Question

1. Solve for the optimal values of C1 and C2 in the following optimization problem: MaxC1,C2 C₁^1/2 + βC₂^1/2

s.t. C₁ + C₂ /1 + r = Y₁ + Y₂/1 + r

Hint: ∂C^1/2 /∂C = 1/2C^−1/2

When r goes up, how does C1 change? Does it increase or decrease?

Answer 1

SOLUTION:

Given That the data is  

  the optimal values of C1 and C2 in the following optimization problem: Max C1,C2 C₁^1/2+ βC₂^1/2

^AND

s.t. C₁ + C₂ /1 + r = Y₁ + Y₂/1 + r

Hint: ∂C^1/2 /∂C = 1/2C^−1/2

^SO  

==> According to this problem, , the consumer receives an income in periods 1 and 2 denoted by y1 and y2. now in the first period, the consumer chooses to consume or save his income:

c1 + s1 = y1,

where c1 is the consumed and s1 is the saved. In the next period, the consumer receives both his period income and principal plus interest on its savings. Therefore, the consumer now consume: c2 = y2 + (1 + r)s1; where c2 is the consumed and r is the (real) rate of interest. Kindly note as consumer has finite lifetime, there will be no saving in 2 period which is represented by

C1 + C2 /1 + r = Y1 + Y2 /1 + r

It states that the present value of consumption equals the present value of income. Now, 1/1+r = p2/p1 as the market discount factor is the relative price of the 2 period consumption. Substituting the values we get:

c1+c2*p2/p2 = W/p1

Lets move to the utility function now, which is C1^1/2 + βC2^1/2 s.t C1 + C2 /1 + r = Y1 + Y2 /1 + r

Applying Lagrangian formula,

L = C1^1/2 + βC2^1/2+{y1 + (1/1+r)y2 -c1 - (1/1+r)c2}

First order

c2/c1 = 1/1+r

It shows that the MRS equals the market discount factor. The budget line is c2 = (1 + r)y1 + y2 - (1 + r)c1

slope is -(1+r)<0

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******* dc2/dc1 = (c1) /(c2) < 0: This implies thas as slope become flatter as c1 increases .*******