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suneja
utility fn: U=min{4x,2x+y} X=chocolates, Y=ice cream. Given consumption; 15chocoltes and 10 ice creams. price of per unit choco is 10 and price of ice cream=? wats her total pocket money?
20,40 I GUESS BUT I M NOT SURE.
can u explain how did u get dat
since solving utility function we get 2x=y, that means the given consumption is not an optimal consumption. thus you cannot use optimization in this problem. thus you cannot solve it.
U = min(4x, 2x+y) Your example: x = 15, y = 10 Then, U(15,10) = 40 Question: Whats py = ? in order to satisfy U(x,y) = 40 Solution: We assume that we can find py because we assume that the person is maximizing his utility for given prices. First, you must know that px.x + py.y = budget. You want to know what is the price of the good y that allows the person to buy many combinations of goods with the same utility U(x,y)=40 Let's think about the conditions under x and y that makes U(x,y)=40: 1)There are three cases were Utility is never 40: If x < 10, U < 40 if x > 20, U > 40 if x = 10, y<20, U < 40 2)If x = 10, y>20, U(10, y) = 40 always. We cant use this case to find py. There is no restriction under y that we can use to find py. 3)If 10 <= x <= 20, then we must have 2x + y = 40 in order to make U (x, y) = 40 In this case, 0 <= y <= 20, and we now can make assumptions under py. Lets solve for this case: We want to max: px.x + py.y subject to some conditions: x=>10 x<=20 2x+y=40 So, we have (for px =10): L = 10.x + py.y -l1(2x + y - 40) deriving: dL/dx = 0: 10 - l1.2 = 0 -> l1 = 5 dL/dy = 0: py - l1 = 0 -> py = l1 -> py = 5 Biding condition dL/dl1 = 0: 2x+y-40=0 We find that py = 5 is a condition to make U(x,y)=40, if 10 <= x <= 20, 0 <= y <= 20. Now, we'll have the following budget, price and quantities in your example: x=15 y=10 px=10 py=5 py.y + px.x = 5.10 + 10.15 = 200 You can now test and see if py=5 is ok for the budget 200: Ex1) All these cases U(x,y)=40 For 10 <= x <= 20 and 0 <= y <= 20 x=20 y=0 -> px.x + py.y = 200 x=19 y = 2 -> 10.19 + 5.2 = 200 x=18 y = 4 -> 10.18 + 5.4 = 200 .. x=11 y = 18 -> 10.11 + 5.18 = 110 + 90 = 200 x=10 y = 20 -> 10.10 + 5.20 = 200 Ex2) For x>20 or x<10, or x=10 and y>20, U is different from 40. Ex4) Now, suppose the price was different from py=5, Think that py=20 Then, your budget, for px=10, x=15 and py=20, y=10 would be: budget: 10.15 + 20.10 = 350 But, if you had 350, you could buy only x goods: x=35 y=0 But in this case, your utility would be U(35,0)= min(140, 70) = 70 And you would be in a better position… In the other hand, if py = 5, then your budget is 200, 10.x + 5.y = 200 and if you do the same thing, only buys x: 10.20 + 5.0 = 200 -> U=40 See that from the conditions we have described earlier, the only way this consumer could get more utility would be if he get x>20. But since px=10, and the budget is 200, it is impossible. He can only get x=20. Note that all the cases where py is not 5, you can reorganizer your consumption in order to make more utility than U=40, or maybe you can never get the same utility U=40. But if you could do that, then you would not being rational at first place.. And note that we assumed that the person was maximizing his utility for given prices.