SAU MBA-626: Managerial Economics a. Set MRS = , so Problem Set #3 With Hints: Utility & Optimal Choice Py ' 2X Py b. Solve for Y, so Y = - 2Px X Part I: Cobb-Douglas Utility Functions 1. Utility function: U(X, Y) = X /sy /5 c. Plug Y into budget constraint PX + PY = m, so PX + P P X|= m Budget Constraint: PX + P,Y = m d. Solve for X. so X Px + P, 28: = m a. Calculate MU, (Marginal Utility of X). m X = - 3m 2Px The Marginal Utility of X is the derivative of the Utility Function with respect to X. Px + Py 3Py To take the derivative of the utility function, use the "Power Rule" e. Solve for Y, so Y = 2Px am] 2m 3py 5P,J Power Rule: U(X, Y) = xayb, f. Demand Functions: X = m Y = _m au SP SPy Derivative = MU* = ax = axa-iyb Notice that these demand functions take the form of X = am, Y = , so we just need to Px SPy turn every Utility function into the form U(X, Y) = yay!-, then plug in the numbers for the In words... multiply the entire function by the exponent on X and decrease the exponent on X demand functions...Let's try it for number 2. by 1. For our example... 2. Utility function: U(X,Y) = 2XY* (turn this into U(X, Y) = 2XY4 = xay1-a) U(X, Y) = X /sy=/s, We can do this by dividing by 2 and raising U(X, Y) to the 1/5 power. Derivative = MUx = ax au_ x /s-1x3/s = =x-3/sy=/s or 0.6X-0.40.4 2 uan /5 = x/sy*/s, so a = and (1 -a) =4/5 Budget Constraint: 2X + 4Y = 80 (Note: Px = 2, Py = 4, and m = 80 b. Calculate MUy, (Marginal Utility of Y). a. Calculate the utility maximizing amount of X and Y. Now do the same for Y... We know = "m, Y = -a) so X = 's (80) 4/5(80) au -= 16 y = byayb-1 SPy - = 8, Y = Derivative = MUy = aY au Derivative = MUy = x3 /SY * /5-1 = =x3/sy-3/s or 0.410.60.6 Part II: Perfect Substitutes av 3. X and Y are perfect substitutes. 2 X are worth 5 Y. Budget Constraint: 2X + 4Y = 80 c. Calculate the MRS? a. Calculate the utility maximizing amount of X and Y. The MRS is the Marginal Utility of X divided by the Marginal Utility of Y For Perfect Substitutes, the MRS is the value of X relative to Y. So, if 2 X are worth 5 Y then MUx ex-2 /sy3/5 3x 3Y MRS = 5/2 MRS = MU y X /sY-3/3 = ZX-3/s-3/sy7/s--3/s = 5 x-ly' = 2 X Now compare MRS to If MRS > , the consumer only consumes X, so X = = = - = 40, and Y = 0 d. Derive the demand functions for X (in terms of Px and m) and Y (in terms of Py and m). If MRS