Consider a competitive industry and a price-taking firm that produces in that industry.The market demand and supply functions are estimated to be: Demand: $Q _ { d } = 10,000 - 10,000 P + 1.0 M$ Supply: $Q _ { s } = 80,000 + 10,000 P - 4,000 P _ { I }$ where Q is quantity,P is the price of the product,M is income,and $P _ { I }$ is the input price.The manager of the perfectly competitive firm uses time-series data to obtain the following forecasted values of M and $P _ { I }$ for 2021: $\hat { M } = \$ 50,000 \text { and } \hat { P } _ { I } = \$ 20$ The manager also estimates the average variable cost function to be $A V C = 3.0 - 0.0027 Q + 0.0000009 Q ^ { 2 }$ Total fixed costs will be $2,000 in 2021.Average variable cost reaches its minimum value of _____ units of output.

Question

Quizplus · Accepted Answer

To find the quantity of output at which average variable cost (AVC) reaches its minimum, we differentiate the AVC function with respect to Q and set the derivative equal to zero. The AVC function is given by $AVC = 3.0 - 0.0027Q + 0.0000009Q^2$. Differentiating AVC with respect to Q gives: $\frac{d(AVC)}{dQ} = -0.0027 + 0.0000018Q$. Setting this derivative equal to zero to find the minimum point, we get: $-0.0027 + 0.0000018Q = 0$. Solving for Q, we find: $Q = \frac{0.0027}{0.0000018} = 1500$. Therefore, the average variable cost reaches its minimum value at 1,500 units of output.

Consider a Competitive Industry and a Price-Taking Firm That Produces Qd=10,000−10,000P+1.0MQ _ { d } = 10,000 - 10,000 P + 1.0 MQd​=10,000−10,000P+1.0M

Consider a Competitive Industry and a Price-Taking Firm That Produces $Q _ { d } = 10,000 - 10,000 P + 1.0 M$