A firm's long run total cost curve is given by TC(Q) = 1000Q - 30Q^2 + Q^3. Derive the expression for the long run average cost curve and sketch it. At what quantity is the minimum efficient scale?

To get a firm's long run average cost, you need to divide the total cost by quantity. As such, (1000Q - 30Q² + Q³)/Q = 1000 - 30Q + Q². From this we can see that the long run average cost curve would have a y-intercept of 1000 as there is a value that is not dependant on Q. As Q² is positive, we can tell that the curve will be a convex parabola.

The minimum efficient scale is the quantity at which the firm's long run average cost is minimised. On the graph this will be the point where the parabola bottoms out and has a gradient of 0. The derivative of the average cost curve shows the curves slope at different points so if we find the derivative and set it equal to 0 then we can find the minimum point. AC' = 30 - 2Q so if we set this equal to 0 then we get 0 = 30 - 2Q. Through rearranging and solving, we get a minimum efficient scale at a quantity of 15.