Question

Suppose a consumer’s Utility Function U(x,y) = X^1/2Y^1/2. The consumer wants to choose the bundle (x*, y*) that would maximize utility.

Suppose Px = $5 and Py = $10 and the consumer has $500 to spend.

1. Write the consumer’s budget constraint. Use the budget constraint to write Y in terms of X.
2. Substitute Y from above into the utility function U(x,y) = X^1/2Y^1/2.
3. To solve for the utility maximizing, taking the derivative of U from (b) with respect to X. Set that equal to zero, and solve for X. Then solve for Y.
4. Is the second order condition for maximization satisfied by your solution in (c) above.

Answer 1

Utility function ;

U(x,y) = x^1/2 y^1/2

Price of x, P_x = $5
Price of y, P_y = $10

Income, M = $500

a) The budget constraint will be;

P_x x + P_y y = M
5x + 10y = 500
x + 2y = 100

y in terms of x;

2y = 100-x
y = 50 - x/2

b) Substituting , y = 50 - x/2 in the utility function;

U(x,y) = x^1/2 y^1/2
= x^1/2 (50 - x/2)^1/2
= 7.07 x^1/2 - x/1.41
U = (9.97x^1/2 - x) / 1.41

c) Taking derivative of U and keeping it 0,

U' = d/dx ((9.97x^1/2 - x) / 1.41)
= (9.97/1.41*2x^1/2 ) - 1
U' = 3.53x^1/2 - 1

Keeping it equal to 0;

U' = 0
3.53x^1/2 - 1 = 0
3.53x^1/2 = 1
x^1/2 = 1/3.53
x^1/2 = 0.28
x = 0.0784

Putting it into equation;

y = 50 - x/2

y = 50 - 0.0784/2
y = 49.96

d) The second order condition of, U' = 3.53x^1/2 - 1;

U" = d/d²x (x^1/2 y^1/2 )
= d/dx (3.53x^1/2 - 1)
= 3.53/2x^1/2 - 0
U" = 1.765x^1/2 > 0

Therefore, the utility function is not maximum.