Show that the price leaves you with a smaller profit

Question

Problem

You run a tutoring service for economics students. On half of all days the demand for your service is high, with

Q_H = 10 - 0.2P

Q_H is the number of hours you will work if you choose price P. On the remaining days, demand is low:

Q_L = 5 - 0.2P.

High and low days arrive at random with probabilities of 0.5 each. Your marginal opportunity cost is $10 per hour on all days. You want to choose a price that maximizes your average profit per day, but that price must be the same on all days. Find that price.

a. Show that this price leaves you with a smaller profit than if you can announce different prices early in the morning of each day, after you have learned whether demand will be high or low.

The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.