Demand q for product B depends on its price p and its quality (a product attribute) x: q = (a – bp + dx). Profit is q(p – c); the variable cost increases with its quality: c = x2; and a, b and d are positive constants. You wish to choose the price and attribute value that generates maximum profit. Assume there are no constraints (i. e.: any real value of p, x, and q are possible).

a. Write this problem mathematically as a standard optimization problem.
b. Write the first order necessary optimality conditions for this problem (you do not need to solve them explicitly).
c. Is this point a local maximum? Prove it.
d. Is this point a global maximum? Prove it.
e. What happens when you use the starting point [0,20] instead of [0,0]? Why does this happen?

Areal estate agent is considering changing her cell phone plan. there are three plans to choose from, all of which involve a monthly service charge of $20. plan a has a cost of $.42 a minute for daytime calls and $.17 a minute for evening calls. plan b has a charge of $.52 a minute for daytime calls and $.15 a minute for evening calls. plan c has a flat rate of $80 with 275 minutes of calls allowed per month and a charge of $.38 per minute beyond that, day or evening. a. determine the total charge under each plan for this case: 150 minutes of day calls and 70 minutes of evening calls in a month. (do not round intermediate calculations. round your answer to 2 decimal places. omit the "$" sign in your response.)c. if the agent will use the service for daytime calls, over what range of call minutes will each plan be optimal? (round each answer to the nearest whole number. include the indifference point itself in each answer.)d. suppose that the agent expects both daytime and evening calls. at what point (i. e., percentage of total call minutes used for daytime calls) would she be indifferent between plans a and b?

Three years ago, stock tek purchased some five-year macrs property for $82,600. today, it is selling this property for $31,500. how much tax will the company owe on this sale if the tax rate is 34 percent? the macrs allowance percentages are as follows, commencing with year 1: 20.00, 32.00, 19.20, 11.52, 11.52, and 5.76 percent.

You are a newspaper publisher. you are in the middle of a one-year rental contract for your factory that requires you to pay $500,000 per month, and you have contractual labor obligations of $1 million per month that you can't get out of. you also have a marginal printing cost of $.25 per paper as well as a marginal delivery cost of $.10 per paper. if sales fall by 20 percent from 1 million papers per month to 800,000 papers per month, what happens to the afc per paper?