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Scott consumes coffee and tea. The local store sells coffee for [tex]$2(x^{1/2})[/tex] per serving. Tea is $1 regardless of the quantity purchased. Scott derives utility from the caffeine in each of the goods. The caffeine in one serving of coffee is three times greater than in one serving of tea.

A. Write down Scott's utility function [tex]u(x, y)[/tex] where [tex]x[/tex] represents coffee and [tex]y[/tex] represents tea.

B. Scott has a budget of $16. What is the optimal bundle? Draw a diagram of your solution, which must include the budget constraint, indifference curve, and the optimal point. Clearly label the intercepts.

C. Suppose the price of coffee is [tex]a(x^{1/2})[/tex] dollars per serving, the price of tea is [tex]b[/tex] dollars per serving, and Scott's budget is [tex]n[/tex] dollars. Derive Scott's demand functions [tex]x(a, b, n)[/tex] and [tex]y(a, b, n)[/tex].

Answer :

A. Scott's utility function u(x,y) is given by u(x, y) = 3x + y. Here, x denotes the quantity of coffee and y denotes the quantity of tea consumed by Scott.

B. The optimal bundle of goods can be obtained by solving the following optimization problem:

Maximise U(x,y) = 3x + y subject to the budget constraint 2(x^(1/2)) + y = 16

Solving the budget constraint for y, we get: y = 16 - 2(x^(1/2))

Substituting this expression for y in the utility function, we get:U(x) = 3x + 16 - 2(x^(1/2))

Now, we maximize U(x) using differentiation: dU/dx = 3 - x^(-1/2) = 0

Solving this for x, we get x = 9.
Hence, y = 16 - 2(9^(1/2)) = 4.

Solving the optimization problem gives us the optimal bundle of 9 servings of coffee and 4 servings of tea. This can be shown graphically using a diagram with the budget constraint, indifference curve and the optimal point as shown below:
The intercepts are: x = (16/2)^2 = 64 and y = 16.

C. Scott's demand functions x(a,b,n) and y(a,b,n) can be derived by solving the optimization problem with the budget constraint:

ax^(1/2) + by = n

Solving this for y, we get:

y = (n - ax^(1/2))/b

Substituting this expression for y in the utility function, we get:

U(x) = 3x + (n - ax^(1/2))/b

Maximizing this function using differentiation, we get: dU/dx = 3 - (a/2bx)^(1/2) = 0

Solving this for x, we get x = (a/2b)^2, and y = (n - ax^(1/2))/b.

Hence, Scott's demand functions are:

x(a,b,n) = (a/2b)^2y(a,b,n) = (n - ax^(1/2))/b


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