Help me to solve the following

QU ESTION ON E (a) Alex Kornbo entered into a five year written lease agreement for office premises with Mercy Omagwa. After the end of the lease period, the parties agreed that Mercy Omagwa would continue occupying the office premises for another five years. On that basis, Mercy Omagwa made extensive renovations to the office premises. She installed new carpets, painted the premises and put in a new heating system. Alex Kombo later changed his mind and told Mercy Omagwa to vacate the office premises claiming that the lease agreement has expired. With specific reference to promissory estoppel in the law of contract, advise Mercy Omagwa on her legal rights. (10 marks) Theory: 1) Take the function: y = f(x;a) = a' -2x - ax . a) Find the first order condition for a maximum/minimum. b) Is this a maximum or a minimum? c) Solve for x" . d) Explicitly solve for dx and da e) Now re-write the first order condition and use the implicit da function theorem to find dx da f) Use the envelope theorem to find " and explain briefly da why this theorem allows you to simplify your calculation from (d). g) What is dy ? Would your answer differ depending on whether x" was a maximum or minimum for this problem? 2) Let a consumer's utility be given by U(x, y, z) = In(x) + In(y) + In(2 + z). The consumer has a budget constraint of x + 3y+2z =18. a) What are the optimal choices for this consumer? b) What would the choices be if the budget constraint were increased from 18 to 19? c) What is the consumer's utility in each case? d) How do you interpret the Lagrange multiplier in (b)? e) Show how you could use the Lagrange multiplier (and the envelope theorem for constrained problems) to estimate the change in consumer utility caused by increasing her budget from 18 to 19 without having to resolve for x, y, and z in (c). 3) Take the problem max z = - subject to x + 4y =90. a) Set up the Lagrangian and find xty the first order conditions b) Solve explicitly for x, y, z and 1, the Lagrangian multiplier. c) Set up the dual problem of (a), i.e., minimize the primal constraint function subject to the primal objective function being equal to the value of z" obtained in (b). d) Solve explicitly for x, y, and 1" , the Lagrangian from the dual. e) What is the relationship between 1 and 1" , and can you provide an economic explanation for this?plicaon: Minimum wage articles 1) 2) 3) 4] 5] J ohn Kennan describes a 1915 Bureau of Labor Statistics report that studies a natural experiment affecting retail stores in Oregon. Ease your answers to the following questions on Kennanis description of the report. {a} What is the policy change that the report studies?l {h} What are the experimental groups?' (c) 1|What is the control group? (d) Why is the control group necessary\";I (e) Are there any problems with this control group? {1'} 1What are the results, and what do they suggest about the effects of minimum wages on employment? Richard Freeman breaks the world into the BR's and 811's. Applying this logic to a change in the minimum wage in a competitive mode] of labor supply and demand. {a} give an example of a labor market where you think that the BR's are most likely to be correct, and (b) give an example where the SR's are most likely to be correct. What is your reasoning in each case? Illustrate your answers with diagrams that show clearly what features of supply and demand curves make the diagrams correspond to BR's and SR's. Card and Krueger (p. 7171) emphasize that one strength of their analysis is that the implementation of the New Jersey minimum wage occurred during a recession. (a) Consider the case where the minimtnn wage was implemented right before an economic expansion instead Draw a diagram that clearly shows how an economic expansion affecting both New Jersey and Pennsylvania might have invalidated the results of the 'natural experiment? (h) Someone points out to you that, according to Table 3 of (ELK. fast food employment fell in PA but it barely budged in NJ. Hence, they say, ifwejust look at NJ, it seems pretty clear that there is no support for a monopsony scenario. How would you respond to this criticism within the rnatural experiment' framework. Draw a carefully labeled diagram that illustrates your point. A NJ legislator, after hearing of the C1851: results says to her colleagues, \"This is good news: the minimum wage increase raised fast food employment. We should pass further increases to get more teenagers jobs.\" Taking her initial premise as correct the minimum wage increase did raise employment analyze her conclusion with a diagram that shows under what circumstances she would be right and what circmnstanees she would be wrong. Richard Freeman distinguishes between the short and long run impacts of minimum wage increases, implying that the long run impacts are likely to he larger [and more negative}. List one or two specic economic reasons why the long rim response would be larger. Consider that labor is not the only factor in fast food restaurants' production functions