Questions: A firm's fixed costs are 30 and the variable costs are (Q+7) per unit. The demand function is P+ 2Q=40 公司的固定成本为 30 , 可变成本为每单位 (Q+7) 。 需求函数为 P+2 Q=40 a. What is the total cost function? 什么是总成本函数? b. What is the total revenue function? 什么是总收益函数? c. For what values of Q does the firm break-even? 对于什么 Q 值，公司可以实现收支平衡？ d. What is the profit function? 什么是利润函数? e. Find the value of Q that profit is maximum. 求利润最大的 Q 值。 f. What is the value of maximum profit? 最大利润的价值是多少?

Solution Steps

a. The total cost function is the sum of fixed costs and variable costs. The fixed cost is given as 30, and the variable cost per unit is \(Q + 7\). Therefore, the total cost function is \(TC = 30 + Q(Q + 7)\).

b. The total revenue function is derived from the demand function. The demand function is given as \(P + 2Q = 40\). Solving for \(P\), we get \(P = 40 - 2Q\). The total revenue is then \(TR = P \times Q = (40 - 2Q) \times Q\).

c. The break-even point occurs when total revenue equals total cost. Set the total revenue function equal to the total cost function and solve for \(Q\).

The total cost function \( TC \) is calculated as the sum of fixed costs and variable costs. Given the fixed cost is 30 and the variable cost per unit is \( Q + 7 \), the total cost function can be expressed as: \[ TC = 30 + Q(Q + 7) = Q^2 + 7Q + 30 \]

The total revenue function \( TR \) is derived from the demand function. The demand function is given by \( P + 2Q = 40 \), which can be rearranged to find \( P \): \[ P = 40 - 2Q \] Thus, the total revenue function is: \[ TR = P \times Q = (40 - 2Q) \times Q = 40Q - 2Q^2 \]

The break-even point occurs when total revenue equals total cost, i.e., \( TR = TC \). Setting the two equations equal gives: \[ 40Q - 2Q^2 = Q^2 + 7Q + 30 \] Rearranging this equation leads to: \[ -3Q^2 + 33Q - 30 = 0 \] Factoring or using the quadratic formula, we find the break-even points: \[ Q = 1 \quad \text{and} \quad Q = 10 \]

Final Answer