1,0].legend()
axs[1,1].plot(qtry,gaptry)
axs[0,0].set_title('Project quality thresholds')
axs[0,1].set_title('Success rate')
axs[1,0].set_title('Mass of projects')
axs[1,0].set_title('Mass of projects')
axs[1,1].set_title(r'Fixed point gap: $q_D-\frac{r_H}{\eta_D} (1+\frac{\sigma}{\rho})$')
axs[1,1].axhline(y=0, color='r', linestyle='dotted')
# axs[0,0].set_xlabel('qD')
# axs[0,1].set_xlabel('qD')
axs[1,0].set_xlabel(r'$q_D$')
axs[1,1].set_xlabel(r'$q_D$')
axs[0,0].set_xlim([0.15,0.56])
axs[0,1].set_xlim([0.15,0.56])
axs[1,0].set_xlim([0.15,0.56])
axs[1,1].set_xlim([0.15,0.56])
