Question

Question 5 4 pts Your utility function over the goods X and Z takes the following form: You want to maximize your utility subject to your budget constraint. Assume that the price of X is $3 per unit and the price of Z is $6 per unit, and that the total income you have to spend on X and Z is $720 The consumption bundle that will maximize your utility subject to your budget constraint is X- 240 and Z-0 (enter only numbers in the blanks, and please round to the nearest integer if necessary). Hint: First solve for the marginal rate of substitution (MRS). Setting that equal to the negative price ratio will give you an optimal ratio of X and Z to consume, which you can use in combination with the budget constraint to determine the exact optimal quantities of X and Z to consume

Answer 1

The two goods are given to be good x and good z The price of good x (p_x) is $ 3 while the price of good z (P_z) is $ 6 income of the consumer is given as $ 720 (M) The bubget constraint is P_x X + P_z Z = M 3x + 6z x + 2z = 240 (1) Now, Marginal rate of substitution (MRS) between two goods is the ratio of the marginal utilities (M U) of the two goods MU is calculated by differentiating the utility function with respect to the goods utility function is given as u (x, z) = x^0.5 + z^0.5 Thus, MRS_x, z = MU_x/MU_z = 0/ x^-0.5/0/5 x^-0.5 = z^0.5/x^0.5 \r\n = squareroot z/squareroot x In equilibrium, MRS is equal to the price ratio MRS (MU_x/MU_z) = -P_X/P_z squareroot z/x = -3/6 squaring on both side z/x = 9-36 z/x = 1/4 x = 4z (2) Put the values os x from equation (2) in (1)