Given U = 18q + 7q2 − (1/3)q3, find the utility-maximizing quantity and the point at which the Law of Diminishing Marginal Utility sets in.

$$MU = \frac{dU}{dq} = \frac{d}{dq}\left(18q + 7q^2 - \frac{1}{3}q^3\right)$$ $$MU = 18 + 14q - q^2$$

Set the MU function to zero and solve for $q$.

$$18 + 14q - q^2 = 0$$

Rearrange the quadratic equation:

$$q^2 - 14q - 18 = 0$$

$$q = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(-18)}}{2(1)}$$ $$q = \frac{14 \pm \sqrt{196 + 72}}{2}$$ $$q = \frac{14 \pm \sqrt{268}}{2}$$ $$q = \frac{14 \pm 16.3707}{2}$$

We consider only the positive value for quantity:

$$q_1 = \frac{14 + 16.3707}{2} = \frac{30.3707}{2} \approx 15.185$$ $$q_2 = \frac{14 - 16.3707}{2} = \frac{-2.3707}{2} \approx -1.185 \quad (\text{not economically relevant})$$

The utility maximizing quantity is approximately 15.19 units.

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.

$$MU = 18 + 14q - q^2$$ $$\frac{dMU}{dq} = \frac{d}{dq}(18 + 14q - q^2)$$ $$\frac{dMU}{dq} = 14 - 2q$$

$$14 - 2q = 0$$ $$2q = 14$$ $$q = 7$$

The Law of Diminishing Marginal Utility sets in at 7 units of consumption.

The utility maximizing quantity is $\boxed{\text{15.19 units}}$.
The Law of Diminishing Marginal Utility sets in at $\boxed{\text{7 units}}$.