Question 1 (20 Marks) Joey likes beer and whisky and spends...

Question 1 (20 Marks) Joey likes beer and whisky and spends £75 in total on drink. His utility function for whisky (w) and beer (b) is U = 5w + b. He can buy beer at £2 a can but the liquor store tries to discourage overconsumption of spirits by charging higher prices for larger quantities: the cost in dollars of w bottles of whisky works out at w^2. a. How many cans of beer and how many bottles of whisky does Joey consume? [10 Marks] b. Illustrate your answer with a clearly labelled diagram showing Joey's budget constraint and his indifference curve passing through the optimal consumption bundle. [5 Marks] c. Joey's friend Ross shops at the same liquor store and spends the same amount on drink. Jim, however, insists on drinking beer and whisky together. His utility function is U = min{5w, b} How much beer and whisky does Jim buy? [5 Marks]

Transcript

00:01 In this question, we are given a scenario where it is said that joey likes beer and whiskey and spends around $75 on the purchase of both the commodities.

00:14 His utility function, u, is given as 5 whiskey and 1 beer.

00:23 Let's suppose if we can buy a beer at a cost of $2, but a liquor store discourages overconsumption of spirits by charging a higher price for larger quantities.

00:38 Now, the cost in dollars of whiskey or bottle is also available in the form of function.

00:45 So, the price of whiskey is available as w square.

00:50 So, the question states, how many cans of beer and whiskey will joey consume? we can also illustrate this given equation in the form of indifference curve analysis.

01:01 Next, in the second part, it states if there is another person who is a friend of joey, they shop in the same liquor store and spend the same amount of drinks.

01:15 Third, it states if a gym, however, insists on drinking beer and whiskey together and his utility function is available to us, which is around utility function equals minimum 5 w and 1 b.

01:34 So, how much will be the beer and whiskey does jim buy with his given utility function? we have to find that out.

01:43 So, this is our given information.

01:51 Here in this situation, we are available with joe's utility function and with the given information regarding its expenditure on both the beer and whiskey, we can calculate, we can assign his budget equation as 2b plus w square equals 75.

02:19 So, in order to find the optimal consumption bundle, we need to maximize his utility function joey's utility function subject to the constraint, this constraint, which is 2b plus w square equals 75 so that we can find the optimum consumption bundle for beer for joey.

02:51 Using the budget constraint equation, we can express b in terms of w.

02:56 So, b will be equal to 75 minus w square upon 2.

03:02 Now, substituting the value of b in joey's utility function, which is this.

03:08 So, u will be 5w plus 75 minus w square upon 2.

03:16 Therefore, to maximize the u utility function, we can take the derivative with respect to w and set it equal to 0.

03:25 Therefore, the derivative, the first difference will be equals 5 minus w upon 2, which is equals to 0.

03:38 Therefore, 5 will be equal to w upon 2 or we can say w is equals to 10.

03:49 Now, we can find the corresponding value of b using the budget constraint equation.

03:58 Let's put it as star.

04:01 So, putting the value of w, the star equation, we can find the value of the b.

04:12 So, b will be 75 minus 10 square upon 2, which is equal to 25.

04:22 So, joey consumes 10 cans of beer and 25 bottles of whiskey given his total spending of $75.

04:44 Alright, let's move on to the next part of the question, which is to illustrate the joey's optimal consumption model, we can draw a diagram with beer on the x -axis, beer and whiskey on the y -axis.

05:16 The budget constraint is 2b plus w square equals 75.

05:25 The indifference curve represents the joey utility function of u equals 5w plus b...

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