How to use use the Excel trendline method to estimate a regression equation of the form 

y = a + bx + (cx)^2? 

 Louisa is managing a training program for a major bank. She designs the following experiment to assess the effect of sleep on learning. A group of trainees is given a 3-hour session on some task, followed by a 2-hour test on what they have just learned. The trainees are then sent home and told to keep track of how many hours they sleep that night. The next morning the trainees are tested again and their performance relative to the previous day is recorded. The following data is generated. The top row shows hours of sleep for a given employee and the bottom row shows the next day's relative test performance.

(a) ) Use the trendline approach in Excel to run a linear regression of the form y = a + bx where y is the relative test score and x is hours of sleep.  (Put sleep in Column A and score in Column B.) Cut and paste the Excel diagram into the space below.

(b) What is the estimated effect of one additional hour of sleep? What is the R²?

(c)  Now use the Excel trendline method to estimate a regression equation of the form y = a + bx + cx².  Report the estimated equation and the R². Cut and paste the diagram into the space below. If you were asked to predict the score associated with 10 hours of sleep, would you use this quadratic regression or the linear regression in a)? What relative score would you predict? Explain.

Question #2

Wheat is produced according to the production function Q = 5(K^0.8 L^0.2) where Q denotes output, K denotes capital and L denotes labor.

(a)   Does this production function exhibit increasing, decreasing or constant returns to scale? Show the mathematics you used to arrive at your answer.

(b)   With capital fixed at 10 units, derive an expression for the MP of labor and AP of labor. Graph your expressions for MP_L and AP_L (L on the horizontal axis).

(c)     If w =10 and r = $5, find cost minimizing combination of K and L to produce 1000 units of Q.  Draw a graph to illustrate your result.

 Question #3 Suppose newsprint is produced in a perfectly competitive market by many identical firms. Each firm (including potential entrants) has a total variable cost VC(q) = 40q + 0.5q². Each firm's fixed cost is equal to $50.

a)  Assuming that the fixed cost is entirely non-sunk (i.e. recoverable), calculate the price below which the firms will not produce any output in the short run.

b) Assume that there are 12 identical firms in this industry. Currently the market demand for newsprint is Q_d = 360 - 2P.  What is the short-run equilibrium price?