The Equivalence of Centralized and Decentralized Equilibrium

The Equivalence of Centralized and Decentralized Equilibrium - Consider an economy occupied by two households (i = A, B) who are facing the two-period consumption-saving choices. Each household i = A, B is facing the following utility maximization problem {64 subject to co+4= where Yo and yl are household i\'s exogenous income in period t 0,1, c0 and cl are household i\'s consumption in period t = 0,1, sl is household i\'s saving, r is the real interest rate, and 0

Solution

Consider the given intertemporal consumption problem given in the question.

So, the 1^st period budget line is given by, “C0i + S1i = Y0i” and the 2^nd period budget line is given by, “C1i = Y1i + (1+r)*S1i.

=> life budget constraint of the ith consumer is given below.

=> C1i = Y1i + (1+r)*S1i, just put “C0i + S1i = Y0i, => S1i = Y0i – C0i”.

=> C1i = Y1i + (1+r)*(Y0i – C0i) = Y1i + (1+r)*Y0i – (1+r)*C0i.

=> C1i/(1+r) = Y1i/(1+r) + Y0i – C0i, => C0i + C1i/(1+r) = Y0i + Y1i/(1+r).

=> C0i + C1i/(1+r) = Y0i + Y1i/(1+r), be the life time budget line here of the ith individual.

So, the lagrange function is given below.

=> L = U(C0i) + *U(C1i) + [Y0i + Y1i/(1+r) – C0i – C1i/(1+r)].

So, the FOC of maximization is given below.

=> L/C0i = L/C1i = 0.

=> L/C0i = 0, => U0/C0i + *(– 1) = 0, => U0/C0i = ………………..(1).

=> L/C1i = 0, => *U1/C1i + *(– 1/1+r) = 0, => *U1/C1i = /(1+r)……………….(2).

Now, by dividing (1) by (2) we will get the “Euler Equation”.

=> [U0/C0i] / *[U1/C1i] = /[ /(1+r)] = (1+r).

=> [U0/C0i] = (1+r)**[U1/C1i]…………………….(3).

Here equation “3” be the “Euler Equation”.

b).

As we know that as “S1i” increasing, => “C0i” will falls, => “U(C0i)” will fall, => “U0/C0i”, be the MC here. On the other hand as “S1i” increasing, => “C1i” will increases, => “U(C1i)” will also increases, => “U1/C1i”, be the MB here. Now, at the equilibrium “MB=MC”, => at the equilibrium the following condition will hold.

=> MC = MB, => [U0/C0i] = (1+r)**[U1/C1i].