Microconomic Theory - Utility and Preference
In a one-person-2-item-one budget world, a consumer has a an indifference curve, U, where U= XY = 24, X is quantity of good X and Y is quantity of good Y. (X and Y are positive integers)
The consumer's total income, I, is $24.
The price of good X, Px= $3
The price of good Y, Py= $2
Question: At the consumer's optimum point, what is the marginal rate of substitution between good X and good Y or MRS xy ? ( The MRS xy is just a fancy name for the slope of the indifference curve) :-)
There are several ways to approach this: Graphically, algebraically, or using the Lagrange multiplier.
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1 solution
Lagrangian multiplier method: Generalized the entire situation. We want to maximize utility subject to a budget/income constraint. Therefore,
L= max U =f(X,Y) subject to I= f( Px,Py,X,Y) L= XY + lambda (24-3X-2Y) where Px=3 and Py=2 and income,I=24 dL/dx= y-lambda3=0 dL/dy=x-lambda2=0 dL/dlambda=24-3x-2y=0
Now, solve these 3 simultaneous equations y-3lambda=0 eq 1 x-2lambda=0 eq 2 24-3x-2y=0 eq 3 Optimum bundle is y=6, x=4 Next, find MRSxy MRSxy=MUx/MUy where MU marginal utility of each respective good. Given U = XY MUx= dU/dx=Y MUy=dU/dy=X Therefore, MRSxy=Y/X At point ( 4,6), MRSxy= 6/4=3/2=1.5