Question

. Given that the slope of an indifference curve at any point is the “marginal rate of substitution” between Y and X explain/prove that MRS is equal to the ratio of the consumer’s marginal evaluation of good X to his/her marginal evaluation of good Y (i.e. MRS = -MUx/MUy). (2 pts)

Answer 1

Consider the given problem here “MRS” is the rate at which consumer will reduce the consumption of one good in order to increase the consumption of another good while maintaining same level of utility, mathematically it’s a ratio of “MU” of goods.

So, let’s assume that the utility function is, U=U(X,Y), where “X” and “Y” both are good consumed by the individual. Now to derive the “MRS” here we need to differentiate the utility function totally.

=> dU = (δU/δX)*dX + (δU/δY)*dY, now because the consumer will maintain the same level of utility, => dU=0.

=> (δU/δX)*dX + (δU/δY)*dY = 0, => MUx*dX + MUy*dY = 0, => MUy*dY = (-1)*MUx*dX.

=> “dY/dX = (-1)*MUx/MUy”. So, the slope of the utility function or the indifference curve is “dY/dX = (-1)*MUx/MUy”, which the MRS also.

SO, the MRS is given by, "(-1)*MUx/MUy".