Homework 4 - Consumer TheoryQuestions 1-3 below refer to a consumer with preferences over consumptionwhen young and old that is given by either of the following utility functions(α = .5). Either representation of preferences giv es rise to the marginal rate ofsubstitution function below:U(c1,c2)=cα1c1−α2U(c1,c2)=α log(c1)+(1− α)log(c2)MRS(c1,c2)=−U1(c1,c2)U2(c1,c2)= −αc2(1 − α)c11. Calculate the marginal rate of substitution between c1and c2when (i)(c1,c2)=(1, 1), (ii) (c1,c2)=(1, 2) and (iii) (c1,c2)=(2, 1).2. Suppose that the consumer has a wage income when young of w1=10and has no wage income when old.a. What are the utility maximizing values for (c1,c2,a2) when the realinterest rate r =0?[Note: The real interest rate in this case is 0%.]b. Repeat part a for the case when r =1.[Note: The real interest rate in this case is 100%.]c. Graph the consumption choices and budget sets in part a-b on a singlegraph.3. Now imagine that the consumer can work both when young and whenold. The wages earned when young are w1=10andwhenoldarew2=5.The budget constraints can then be written as either of the two lines indicatedbelow:c1+ a2≤ w1and c2≤ w2+ a2(1 + r2)c1+ c2/(1 + r2) ≤ w1+ w2/(1 + r2)a. What are the utility maximizing values for (c1,c2,a2) when the realinterest rate r2=0?b. Repeat part a. when r2=1.c. Graph the consumption choices and budget sets in part a-b on a singlegraph.4. Assume a consumer has the utility function below and makes best choicesas described in Chapter 4.1U(c1,c2, ..., c50)=α1log(c1)+α2log(c2)+···+ α50log(c50)Also assume that (i) α1= α2= ···= α50=1/50, (ii) the real interest rateis r = 0 and (iii) the agent starts out with no financial wealth in mo del period1.(a) Calculate the best consumption choices (c1,c2, ..., c50) when the agent’searnings profile is w1= w2= ··· = w40=1andw41= w42= ··· = w50=0.Discuss the logic that underlies these best choices.(b) Calculate the best consumption choices (c1,c2, ..., c50) when the agent’searnings profile is w1= w3= ···= w39=2andw2= w4= ···= w40=0andw41= w42= ···= w50= 0. Discuss the logic that underlies these best choices .5. Read the (attached) article ”Paying the Displaced Proves no Panacea”.Answer the following questions.(a) Is Kevin Sather’s behavior potentially consistent with the utility maxi-mization theory of consumer behavior discussed in class? Explain.(b) What would you predict that Kevin Sather would do if the city of Min-neapolis simply gave him equal payments over time instead of a single lump-sumpayment? You should assume that either payment method has the same presentvalue. Explain your reasoning carefully.Hint: You may want to make a number of BIG assumptions in your analysis.Be sure to list any important assumptions that you use to answer the