Question

3. Given the following information; solve the consumer's problem by finding the optimal demand functions for X and Y: U(X,Y)= 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

UCKY) = 4x + y lo Be = $5 Py = $5 and im doo $100, momuzako problem Budget constraint BxX + By Y e Mish -) 75x+54 3 100 1 mant monnization of lithty u subject to the Constraint imposed by the budget and prices. That is max. (4x+4) sit 5x1542100 / Here, both goods x I unit of x i and perfectly y are substitutes.. Substituted by 4 units That is I uit of x gives i dtimes the utility that I unit of ý gives. mux= 2 = 4 c muy = 20 = 1 and you x if consumer Consumes only x it mx > m mux > muy Oy and only good y if my mx . Consumer Consumes any or Y if amount of x mix = muy . . . .

MUх Camlin Page Date 1 и DX nu 5 moy st ndo D S consumes consumes He spends whole Xoco 80, X* = M only x income (m) and no units of y. on x . = 0. x*- $100 = 20 $5 - Tx* = 20 and Ty*=0 seront ut 80 och COCO) Cyx) TacIndifference - cuore B2 Rudged one on ot Jilter altrud uit t e E optimal bundle. that The indifference cuine drawn in the graph, CIC) conespording to utility = 80. This is the highest et Inditt Indifference cove (highest utility can be acherred with given income and prices at happens when x*=20 and y*=o. therefore, (xx, y) = (20,0) is the optimal bundle. d yx=0.

(x,y) = (0,20). let's consides another bundle It les on the budget line BL. At this bundle, utility =20 200 But at the same mome and prices, the consumer can attain higher utility by consuming more X and ley y. let's compare soith optimal bundle: U (2010) = 80 and U60120)=20 At the same income, higher ability can be attained hu the consumer. That is why (x,y) = (0120) is not the in ne optimal bundle.

D Income doubles But prices and M = 2000 utility function are unchanged pon xun - P = $5 , Py = $5. mby = 1 2 MQ4 1802 onx. So, still the consumer Spends total income xt $200 - 40 ; y* - 0 2 1x*=40 and y*=0