To solve the optimization problem, let's define the objective function and the constraint equation in a more explicit manner:
Objective function:
f(C1, C2) = √C1 + β√C2
Constraint equation:
C1 + C2/(1+r) = Y1 + Y2/(1+r)
Now, let's proceed with solving the problem.
Step 1: Rewrite the constraint equation
Multiply both sides of the constraint equation by (1+r) to eliminate the denominator:
(1+r)(C1 + C2/(1+r)) = (1+r)(Y1 + Y2/(1+r))
C1 + C2 + C2r/(1+r) = (1+r)Y1 + Y2
Step 2: Express C1 in terms of C2
C1 = (1+r)Y1 + Y2 - C2 - C2r/(1+r)
Step 3: Substitute the expression for C1 into the objective function
f(C2) = √[(1+r)Y1 + Y2 - C2 - C2r/(1+r)] + β√C2
Step 4: Differentiate the objective function with respect to C2
f'(C2) = -1/2[(1+r)Y1 + Y2 - C2 - C2r/(1+r)]^(-1/2) - 1/2βC2^(-1/2)
Step 5: Set the derivative equal to zero and solve for C2
-1/2[(1+r)Y1 + Y2 - C2 - C2r/(1+r)]^(-1/2) - 1/2βC2^(-1/2) = 0
Simplifying the equation:
[(1+r)Y1 + Y2 - C2 - C2r/(1+r)]^(-1/2) = -βC2^(-1/2)
Squaring both sides:
[(1+r)Y1 + Y2 - C2 - C2r/(1+r)]^(-1) = β^2C2^(-1)
Taking the reciprocal of both sides:
[(1+r)Y1 + Y2 - C2 - C2r/(1+r)] = β^2C2
Simplifying further:
(1+r)Y1 + Y2 - C2 - C2r/(1+r) = β^2C2
Rearranging the equation:
(1+r)Y1 + Y2 = (1+β^2)C2 + C2 + C2r/(1+r)
Step 6: Express C2 in terms of Y1, Y2, and r
C2 = [(1+r)Y1 + Y2]/[(1+β^2) + (1+r)(1+β^2)]
Step 7: Substitute the expression for C2 back into the constraint equation to solve for C1
C1 = Y1 + Y2 - C2 - C2r/(1+r)
With the obtained expressions for C1 and C2, you have the optimal values for C1 and C2 in the given optimization problem.
♥️
