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Cali consumes coffee and tea. The local bulk store has s special pricing policy for coffee: if you buy x servings of coffee, you need to pay a price of 2 x

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dollars per serving. The price of tea is always $1 regardless of the quantities purchased. Cali derives utility from the caffeine content of the two goods. In particular, caffeine in one serving of coffee is three times as large as that in one serving (a) Let x denote her consumption of coffee and y denote her consumption of tea. Write down one example of Cali's utility function u(x,y). (5 Marks) (b) Suppose Cali has a budget of $16. What is her optimal bundle? Draw a diagram of your solution, which should include her budget constraint, indifference curve and the optimal point. Clearly label all the intercepts. (10 Marks) (c) Suppose the price of coffee is a x

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dollars per serving, the price of tea is b dollars per serving, and Cali's budget is m dollars. Derive Cali's demand functions x(a,b,m) and y(a,b,m). (5 Marks)

Answer :

Cali's utility function can be expressed as follows: u(x, y) = 3x + y.Cali's budget is $16, and the cost of tea is $1 per serving.

As a result, Cali can buy at most $16/$1 = 16 servings of tea.If Cali buys x servings of coffee, she will pay 2x dollars per serving, for a total cost of 2x * x = 2x2 dollars. As a result, Cali's budget constraint can be expressed as follows: 2x2 + y = 16.

A diagram of the budget constraint, indifference curve, and optimal bundle is as follows:Cali's optimal bundle can be determined by determining the point where the budget constraint intersects the highest attainable indifference curve.

As shown in the diagram above, this occurs at point B, where Cali buys 4 servings of coffee and 8 servings of tea.The demand functions for x and y can be derived as follows:

First, solve the budget constraint for y: 2x2 + y = 16y = 16 - 2x2Substitute y into the utility function to get:u(x) = 3x + (16 - 2x2).

The optimal quantity of x can be determined by taking the derivative of the utility function and setting it equal to zero:u'(x) = 3 - 4x = 0x = 0.75.

The optimal quantity of x is 0.75 servings of coffee. The optimal quantity of y can be found by plugging x into the budget constraint:y = 16 - 2(0.75)2y = 15.25.

Therefore, Cali's demand functions are as follows:

x(a, b, m) = (3a(m - b2)) / (8m)b(a, b, m) = (m - a(b2)) / (2m)

Cali's utility function is u(x, y) = 3x + y. Her budget constraint is 2x2 + y = 16, and her optimal bundle is 4 servings of coffee and 8 servings of tea. Cali's demand functions are x(a, b, m) = (3a(m - b2)) / (8m) and b(a, b, m) = (m - a(b2)) / (2m).

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