1. Chipo has the following utility function of 2 goods Pies (X) and fanta (Y):
U= log X + log Y.
(a) show that the consumer maximizes utility subject to the budget constraint.
(b) derive the demand functions of good X and good Y.
2. You are given the individual demand curves for food for 2 households. These are household A and household B. The demand for household A for food is given by equation:
Qa= 1430 - 55p
The demand for food for household B is given by the equation:
Qb= 1470 - 70p
Where Qa= demand for food in kg by household A
Qb= demand for food in kg by household B
Calculate the market demand for food
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1. (a) To show that the consumer maximizes utility subject to the budget constraint, we need to find the optimal combination of goods that maximizes utility while considering the budget constraint.
The budget constraint is given by:
Px * X + Py * Y = I
where Px is the price of good X, Py is the price of good Y, and I is the consumer's income.
The utility function is given by:
U = log X + log Y
To maximize utility, we need to find the values of X and Y that satisfy the budget constraint and maximize the utility function.
We can use the method of Lagrange multipliers to solve this problem. The Lagrangian function is defined as:
L = log X + log Y - λ(Px * X + Py * Y - I)
Taking the partial derivative of L with respect to X, Y, and λ, we get:
∂L/∂X = 1/X - λPx
∂L/∂Y = 1/Y - λPy
∂L/∂λ = Px * X + Py * Y - I
Setting the partial derivatives equal to zero and solving the resulting system of equations will give us the optimal values of X, Y, and λ.
(b) To derive the demand functions of goods X and Y, we need to solve for X and Y in terms of Px, Py, and I.
Using the solution from part (a), we find that the demand function for good X is:
X = (λPx)^(-1)
And the demand function for good Y is:
Y = (λPy)^(-1)
2. To calculate the market demand for food, we need to add up the individual demands for food by each household.
The individual demand for food for household A is given by the equation:
Qa = 1430 - 55p
The individual demand for food for household B is given by the equation:
Qb = 1470 - 70p
To calculate the market demand, we add up the individual demands:
Qm = Qa + Qb
Qm = (1430 - 55p) + (1470 - 70p)
Qm = 2900 - 125p
So, the market demand for food is given by the equation:
Qm = 2900 - 125p