$5.$ Tamika and Catalina work as plumbers, each serving the same neighborhood. Tamika is a careful worker who utilizes the best replacement parts on her jobs. Catalina worries less about the quality of her jobs. She tries to complete them quickly and often uses low$-$cost replacement parts. Tamika believes that if customers knew the quality of her work, she could charge $$100$ for a typical job. If customers recognized that Catalina$$s work was of lower quality, she would be able to charge only $$60$ for a typical job. Because customers cannot judge the quality of the two plumbers$$ work, they consider it equally likely that the work is high quality $($i$.$e$.,$ a $$100$ job$)$ or low quality $($i$.$e$.,$ a $$60$ job$).$ Thus, both Tamika and Cataline are able to charge $$80$ for a typical job. Recently, Tamika placed an advertisement in the local paper emphasizing the high quality of her work. She was angered and dismayed when Catalina placed a similar advertisement the following week. $(30$ points$)$

a$.$ Describe the problem Tamika faces in economic terms, using the specific concepts and language discussed in class. Do you expect that Tamika$$s advertisement will help her to better compete with Catalina? Why or why not?

b$.$ A friend has suggested that Tamika offer a warranty on her work. In researching this suggestion, Tamika has concluded that on average her obligations under a warranty will be $$10$ per job per year of the warranty. Catalina$$s costs will be higher, $$15$ per job per year. Should Tamika offer a warranty? How long of a warranty should she offer, assuming the only benefit to Tamika from offering the warranty is to distinguish her work from Catalina$$s work? If Tamika offers her optimal warranty, should Catalina offer any warranty? Why or why not? $($Note: Be sure to answer everything that is asked in this part of the question. Try to get a specific answer as to length of warranty, if you can.$)$