In [10]:

import numpy as np
import matplotlib.pyplot as plt

PS 1¶

1. Two period consumption savings problem with the utility function¶

$$U(c)=-\frac{1}{a}e^{-ac}$$

1.A.1¶

A.1.1 Verify increasing and concave
A.1.2 Draw a picture of the utility function over positive c

1.A.1.1¶

$$U(c)= -\frac{1}{a}e^{-ac}$$$$U(c)\prime = e^{-ac}, \ \ a>0$$$$U(c)\prime\prime =-ae^{-ac}, \ \ a<0$$

1.A.1.2¶

In [37]:

def u(a,c):
    solu = (-1/a)*np.exp(-a*c)
    return solu
def u_1d(a,c):
    solu = np.exp(-a*c)
    return solu
def u_2d(a,c):
    solu = -a*np.exp(-a*c)

In [41]:

x = np.linspace(0,10,101)
y1 = np.zeros(len(x))
y2 = np.zeros(len(x))
y3 = np.zeros(len(x))

for idx,i in enumerate(x):
        y1[idx] = u(1, i)
        y2[idx] = u_1d(1, i)
        y3[idx] = u_2d(1, i)
plt.plot(x,y1)
plt.plot(x, y2, label=r'$U\prime$')
plt.plot(x, y3, label=r'$U\prime\prime$')
plt.ylabel('U(c)')
plt.xlabel('c')
plt.axhline(u(1,0), color='red', label=r"$\frac{-1}{a}$")
plt.legend();

1.B¶

Set up the maximization problem, derive the exact form of the Euler equation with this utility function, and derive a closed-form solution for optimal consumption at time 1. Assume that capital markets are perfect.

$$c_1= y_1+s_1 \tag{3}$$$$c_2 = y_2 + s_1(1+r)\tag{4}$$$$\max_{c1,c2,s1}\beta^{t-1}\sum^{2}_{t=1}U(c_t)$$$$L= -\frac{1}{a}e^{-a c_1}+-\frac{\beta}{a}e^{-a c_2}+\lambda[y_1+s_1-c_1]+\varphi[y_2+(1+r)s_1-c_2]$$

ANSWER

In [95]:

def ibc(y1, y2, r, k=1):
    c1_1 = (1+r)/(2+r)
    c1_2 = (y1+(y2/(1+r)))-(k/(1+r))
    c1 = c1_1*c1_2
    
    c2_1 = (1+r)/(2+r)
    c2_2 = (y1+(y2/(1+r)))+k
    c2 = c2_1*c2_2
    return c1,c2

In [210]:

x = np.linspace(0,.5,10)
y = np.zeros(len(x))
y1, y2,r = [20,20, .1]
for idx,i in enumerate(x):
        ans1, ans2 = ibc(y1,y2,i, k=2)    
        plt.scatter(ans1,ans2, label='r= {:.2f}'.format(i))
        plt.legend()
        plt.ylabel(r'$C_2$')
        plt.xlabel(r'$C_1$')

In [214]:

x = np.linspace(0,5,6)
y = np.zeros(len(x))
y1, y2,r = [20,20, .25]
for idx,i in enumerate(x):
        ans1, ans2 = ibc(y1,y2,r, k=i)    
        plt.scatter(ans1,ans2, label='k= {}'.format(i))
        plt.legend()
        plt.ylabel(r'$C_2$')
        plt.xlabel(r'$C_1$')