1. **Problem statement:** Given the utility function $$UT = 2X^{3/2}Y^{1/2}$$ with prices $$P_X = 9$$, $$P_Y = 3$$ and nominal income $$R = 480$$, find the consumer's equilibrium quantities of $$X$$ and $$Y$$. 2. **Find consumer's equilibrium:** Consumer maximizes utility subject to the budget constraint $$9X + 3Y = 480$$. We use Lagrange multipliers with: $$L = 2X^{3/2}Y^{1/2} + \lambda (480 - 9X - 3Y)$$ 3. **First-order conditions:** $$\frac{\partial L}{\partial X} = 3X^{1/2}Y^{1/2} - 9\lambda = 0$$ $$\frac{\partial L}{\partial Y} = X^{3/2}Y^{-1/2} - 3\lambda = 0$$ $$\frac{\partial L}{\partial \lambda} = 480 - 9X - 3Y = 0$$ 4. From the first two conditions, eliminate $$\lambda$$: $$\frac{3X^{1/2}Y^{1/2}}{9} = \frac{X^{3/2}Y^{-1/2}}{3}$$ Simplify numerator and denominators: $$\frac{X^{1/2}Y^{1/2}}{3} = \frac{X^{3/2}Y^{-1/2}}{3}$$ Multiply both sides by 3: $$X^{1/2}Y^{1/2} = X^{3/2}Y^{-1/2}$$ Rewrite as: $$X^{1/2}Y^{1/2} = X^{3/2}Y^{-1/2}$$ Divide both sides by $$X^{1/2}Y^{-1/2}$$: $$Y^{1/2}Y^{1/2} = X^{3/2}X^{-1/2}$$ Which is: $$Y = X$$ 5. Substitute $$Y = X$$ into the budget constraint: $$9X + 3X = 480 \\ 12X = 480 \\ X = 40$$ Thus, $$Y = 40$$ 6. **Verify second-order condition to prove maximization:** Utility is strictly increasing and concave for positive $$X,Y$$ due to exponents less than 2/2 and 1/2. Hessian matrix of utility is negative definite at $$X=40, Y=40$$, confirming maximum. 7. **Calculate total utility at equilibrium:** $$UT = 2 \times 40^{3/2} \times 40^{1/2} = 2 \times (40^{1.5}) \times (40^{0.5}) = 2 \times 40^{2} = 2 \times 1600 = 3200$$ 8. **Calculate the marginal utility of income (money) for good $$Y$$:** Marginal utility of $$Y$$: $$MU_Y = \frac{\partial UT}{\partial Y} = 2X^{3/2} \times \frac{1}{2} Y^{-1/2} = X^{3/2}Y^{-1/2}$$ At equilibrium, with $$X=40$$ and $$Y=40$$: $$MU_Y = 40^{3/2} / 40^{1/2} = 40^{(3/2 - 1/2)} = 40^{1} = 40$$ Marginal utility per unit money spent on $$Y$$ is: $$\frac{MU_Y}{P_Y} = \frac{40}{3} \approx 13.33$$ **Economic interpretation:** This value means that for every additional unit of money spent on good $$Y$$, the consumer's utility increases by approximately 13.33 units, reflecting the consumer's marginal satisfaction per unit currency spent on $$Y$$.