Question 1:
What is the price elasticity of demand for a product that experiences a 10% increase in price and a 15% decrease in quantity demanded?
Answer:
Price elasticity of demand = percentage change in quantity demanded / percentage change in price
In this case, the percentage change in quantity demanded is -15% (a decrease of 15%), and the percentage change in price is +10% (an increase of 10%).
Thus, the price elasticity of demand = -15% / 10% = -1.5
Question 2: A monopolist faces the following demand function: Q = 100 - 2P, where Q is quantity demanded and P is price. If the monopolist's total cost function is given by C(Q) = 200 + 5Q, what is the profit-maximizing price and quantity?
Answer:
The monopolist's profit is equal to total revenue minus total cost. Total revenue is given by P * Q, so the monopolist's profit function is:
π(Q) = P(Q) * Q - C(Q) = (100 - Q/2) * Q - (200 + 5Q) = -Q^2/2 + 95Q - 200
To find the profit-maximizing quantity, we take the derivative of the profit function with respect to Q and set it equal to zero:
dπ/dQ = -Q + 95 = 0 Q = 95
To find the profit-maximizing price, we substitute the value of Q into the demand function:
Q = 100 - 2P 95 = 100 - 2P P = 2.5 Therefore, the profit-maximizing price is $2.50 per unit and the profitmaximizing quantity is 95 units.
Question 3: Suppose a firm has a production function given by Q = L^(1/3) * K^(2/3), where Q is output, L is labor input, and K is capital input. If the wage rate is $10 per unit of labor and the rental rate for capital is $20 per unit of capital, what is the firm's cost function?
Answer:
The firm's cost function is given by C(w, r, Q) = wL + rK, where w is the wage rate, r is the rental rate for capital, and Q is the level of output. To find the cost function, we need to express the firm's inputs in terms of its output:
Q = L^(1/3) * K^(2/3) L = Q^3/K^2/3 K = Q^3/L^1/3
Substituting these expressions into the cost function, we get:
C = wL + rK = w(Q^3/K^2/3) + r(Q^3/L^1/3) = 10Q^3/K^2/3 +
20Q^3/L^1/3
Simplifying this expression, we get:
C(Q) = 10Q^(2/3) * 20Q^(1/3) = 200Q
Therefore, the firm's cost function is C(Q) = 200Q.
Question 4:
Suppose a consumer's utility function is given by U(x,y) = x^2y^3, where x is the quantity of good X consumed and y is the quantity of good Y consumed. If the price of good X is $2 per unit and the price of good Y is $3 per unit, what is the consumer's optimal bundle of goods?
Answer:
The consumer's utility maximization problem can be expressed as: max U(x,y) subject to the budget constraint Px * x + Py * y = M, where Px and Py are the prices of goods X and Y respectively, M is the consumer's income, and x and y are the quantities of goods X and Y consumed.
To solve this problem, we use the Lagrangian method and set up the following equation:
L(x,y,λ) = x^2y^3 + λ(M - Px * x - Py * y)
Taking the partial derivatives of L with respect to x, y, and λ and setting them equal to zero, we get:
∂L/∂x = 2xy^3 - λPx = 0 ∂L/∂y = 3x^2y^2 - λPy = 0 ∂L/∂λ = M - Px * x - Py * y = 0
Solving for λ in the first two equations and setting them equal to each other, we get:
2xy^3/Px = 3x^2y^2/Py 2Py = 3Px
Substituting this into the budget constraint, we get:
Px * x + (2/3)Px * y = M y = (3/2)(M/Px - x)
Substituting this into the first equation, we get:
2x(3/2)(M/Px - x)^3/Px = λPx
Simplifying this expression, we get:
6x(M/Px - x)^3 = λPx^2
Solving for x, we get:
x = M/5Px
Substituting this into the equation for y, we get:
y = (3/2)(M/Px - M/5Px) = (9/10)M/Py
Therefore, the consumer's optimal bundle of goods is (x,y) = (M/5Px, (9/10)M/Py).