SAU: MBA 626: Managerial Economics Slope of Engel = 80 Problem Set #4: Utility & Engel -=5 16 1. Utility function: U(X, Y) = 2XY4 Budget Constraint: 2X + 4Y = 80 c. Calculate the slope of the Engel curve for Y. m = Slope of Engel(Y) a. Calculate the utility maximizing amount of X and Y. 80 = Slope of Engel (12) 80 20 Step 1: Divide the utility function by the coefficient out front (2). U(X, Y) = XY4 Slope of Engel = 12 3 - = 6.67 Step 2: Add the exponents on X and Y (1+4=5). Step 3: Raise the utility function to the power of the inverse of Step 2. [U (X, Y) = XY4]/s = x / sy*/5 Step 4: X* = am - 75(80) -= 8, Y* = (1-a)m _75(80) - = 16 4. X and Y are perfect substitutes. 2 X are worth 5 Y. PI 2 PY Budget Constraint: 2X + 4Y = 80 Or, Step 4: MRS = x , MRS = AX _ = Y = 2X a. Calculate the utility maximizing amount of X and Y. Step 5: Plug Y = 2X into 2X + 4Y = 80, 2X + 4(2X) = 80, 10X = 80, X = 8, Y = 2(8)=16 If MRS > Py *,then X* = - and Y* = 0 Px b. Calculate the slope of the Engel curve for X. If MRS *,then X* = 0 and Y* = m PY x* = am, som = "x X Since 2 X are worth 5 Y PX 80 Slope = 2 - = 10 Px a 1/5 MRS = = and = =, SO MRS > , so X* = = 40 and Y* = 0 b. Calculate the slope of the Engel curve for whichever good is consumed. 2. X and Y are perfect complements. 4 units of X are consumed with 3 units of Y. m = Slope of Engel (X) Budget Constraint: 2X + 4Y = 80. 80 = Slope of Engel(40) 80 Slope of Engel = 40 = 2 a. Calculate the utility maximizing amount of X and Y. Step 1: Since 4 units of X go with 3 units of Y, 3X = 4Y 1 Step 2: Solve for Y, so Y = _X 5. Utility Function: U(X. Y) = x /zyl/2. Budget Constraint: 4X + 2Y = 72 Step 3: Plug Y = =X into Budget Constraint, 2X + 4 (-x ) = 80 a. Calculate the utility maximizing amount of X and Y. Step 4: Solve for X and Y: 2X + 3X = 80, so X = 16, Y = = (16) = 12 b. Calculate the slope of the Engel curve for X. b. Calculate the slope of the Engel curve for X. m = Slope of Engel (X) 80 = Slope of Engel (12)