This economics question tests your understanding of economic models and analysis. The step-by-step answer below applies the relevant framework and explains the reasoning.
ScanSolve AI Answer
Another one Abona — let's solve it. The given utility function is: U = 18q + 7q^2 - (1)/(3)q^3 i. The utility maximizing quantity Step 1: To find the utility-maximizing quantity, we first need to derive the Marginal Utility (MU) function by taking the first derivative of the Total Utility (U) function with respect to quantity (q). MU = (dU)/(dq) = (d)/(dq)(18q + 7q^2 - (1)/(3)q^3) MU = 18 + 14q - q^2 Step 2: Utility is maximized when Marginal Utility (MU) is equal to zero. Set the MU function to zero and solve for q. 18 + 14q - q^2 = 0 Rearrange the quadratic equation: q^2 - 14q - 18 = 0 Step 3: Use the quadratic formula q = -b ± sqrt(b^2 - 4ac)2a to solve for q, where a=1, b=-14, and c=-18. q = -(-14) ± sqrt((-14)^2 - 4(1)(-18))2(1) q = 14 ± sqrt(196 + 72)2 q = 14 ± sqrt(268)2 q = (14 ± 16.3707)/(2) We consider only the positive value for quantity: q_1 = (14 + 16.3707)/(2) = (30.3707)/(2) ≈ 15.185 q_2 = (14 - 16.3707)/(2) = (-2.3707)/(2) ≈ -1.185 (not economically relevant) The utility maximizing quantity is approximately 15.19 units. ii. At what unit of consumption does the Law of Diminishing Marginal Utility (DMU) set in Step 4: The Law of Diminishing Marginal Utility sets in when Marginal Utility (MU) starts to decrease. This occurs at the point where the MU function reaches its maximum. To find this point, we take the derivative of the MU function with respect to q and set it to zero. MU = 18 + 14q - q^2 (dMU)/(dq) = (d)/(dq)(18 + 14q - q^2) (dMU)/(dq) = 14 - 2q Step 5: Set (dMU)/(dq) = 0 to find the quantity where MU is maximized (and thus DMU begins). 14 - 2q = 0 2q = 14 q = 7 The Law of Diminishing Marginal Utility sets in at 7 units of consumption. The utility maximizing quantity is 15.19 units. The Law of Diminishing Marginal Utility sets in at 7 units. Send me the next one 📸