Final answer:
The marginal rate of substitution (MRS) measures the rate at which a consumer is willing to trade one good for another while keeping the same level of utility. The MRS for x with y in the provided utility function is equal to 1 divided by twice the value of x.
Step-by-step explanation:
The marginal rate of substitution (MRS) measures the rate at which a consumer is willing to trade one good for another while keeping the same level of utility. In this case, the MRS for x with y can be calculated by differentiating the utility function with respect to x and y and taking the ratio of the two derivatives. The utility function provided is u = 3x^2y^4.
Taking the partial derivatives with respect to x and y, we get du/dx = 6xy^4 and du/dy = 12x^2y^3. The MRS is then given by the ratio of these derivatives: MRS = (du/dx) / (du/dy) = (6xy^4) / (12x^2y^3) = 1 / 2x.
Therefore, the marginal rate of substitution for x with y is equal to 1 divided by twice the value of x.