Question

Consider an economy occupied by many households with two types denoted by i, (i- A, B) who are facing the two-period consumption problem. Each household i- A, B is facing the following utility maximization problem: max subject to ci bi(1+r)bo where yl and yẳ are household is exogenous income in period t 1, 2 . CI and då are household i's consumption in period t = 1.2. , bị is household i's bond holdings of which bo is exogenously given, r is the real interest rate, ad 0 < β < 1 is the household discount factor. (a) Set up and solve the household's problem. What are the equilibrium conditions of the model? household (b1, b2) NOW: Assume each household's endowments are given by the following: (b) Solve for the household's optimal consumption plan (c1, c2) and bond holdings for each (21,0) y2(0,1 +9) Furthermore assume that there are precisely N individuals of type A and M individuals of type B. (c) What are the market clearing conditions of the economy? (d) Now solve for each type's consumption in each period as a function of g, N and M

Answer 1

There are 2 types of households - A & B

a)

We need to maximize the utility of each household subject to two given budget constraints. For easy notations we are not considering the sign i in each equation. We will incorporate the value of i while considering cases of the specific type of household. The budget constraints for each household are:

$c_{1} &plus; b_{1} = y_{1} &plus; (1&plus;r)b_{0}$  

and

$c_{2} = y_{2} &plus; (1&plus;r)b_{1}$

look, we can find the value of b₁ from second budget constraint and substitute it in first budget constraint to obtain :

$c_{1} &plus; \frac{c_{2}}{1&plus;r} = y_{1} &plus; \frac{y_{2}}{1&plus;r} &plus; (1&plus;r)b_{0}$

If you wish to see the derivation of this equation, refer to the following image:

This equation of budget constraint hence obtained is called the lifetime budget constraint of each individual. Rather than solving the optimization exercise for two budget constraints, we only have one now. It makes the calculation easy.

The next exercise is to maximize the utility of each household based on the lifetime budget constraint.

$max._{(c_{1}, c_{2}, b_{1})} log(c_{1}) &plus; \beta log(c_{2})$

subject to

$c_{1} &plus; \frac{c_{2}}{1&plus;r} = y_{1} &plus; \frac{y_{2}}{1&plus;r} &plus; (1&plus;r)b_{0}$

Forming Lagrangian to solve the above optimization exercise:

$\L = log(c_{1}) &plus; \beta log(c_{2}) -\lambda (c_{1} &plus;\frac{c_{2}}{1&plus;r} - y_{1} &plus; \frac{y_{2}}{1&plus;r} &plus; (1&plus;r)b_{0}))$

b)

differentiating the $\L$  with respect to c₁, c₂ and $\lambda$ respectively and then putting first order derivative of each to 0 (solving for first-order conditions) gives:

Note that we substituted b₁ while calculating the lifetime budget constraint. Taking the value of b₁ from there and substituting in it the value of c₂, we obtain :

Notice that b₀ is exogenously given. Also, as a person lives only for 2 periods the given question, he won't invest in bonds period 2 (b₂ = 0). Therefore, we have solved for part a and b of the above question.

c)

Now, bringing the type of households into consideration. We obtain :

Differentiating each household of A and B, we get :

for household A and

for household B

as there is N number of household A and M number of household B, The optimization exercise above will run N times for each household in type A and M times for each household in type B.

generating the market clearing conditions given by

$C_{1}^{A} = N. c_{1}^{iA}$

$C_{2}^{A} = N. c_{2}^{iA}$

$C_{1}^{B} = M. c_{1}^{iB}$

$C_{2}^{B} = M. c_{2}^{iB}$

d)

Putting the values of  $c_{1}^{iA}, c_{2}^{iA}, c_{1}^{iB}, c_{2}^{iB}$ calculated above into these equations, we get the final answer.