Question

Part 1:

Find the basket of goods and that will maximize your utility function $LaTeX: \sqrt{x}\sqrt{y}$ such that the total of and is 10.

Part 2:

Find the basket of goods and that will maximize your utility function $LaTeX: x^{\frac{2}{3}}y^{\frac{1}{3}}$ such that the total of and is 12.

Answer 1

Ans. Part 1 the basket of goods X and Y is ( 5, 5) units

U = squareroot x. squareroot y = x^1/2.y^1/2 (1) such that x + y = 10 (2) we use language to solve for x and y let F(x, y) = x^1/2.y^1/2 - lambda(x + y - 10) Diff w.r.t x, y and lambda, we have f_x = 1/2 x^-1/2.y^1/2 - lambda = 0 f_y = 1/2 x^1/2.y^-1/2 - lambda = 0 f_lambda = -(x + y - 10) = 0 1/2 x^-1/2.y^1/2 = lambda 1/2 x^1/2.y^-1/2 = lambda x + y = 10 (3) equate the following equation for lambda, we have 1/2x^-1/2.y^1/2 = 1/2x^1/2.y^-1/2 y = x \r\n the y = x in equation (3) y + y = 10 2y = 10 y = 5 units and x = 5 units Hence, the basket of goods x and y that will maximize utility function squareroot x. squareroot y is (5, 5) \r\n U = x^2/3.y^1/3 (1) such that: x + y = 12 (2) we use languages to solve for x and y

Ans. Part 2 the basket of goods X and Y is ( 8, 4) units