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
1. (a) To show that the consumer maximizes utility subject to the budget constraint, we need to set up the consumer's optimization problem. The consumer maximizes utility subject to the budget constraint by choosing the quantities of goods X and Y that maximize their utility, given their income and the prices of the goods.
Let's assume the consumer has an income of I, and the prices of goods X and Y are Px and Py, respectively. The consumer's budget constraint can be expressed as:
Px*X + Py*Y = I
To maximize utility, the consumer needs to find the combination of X and Y that maximizes the utility function U = log X + log Y, while satisfying the budget constraint.
To solve this problem, we can use the method of Lagrange multipliers. The Lagrangian function is:
L = log X + log Y + λ(I - Px*X - Py*Y)
Where λ is the Lagrange multiplier.
To find the maximum utility, we need to take the first-order partial derivatives of the Lagrangian function with respect to X, Y, and λ, and set them equal to zero:
∂L/∂X = 1/X - λ*Px = 0
∂L/∂Y = 1/Y - λ*Py = 0
∂L/∂λ = I - Px*X - Py*Y = 0
Solving these equations simultaneously will give us the optimal quantities of X and Y that maximize utility subject to the budget constraint.
(b) To derive the demand functions of good X and good Y, we need to solve for X and Y in the optimization problem above.
From the first equation, we can solve for X:
1/X = λ*Px
X = 1/(λ*Px)
From the second equation, we can solve for Y:
1/Y = λ*Py
Y = 1/(λ*Py)
Substituting the values of X and Y into the budget constraint equation, we get:
Px/(λ*Px) + Py/(λ*Py) = I
1/λ = I
λ = 1/I
Substituting λ back into the expressions for X and Y, we get the demand functions:
X = 1/(Px*I)
Y = 1/(Py*I)
So, the demand functions of good X and good Y are X = 1/(Px*I) and Y = 1/(Py*I), respectively.
2. To calculate the market demand for food, we need to sum up the individual demands for food by household A and household B.
The demand function for household A is given as Qa = 1430 - 55p, where Qa represents the demand for food in kg by household A and p represents the price of food.
Similarly, the demand function for household B is given as Qb = 1470 - 70p, where Qb represents the demand for food in kg by household B and p represents the price of food.
To calculate the market demand, we need to add the individual demands of household A and household B together:
Qm = Qa + Qb
Qm = (1430 - 55p) + (1470 - 70p)
Qm = 2900 - 125p
Therefore, the market demand for food is given by Qm = 2900 - 125p, where Qm represents the market demand for food in kg and p represents the price of food.