Problem 1

A company selling a perishable good (so there's no need to worry about inventory) has been doing some research on the demand and its sensitivity to price. They've been pricing their product at $5.00 for a long time and have seen demand fluctuate randomly. The most recent 30 weeks of sales data are contained in the table below:

Week	Sales		Week	Sales		Week	Sales
1	417		11	413		21	352
2	376		12	366		22	398
3	406		13	397		23	404
4	348		14	425		24	405
5	402		15	418		25	468
6	400		16	417		26	379
7	400		17	430		27	408
8	345		18	385		28	443
9	380		19	427		29	359
10	431		20	407		30	432

The problem is that they're not sure how their profits would respond if they were to try different prices. The company agrees to explore a few different prices. The following table summarizes the prices they tried and the demand they observed.

Price	Observed Demand
$4.00	494
$6.00	344

In such a small price window, it's generally safe to assume that the demand is a linear function of the price with an additive stochastic element with a mean of zero:

demand=0+1price+

To understand how to properly price and supply their product, the company needs to know the mean and standard deviation at all prices.

The mean is given by completing the regression and using the equation:

demand=^0+^1price

Your task in the Regression assignment was to find the estimates, ^0and ^1.

For this problem, your task is to find the optimum price the company should charge and the quantity the company should have on hand while taking into account the randomness. The optimum price and quantity are the price and starting quantity that maximize the expected profits. There are 3 different sources of randomness. There's randomness in the slope estimate, the intercept estimate, and the natural randomness in demand.

Based on the data, you can simulate demand at any price using the equation below:

^0+^1p+tn2MSE1+1n+(pp)2(pip)2

In this equation,p is the price at which you'd like to simulate the demand,MSE is the calculated mean squared error from your regression,n is the number of data points in your regression,p is the average price of the data in your regression, andpi refers to the individual prices in the data used in your regression. The termtn2 is the only random element in the equation -- it's a random draw from the t-distribution with n2 degrees of freedom. Random draws (i.e., simulations) can be generated in Excel using the formula:

=T.INV(RAND(),n-2)

(you'll need a real number instead of just "n" for Excel to work properly with this formula -- see the paragraph above for a description of what "n" means).

Your second task is to answer questions about the randomness in demand.

Problem 1:

Using the procedure outlined above, simulate 2000 values of demand. What is the mean value of demand for a price of $6.00?