The MRS and the Cobb-Douglas

1 Utility Maximization StepsMPP 801 Fall, 2007 The MRS and the cobb -DouglasConsider a two-good world,xandy. Our consumer, Skippy, wishes to maximize utility, denotedU(x,y).Her problem is then to Maximize:U=U(x,y)subject to the constraintB=pxx+pyyUnless there is aCorner Solution, the solution will occur where the highest indifference curve is tangent tothe budget constraint. Equivalent to that is the statement:The Marginal Rate of Substitution equals theprice ratio,orMRS=pxpyThis rule, combined with the budget constraint, give us a two-step procedure forfinding the solution to theutility maximization , in order to solve the problem, we need more information about theMRS. As it turns out, everyutility function has its ownMRS, which can easily be found using calculus. However, if we restrict ourselvesto some of the more common utility functions,we can adopt some short-cuts to arrive at example, if the utility function isU=xythenMRS=yxThis is a special case of the " Cobb-Douglas " utility function, which has the form:U=xaybwhereaandbare two constants.

2 In this case the marginal rate of substitution for the Cobb-Douglas utilityfunction isMRS= ab yx regardless of the values the utility max problemConsider our earlier example of "Skippy" whereU=xyMRS=yxSuppose Skippy s budget information is as follows:B= 100,px=1,py=1. Her budget constraint isB=pxx+pyy100 =x+y1 Step 1 Set MRS equal to price ratioMRS=pxpyyx=11y=xthis relationship must hold at the utility maximizing 2 Substitute step 1 into budget constraintSincey=x, the budget constraint becomes100 =x+y=x+x=2xSolving for x yieldsx=1002=50 Thereforey=50andu= (50)(50) = 2500 Change the price of xNow suppose the price ofxfalls ,MRS=pxpyyx= this new relationship into the budget constraint100 =x+y100 =x+12x100 = Solution to Cobb-Douglas UtilityUsing the general form of the cobb -DouglasU=xaybwhereMRS=aybxand the budget constraint in the formB=pxx+pyy2where the price ratio ispx/py,thefirst rule of utility maximization yieldsMRS=pxpyaybx=pxpyy=bapxpyxSubstitu ting into the budget constraint yieldsB=pxx+py bapxpyx B=pxx+bapxxB= a+ba pxx(seefootnoteforalgebra)

3 X = aa+b BpxSimilarly, we canfindyby the same method, which gives usy = ba+b BpyThe solutions forxandyare called the consumer s DEMAND that in ourfirst example whereU=xy,thevaluesofaandbarea=b=1substi tuting intox andy we getx = 11+1 Bpxx =B2pxandy =B2pyUse the values ofpx,py,andBto test to see if these equations give you the solutions in example we substitute the answers backinto the utility function, we getU=xy= B2px B2py U=B24pxpyThis gives you the utility number directly from the budget and prices. If you re-arrange this expressionto getBby itself, you getB=p4pxpyUYou can use this equation to calculate the amount of budget is needed if you know prices AND the desiredutility number (Helpful for CV and EV)0 The trick used here is as follows:x+bax=aax+bax= aa+ba x=a+bax3