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 express utility as a function of the budget constraint and then optimize it.
The budget constraint is given as:
Pies (X) * Quantity of X + Fanta (Y) * Quantity of Y = Total Income (I)
Utility function: U = log X + log Y
To find the maximum utility, we can use the Lagrange multiplier method. We construct a Lagrangian function:
L = log X + log Y + λ(I - P_X * X - P_Y * Y)
where λ is the Lagrange multiplier.
Next, we take the partial derivatives of L with respect to X, Y, and λ, equate them to zero, and solve the resulting system of equations.
∂L/∂X = 1/X - λP_X = 0
∂L/∂Y = 1/Y - λP_Y = 0
∂L/∂λ = I - P_X * X - P_Y * Y = 0
From the first equation, 1/X = λP_X, we can rearrange it to find X as a function of λ and P_X: X = 1/(λP_X)
Similarly, from the second equation, 1/Y = λP_Y, we can rearrange it to find Y as a function of λ and P_Y: Y = 1/(λP_Y)
Now, substitute these expressions for X and Y into the third equation:
I - P_X * X - P_Y * Y = 0
I - P_X * (1/(λP_X)) - P_Y * (1/(λP_Y)) = 0
I - 1/λ - 1/λ = 0
I - 2/λ = 0
λ = 2/I
Substituting this value of λ back into X and Y:
X = 1/((2/I) * P_X) = I/(2 * P_X)
Y = 1/((2/I) * P_Y) = I/(2 * P_Y)
Therefore, we have derived the demand functions for good X and good Y as:
X = I/(2 * P_X)
Y = I/(2 * P_Y)
(b) The demand functions for good X and good Y are:
X = I/(2 * P_X)
Y = I/(2 * P_Y)
These equations show how the quantity demanded of each good depends on the income of the consumer (I) and the prices of the goods (P_X, P_Y).