Question

Professor Sprout decides to leave Hogwarts and run a business of mandrakes which have commercial due to its health restorative properties. She reads a research article in the Daily Prophet that provides the following production function (as a function of labor): ?(?) = ??? + ??? ? − ?. ?? ? Before she enters into business, she asks you to help her to figure out if it is feasible for her to enter in the market with 14 workers. I. (1 pt.) Find the average product of labor [APL] and marginal product of labor [MPL] formulas. II. (2 pts) Graph the total product curve, indicate the inflection point and the possible maximum value of production III. (2 pts) Graph the MP and AP curves and verify that AP = MP when AP is at its maximum IV. (1 pt.) What would be your advice? Would you allow her to enter into business or not? Why or why not?

Answer 1

We use the standard formulae of APL and MPL to calculate them using the given production function.

We advice Professor Sprout to undertake the business as it is having positive production and APL isn't maximum at 14 workers yet.

- Glven: We have the production function: 8 - 10L + 2042 -0.6 I we know that APL = 8 and MPL = da production Hence, substituting the giver function, we get - APL = 10L + 202² - 0.6L3 - APL = 10 + 2OL - 0.6L² IMPLE ALIOL 2012-0613) • MPL = 10 + 40 L - 1.8L2

11 Total Product Curve is . 8 = 10L + 2012 -0.623 8 - LC 10 +20L -0.614) If we try to factorise this further, we get DEL (L - (20 + 5 424 V - 20 -–L) 2 N-1.2) 8 = L (L+ 0 4987) (L - 33.8261) (If we round off to 4 decimal places) Also, we know that L o . Hence we know that we can't plate for L<o. Now, we have a cubic equation with all three roots known, hence we can plot the graph as

AK Inflection point (33.8261,0 ) At this inflection point, we And that do = 0 as slope of curve is o. 10 + 40L - 1.8120 .. L = - 40 1/1600 + 40x1.8 1.8x2) L = - 40 + 1 1672 - 3.6 L= 40 + 40.89 -3.6 L = 22.4694 LiL = -0.2472

Since we know that Loo at the Inflection point, we can say that. L = 22.4694 is the inflection point 0 (L=22.4694) = 10 (22.4694) + 20 (22.46987 -0.6(22.4694) 3515.644 units iL th) MPL = 10 +40L - 1.8L APL = 10 + 20L -0.61 For AP to be d (APL) - maximum, - 20 - 1.2L =0 L-20-400 - 1. 26 = 16.6667

MPL (L- 503) = 10 + ftoxs) -8% MPL = 176.6667 APL CL-50%) 10+ Boxse) – 6.6 xsongo APL = 176.6667 -> MPL - APL @ All's maximum Value MPL. APL 7 (16.6667,176.6669) 180 150 140 120 LOD bot 40+1 MPL

"At L = 14, 8 = 10 (14) +20(14)2 -0.644) 8= 2413.6 units Also, of is maximum @ 22.4694 workers Hence, although it is feasible to center into the business, it is advisable to have 22 workers to maximise production Also, APL is max @ 16.6667 workers Hence, this also indicates to have more workers.