Question

Solve the optimization problem for a Cobb-Douglas utility function with three goods: U(x1, x2, x3) = x c 1x d 2x f 3 , where c, d, f > 0.

Answer 1

So, here the maximization problem is given by.

=> Max, “U(X1, X2, X3) = X1^c* X2^d* X3^f”, subject to “P1*X1 + P2*X2 + P3*X3 = M”. So, the lagrangian function is given by.

=> L = X1^c*X2^d*X3^f + c[M - P1*X1 - P2*X2 - P3*X3], => the FOC for maximization are given by.

=> dL/dX1 = dL/dX2 = dL/dX3 = dL/dc = 0, where “d” is the “delta” represent the partial differentiation.

=> dL/dX1 = 0, => c*X1^c-1*X2^d*X3^f + c*(- P1) = 0, => c*X1^c-1*X2^d*X3^f = c*P1…..(1).

=> dL/dX2 = 0, => d*X2^d-1*X1^c*X3^f = c*P2………………(2).

=> dL/dX3 = 0, => f*X3^f-1*X1^c*X2^d = c*P3………………(3).

Now, by “1” divided by “2” we have.

=> c*X1^c-1*X2^d*X3^f / d*X2^d-1*X1^c*X3^f = c*P1/c*P2.

=> (c/d)*(X2 / X1) = P1/P2, => X2 = (d*P1/c*P2)*X1…………(4).

Now, by “1” divided by “3” we have.

=> X3 = (f*P1/c*P3)*X1…………(5).

The budget line is given by.

=> P1*X1 + P2*X2 + P3*X3 = M, => P1*X1 + (d*P1/c)*X1 + (f*P1/c)*X1 = M,

=> [1 + (d/c) + (f/c)]*P1*X1 = M, = > X1 = (c/c+d+f)*(M/P1), be the optimum choice of “X1”. Now, if we put the value of “X1” inn “4” and “5” we will get the optimum solution for “X2” and “X3” also.

=> “X2 = (d/c+d+f)*(M/P2)” and “X3 = (f/c+d+f)*(M/P3)”.