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Answer :
In this case, let's assume that the utility function is u(x, y) = 3x + y, where x represents the consumption of coffee and y represents the consumption of tea.
To solve the problem, let's proceed with the given information and derive Cam's demand functions for coffee and tea.
(a) The utility function is u(x, y) = 3x + y, where x represents the consumption of coffee and y represents the consumption of tea.
(b) With a budget of $16, we can set up the budget constraint:
2√x + y = 16
To find the optimal bundle, we need to maximize Cam's utility function subject to the budget constraint. We can use the Lagrange multiplier method to solve this constrained optimization problem.
Let's define the Lagrangian function:
L(x, y, λ) = 3x + y - λ(2√x + y - 16)
To find the optimal point, we take the partial derivatives and set them equal to zero:
∂L/∂x = 3 - λ/√x = 0
∂L/∂y = 1 - λ = 0
2√x + y = 16
From the second equation, λ = 1. Substituting this value into the first equation, we have:
3 - 1/√x = 0
1/√x = 3
√x = 1/3
x = 1/9
Substituting x = 1/9 into the budget constraint, we can solve for y:
2√(1/9) + y = 16
2/3 + y = 16
y = 16 - 2/3
y = 46/3
Therefore, the optimal bundle is x = 1/9 servings of coffee and y = 46/3 servings of tea.
(c) To derive Cam's demand functions x(a, b, m) and y(a, b, m), where a represents the price of coffee per serving, b represents the price of tea per serving, and m represents the budget, we need to generalize the equations.
The budget constraint becomes:
a√x + by = m
Using the Lagrange multiplier method as before, we set up the Lagrangian function:
L(x, y, λ) = 3x + y - λ(a√x + by - m)
Taking the partial derivatives and solving, we can find the values of x and y that maximize the utility function under the budget constraint, resulting in Cam's demand functions for coffee and tea.
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