Question

Joyce's utility function is as follows: U= 10X2Y3 Where, X, is the quantity of good X consumed, Y, is the quantity of good Y consumed and, U, is Joyce's utility function. The general budget constraint for the two goods is a follow: B=PxX + PYY A. Derive Joyce's Marshallian demand equation for good X. Also compute her demand for good X when B= 500, and the price of good X is 1 and 2. Also draw the Marshallian demand curve for X at these prices. Please show all of your work to receive full credit. (10 points) B. What is Joyce's optimal amount of good Y purchased if Py= 1 and Pr=1 and if Pr=2 and Py=1? (10 points) C. Derive the Hicksian demand for good X at these prices. Hint, you need to choose the three correct equations you've derived above and solve simultaneously. Also, draw both demand curves on the same graph. Please show all of your work to receive full credit. (30 points) D. Using the information derived in parts A and B, what is the substitution effect and income effect obtained when changing the price of good x from a value of 1 to a value of 2. Please show all of your work to receive full credit. ( 10 points)

Answer 1

514 (B) Given Py=1, Px=1 and Py=1, Px=2 Foom egn (8), we have the Walrasian Demand function for chood y * = 33 case-I when (Px, Py) = (1, 1), we have yx - Y* = 38 = 36 ie Y* (1,1,B) = 36 When (Px, Py) = (2,1), we have y* _ Y* = 38 = 36 => y* (23 1B)= 3B Thus, Optimal Demand wove: Y* (PxP4, B) = ( 1) Y* (1919B) - 3 y* (2,1,B) = 38 Qale rs) ) Ang Page -(3) oot - Copes - .. .

Solution- on A Given Utility function: Budget line : U=10x²y3. B = Px. X + Py. Y (A) Marshallian demand equation for Good x Step-I Set up the utility Maximization Lagrangian function- We have max U(X,Y) subject to B = Px. X + Py. Y & L= 10x²y3 + x[B- Px. X- Py. Y] Differentiating '' with respect to x, y and respectively = 20xY3 xPx ....... ) a = 30x² 4² - xpy .. .. .. .. (2) d = B- Px.x - Py. Y .... ...... (3) at - Setting the derivates equal to zero . 20xy3 = x Px (from()) .....(4) 30x²12= xpy B = Px.x+Pyy ry ... ...! (from ② ...(5) Now calculating MRSxy by dividing (4) by (5) allax = 20xy3 – Alx allay → ollox - 24 - Px......CH away - 3x Ty on manipulating, px = 2.Yoky Putting the value of Px in eq"(6), we get - B= 2y.py .X + Py. Y 30x²y? 3x 3B=2Y. Py + 3 Y.Py 3 OST () mayo en 5 P

Substituting value of Y in ean (7) - 24 = P x L 3X = $1.24 = x= Py. 24 7 | ** = 28 l......(9) Jyoti's Marshalian demand equation for Good x is- ** (Px, Py, B) = 2B 5px Whem B = 500 Px=1, We have ** = 2(500) - 200 ie ** (1, P4,500) = 200 5 () When B=500, Px=2, we have * = 2(500) - 100 ie **(2,P4,500) 200 5(2) Ms. A **(Px, Py»B) = ? x*(1, P4,500) = 200 SPX X+ (2.py, 500) = 100 -- Waluasian Demand function for Good * 1x*(2,Pv,500) 7-- x* (1,py, Soo) it--- Quantity of 400 0 100 200 300 Page -(2)