Question

The City of Wasatchia is situated in a narrow valley. A single highway, I-16, runs the ten-mile length of the narrow valley Two dentists operate in Wasathia. Annie Adams, DDS, has an office at the far western edge of Wasatchia. Beverly Bodeen, DDS, has an office at the far eastern edge of Wasatchia. The distance between the dentists' offices is ten miles. There are 1000 residents in the town and the residents' homes are evenly spread along the ten miles between the two dentists' offices. Each of the 1000 residents gets a teeth cleaning once per year. Residents like clean teeth, but they dislike travel and high dentist bills. Dr. Adams and Dr. Bodeen are equally good at cleaning teeth. If a resident is located d miles from a dentist, then that resident's willingness to pay for a cleaning from that dentist is given by 100 -td2 (a) Let pa be the price charged by Dr. Adams for a cleaning. Let p be the price charged by Dr. Bodeen for a cleaning. Derive Dr. Adams demand function. (That is, figure an expression for the total number of cleanings Dr. Adams will do in a year as a function of Pa Pb, and t (b) Derive an expression for Dr. Adams's demand elasticity in terms of Pa P, and t. How does elasticity vary with t? Explain the connection between t and elasticity (c) Let t = 2, and suppose the marginal cost of a cleaning is $30, what are the Nash equilibrium prices for the two dentists? What is the variable profit of each dentist? (d) Continue to assume that t 2. Dr. Adams's normal supplier of teeth cleaning goop runs into operational difficulties and cannotfll Dr. Adams's regular order. She quickly finds another supplier but her marginal cost of a cleaning rises to $42. What are the Nash equilibrium prices and quantities for the two dentists? What is the variable profit of each dentist? (e) Return the assumption that marginal cost for both dentists is 830. Road construction hits I-16, and the resulting delays make travel even more costly. We now havet 3.

Answer 1

a) The p(a) and p(b) are the price of services of Adams and Bodeen respectively.

So Demand function of the Adams = p(a) - p(b)/t*100-t*d², t refers to the total number of services which Adams rendered to the public. Here p(b) was excluded in order to derive the separate demand function for the Adams.

b) Elasticity in relation to t refered as p(a) =change in quantiy demanded / change in price*t. Thus it utmost changes the situation of difference between the Elasticity and the t.  

c) According to Nash equilibrium, the variable profits of both Adams and Bodeen are elaborated as p(a)+(pb) = t(30)/100-t*d².

d) When the marginal cost increased to $42. Then the variable profits of both dentists will be p(a) + p(b)= 100-t*d² -42*/p(b).

where price of p(b) will be in equal distribution limit of income where y = p(a)+p(b) - 42, where y is the income.

e) Assuming the marginal cost of both dentists as $30. Then elasticity related to nash equilibrium p(a) + p(b)= 100-t*d² -72*/p(b). The total marginal cost of both dentists as 72.

f) Assuming the variable cost of different direction of both the 2 and 3 related to both dentists, The the variable profits of both the dentists will be p(a) + p(b)= 100-t*d² -5.