econ4334finalexamsample

The Sample Final Exam (Econ 4334 Fall 2023) Reminders 1. You have 165 minutes. Please follow every instruction from proctors. The exam is a closed-book-closed-notes exam; no cheat sheet; no calculator. 2. When you are asked to provide a numerical value of a variable  , you do not have to give a decimal number. For example, if you use a formula to obtain  = 2 1  3 2 + 3 √ 21 , then  = 2 1  3 2 + 3 √ 21 is an acceptable answer. Moreover, if you obtain  3 +  2 +1 = 0 , then saying  that is a solution to the equation is an acceptable answer. 3. When you are asked to derive a result, please go step by step. Each step counts. Stating the result with no reasoning gets no credits. 4. When you are asked to answer a question and explain the answer, answering with no explanation gets no credits. I. Equity premium (15 points) I. There are a risky asset and a risk- free asset. Each unit of the risk-free asset is exchanged for one unit of goods at date 1 and turns into  = 1 +  units of goods at date 2. Each unit of the risky asset is exchanged for  units of goods at date 1 and, at date 2, with probability  it generates   units of goods and with probability 1 −  it generates   (    ) units of goods. It is observed that an investor chooses to buy  units of the risk-free asset and  unit of the risky asset at date 1. For this investor, from consuming   at  ,  = 1  2 , his utility is  (  1   2 ) =  (  1 ) +  (  2 ) , where  (  ) =  1 −  − 1 1 −  with  ≥ 0 and ln  = lim  → 1  1 −  − 1 1 −  . The investor maximizes his expected utility.Suppose  = 0  5 ,   = 2 ,   = 0  8 ,  = 1 , and  = 1  1 . 1. What is the value for the equity premium ∆ ? 2. Find the value for ˆ  such that  (ˆ  ) = ∆ . 3. Suppose you have strong reasons to believe that the investor’s  is equal to 1 . Does that imply that equity dominates bonds in the rate of return (according to the model/theory in section 4)? 4. Let  = 1 and maintain the values of  = 0  5 ,   = 2 ,   = 0  8 , and  = 1  1 . Find the value for  such that  (  ) is equal to the equity premium ∆ . II. Liquidity-provision banking theory (30%) There are three dates, 0, 1, and 2. Each person has one unit of goods at the start of date 0. There 1

are two sorts of technologies: storage and investment. If one person puts  units of goods at date  into storage, then he obtains  units of goods (from storage) at date  +1 for  = 0  1 . If the person puts  units of goods into investment at date 0 and carries all  to date 2, then he obtains  units of goods (from investment) at date 2; if the person puts  into investment at date 0 and liquidates  at date 1, then he obtains  at date 1. At the start of date 1, each person has the probability   to become type  ∈ { 1  2 } . A type-  ∈ { 1  2 } agent only desires to consume at date  and the utility function is  (  ) = −  1 −  . Let  1 =  2 = 0  5 ,  = 2 ,  = 2 , and  = 0  5 . 1. What is the optimal (date-0) investment (in the autarky allocation)? What is the agent’s (date-0) expected utility implied by this investment? 2. Suppose there are two agents, their types are perfectly correlated (i.e., when one becomes type 1, another must be type 2), and the two agents form a coalition a date 0. What is the optimal coalition per-capita investment? What is each agent’s expected utility implied by this investment? 3. Suppose there is a continuum agents, i.e., the set of agents corresponds to the closed interval [0,1]. What is the per-capita investment in the fi rst best allocation? What is each agent’s expected utility implied by this investment? III. Monetary creation (5%) Suppose  = 10% ,  = 0 , and  = 0 . Using T-accounts, show what happens in the banking system when the Fed sells bonds whose value is $100 to the First National Bank. (Answer: LectureSlides, pages 11-21; it su ffi ces to examine a few banks to obtain the pattern in T-accounts and, once the pattern emerges you can apply it to other banks and conclude) IV. Private v.s. public information (5%) There are two dates, 1 and 2. There is an investor who decides whether to invest on a project at date 1. If he invests some   0 then he can build the project at date 1 but he must hire a manage to run the project at date 2. Provided that the project is built at date 1, it will yield some output at date 2. Speci fi cally, if the project succeeds, the output is   0 but if it fails the output is 0. The successful probability of the project depends on the manager’s e ff ort. If the manager makes e ff ort (does not shirk), the successful probability is 90%; otherwise the successful probability is 10%. The manager gets extra bene fi t   0 by being lazy. Provide an action-based contract in the absence of private information. V. Rules vs discretionary decision (10 points) A central bank has full control of the in fl ation rate  ; it concerns both  and output  with the 2

there is the transfer with a person’s utility in the stationary equilibrium when the money supply is fi xed at 10). VII. Foreign exchange rate (5%) Explain why sterilized foreign ex- change intervention does not a ff ect the exchange rate. Remarks: 1. The above questions (except Question III) are selected from Homework sets Review Problems 2. They intend to tell you the format of the midterm exam; they do not imply that the midterm is to use the exact same questions and problems that have been used in Homework sets Review Problems. 4