Question

2. Given the following information; solve the consumer's problem by finding the optimal demand functions for X and Y: U(X,Y)=min(4X,Y) Also, you are given the the following initial market conditions: Px = $5 Py = $5 M = $100 a. Setup the Optimization Problem b. Find X* and Y* c. Graph the solution and explain the economic intuition behind the graph: i.e. What are the conditions met at the optimal bundle? Introduce another consumption bundle and explain why it is NOT the optimal. d. If income doubles, what will the new X* and Y* be?

Answer 1

22 Utility function is givmby; na u (X,Y)= min (4X, Y) Now, initially market condition ist Px = 45 Py = $5 $100 m = a tie it he budget constraint will ber М» Рx x + Руу =) 100 = 5 x + 5y Now, the horizontal inducept of the budget line! [ substitute Y=0 in the budget constraint ? 2) 100-5* :) X=20 Vertical in Ancept of the budget line! Tsubstitute X20 in the budget constaint] 100 = 5y - Y=20 Hince by Joining horizontal and untical in tucept, we can get the buelg et live AB. snow ability is Maimised at a point where --a) CamScanner 4X= Y

And by substituting 4X=Y in the budget construint we get? 5x + y = 100 4 2) 2) 5x +(5* 4x) = 100 25x = 100 -) X w Huice, &* = 4 and y* = 16 e tuce the 'L Shaped in diftsance Care will be tougent to the budget line AB at the point of kink ic at Poist m (3,12) mi IC Scanned with CamScanner

The general form of ability function for pnfect (one plement goods are Uz min (an, by) = min (4X,Y) here, a=4 and b= 1 hence here good X and Y one consmed in t! 1 or 1:4 Proportion. We consider Anod suppost x =3 and y = 12 hence 114 proportion is maitained. At this bulles slope of Ic or MRS be is not equal to slope of budget line. atht and for brudle (3,12) utility leve min (4*3, 12) = min (12, 12] 12 and for bundle .(4,167 utility level is 1. min (4*4,16) = min (16,16) = 16 #ate, Also, when the slope of 'Ic boscos is I not equal to the slope of budget line then the rate at which goods me traded is market (MRS) is not equal to the rate at which the goods are traded in the market. Hace utility is not optimum. cs Scanned with CamScanner

M = 4 , VN of income levce doubles then. . M=$200 and new budg it constraint ist : 200 = 5x +sy Now Utility manimising condition is 4X=Y x=48 in the budget constant wc quta Subs 200 = (5 * 4x) +sx =) 200= 25 x aj X=8 and as, y = 4* =) 8*4 = 32 *=8 and y*= 32 tace now, new x Scanned with CamScanner