Answer to Question 3:

Show mathematically that in equilibrium the marginal rate of substitution between two goods in consumption must equal their relative prices.

Given a utility function of the form

U = U(X , Y , ......... )

the change in total utility as a result of small changes in the quantities of two commodities, X and Y, consumed will equal

dU = ∂U/∂X dX + ∂U/∂Y dY

Since the level of utility must be constant---that is dU = 0 ---along an indifference curve, the above equation can be rearranged to yield

0 = ∂U/∂X dX + ∂U/∂Y dY

which can be further rearranged as

dY/dX = − ∂U/∂X / ∂U/∂Y

Since

dY/dX = − P_x / P_y

is the slope of the budget line, we can combine the two equations immediately above to yield

P_x / P_y = ∂U/∂X / ∂U/∂Y .