Consider the following consumer maximization problem. maximize u(x, y) subject t

Question

Consider the following consumer maximization problem. maximize u(x, y) subject to p_xx + p_yy lessthanorequalto I:x.y Greaterthanorequalto 0. where p_x and p_y are prices of commodities i and y and I is income(budget). We assume u(x, y) = x^2y and let p_x = 2. py = 1 and I = 10. Solve the problem of maximizing u(x. y) subject to 2x + y lessthanorequalto 10. What is the optimal values of x and y? (There are non-negativity constraints x, y Greaterthanorequalto 0 but we can ¡gore these so long as all points satisfying the necessary condition for maximum are positive.) Let (x, y) be the solution to problem b. Letting u = u(x, y):solve the following minimization problem (observe' that minimizing a function f is equivalent to maximizing -f): min 2x + y subject to u(x, y) Greaterthanorequalto u.

Explanation / Answer

a.

Optimal bundle and values:

Let, X = x and Y = y

Equation of the budget line is as below:

2X + Y = 10 …….. (1)

The utility function is as below:

U = X^2Y

Slop of budget line = - (2/1) = - 2

Slope of indifference curve = - (MUX/MUY) = - (2 / X)

As per the condition, - (2 / X) = - 2

                                    X = 1 ………..(2)

Equation 1 and 2 should be solved as below:

2 + Y = 10

Y = 8

Therefore, the optimal bundle would be x = 1 and y = 8

Answer: The optimal value of x = $2 × 1 = $2 and the optimal value of y = $1 × 8 = $8.

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