4. If an individual's utility function for coffee (x) and cream (y) is given by U(x, y) = min (x, 5y), write the demand function for coffee .
7. Assume x and y are the only two goods a person consumes. If after a rise in pX the quantity demanded of y increases. Evaluate the following as true or false and explain. Support your explanation with a graph.
The income effect dominates the substitution effect for good y.
To find the demand function for coffee, we need to determine the quantity of coffee that an individual would demand at different prices, while holding the price of cream constant.
Given the utility function U(x, y) = min(x, 5y), this means that the individual's utility is determined by the minimum between x (quantity of coffee) and 5y (quantity of cream). In other words, the individual values cream more than coffee.
Now, let's consider the situation where the price of coffee is denoted as P and the individual's income is denoted as M. We can express the problem of the consumer as:
Maximize U(x, y) = min(x, 5y)
Subject to the budget constraint: P * x + Price of Cream * cream consumed = M
Since we are interested in finding the demand function for coffee, we need to determine how the quantity of coffee demanded (x) changes with the price of coffee (P).
To do this, we first need to write the individual's optimization problem using Lagrange multipliers. The Lagrangian function is:
L = min(x, 5y) + λ(M - P * x - Price of Cream * cream consumed)
Taking the partial derivative of L with respect to x and y, we have:
∂L/∂x = ∂/∂x min(x, 5y) - λP = 0
∂L/∂y = ∂/∂y min(x, 5y) - 5λPrice of Cream = 0
To find the demand function for coffee, we set ∂L/∂x = 0:
∂/∂x min(x, 5y) - λP = 0
The partial derivative of min(x, 5y) with respect to x depends on the relationship between x and 5y:
If x < 5y, then ∂/∂x min(x, 5y) = 1
If x > 5y, then ∂/∂x min(x, 5y) = 0
If x = 5y, then ∂/∂x min(x, 5y) is undefined
Since the individual would always choose the smaller of x and 5y to maximize utility, we consider the case where x < 5y. Therefore, we have:
1 - λP = 0
Solving for λ, we find:
λ = 1/(P)
Now, we substitute λ back into the derivative equation:
1 - (1/P) * P = 0
Simplifying, we have:
1 - 1 = 0
This implies that the equation is satisfied regardless of the value of coffee.
Therefore, the demand function for coffee is x = x(P), where x(P) can take any value greater than or equal to zero. This means that the individual will demand any quantity of coffee as long as they can afford it and the price is greater than or equal to zero.
To derive the demand function for coffee, we need to find the optimal quantity of coffee that maximizes the utility function U(x, y) = min (x, 5y) given the individual's budget constraint.
The budget constraint represents the limited income available to the individual to purchase coffee and cream:
p₁x + p₂y = m
Where:
- p₁ is the price of coffee,
- p₂ is the price of cream,
- m is the income or budget available to the individual.
To find the demand function for coffee, we will maximize the utility function U(x, y) subject to the budget constraint.
Step 1: Set up the Lagrangian function:
L(x, y, λ) = U(x, y) - λ(p₁x + p₂y - m)
Where:
- λ is the Lagrange multiplier.
Step 2: Differentiate L with respect to x, y, and λ, and set the derivatives equal to zero:
∂L/∂x = ∂U/∂x - λp₁ = 0
∂L/∂y = ∂U/∂y - λp₂ = 0
∂L/∂λ = p₁x + p₂y - m = 0
Step 3: Solve the above equations simultaneously:
From the first equation, we have:
∂U/∂x = λp₁
Since the utility function U(x, y) = min (x, 5y), we consider two cases:
Case 1: When x ≤ 5y
∂U/∂x = λp₁ = 1p₁, as x is the minimum value in this case.
Case 2: When x > 5y
∂U/∂x = λp₁ = 5p₁, as 5y is the minimum value in this case.
Step 4: Derive the demand function by considering the two cases:
Case 1: When x ≤ 5y
Since x is the minimum value, we set x = x₀, where x₀ represents the optimal quantity of coffee. Therefore, the demand function for coffee in this case is:
x₀ = x
Case 2: When x > 5y
Since 5y is the minimum value, we set 5y = x₀, where x₀ represents the optimal quantity of coffee. Therefore, the demand function for coffee in this case is:
x₀ = 5y
Therefore, combining both cases, we can write the demand function for coffee (x) as:
x₀ = min (x, 5y) or
x₀ = min (x, 5(m - p₂y)/p₁)
The demand for coffee will depend on the prices of coffee and cream (p₁ and p₂, respectively), as well as the individual's income (m) and their preference for coffee and cream (as represented by the utility function). The demand function will give the optimal quantity of coffee to purchase given these factors.