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MCAT PROJECT DEFINE PHASE TECHNICAL REPORT

. Create a professionally formatted technical repot (format up to you) to your supervisor communicating the results of the . . . Design Phase of Projectile LLC's process improvement project.

. Your report should contain the following.

1. Executive summary of the problem.

2. Results of your "Process Walk" (interviews, initial discussions). What were some of

the difficulties with data collection that you observed? Are there any process improvement opportunities that jump out of you from your "Go see" process walk / baseline analysis?

3. VOC analysis.

4. SIPOC chart and discussion.

5. Discuss your plants quality levels, ability to deliver and overall cost of (poor)

quality. What are the opportunities that you can pursue? What is the potential cost

savings to be realized?

6. Complete the Catapult QC Report (baseline) and include it in your memo as a figure.

Complete the MCAT Cost of Quality Analysis. Include the full analysis in an appendix

and a SUMMARY of your cost of quality analysis (don't just cut and paste the spreasheet! Make your own summary table with the most important, bottom line information.)

7. Complete the Project Charter and upload it with your tech memo as a separate document. Though do please reference it in your memo.

Consider a pure exchange economy with 2 islands. Each island consists of infinitelylived identical agents whose measure is normalized to one. There is a single consumption good, or fruit, that is non-storable. The representative agent of island i values different consumption streams according to ui(c i ) = X t=0 t (c i t ) 1 , i, with > 0. The total endowment in this economy (i.e., in both islands) in period t is given by a deterministic sequence {et} t=0, with et > 0 for all t. However, due to the location of the two islands, weather conditions are different affecting the fruit yield on each island. Letting each period t denote a season, we assume that in even periods the fruit yield on island 1 is given by e 1 t = a et , a [0, 1], and in odd periods it is given by e 1 t = (1 a) et . Of course, by definition, the endowment of fruit on island 2 must be given by e 2 t = et e 1 t , in all periods. a) For this economy define an Arrow-Debreu equilibrium (ADE) and a Sequential Markets equilibrium (SME). b) Fully characterize (i.e., find a closed-form solution for) the ADE prices. c) Using any method you like, characterize the SME consumption allocation in as much detail as you can.1 In the remaining questions, we will assume that the total endowment in the economy (in both islands) follows the process et = t e, with > 0. d) Given this new parametric specification, provide a closed-form solution for the SME consumption allocation. e) What happens to the consumption allocation you calculated in part (d) when = 1 (no growth in the endowment)? What if ( = 1 and) a = 1? f) Back to the model with general a, ; can you specify parameter values for which agents on island 1 consume more than agents on island 2 in a typical period t? g) For what parameter values do the ADE prices you calculated in part (b) increase as a function of t? Provide some intuition for your result.

This question considers the macroeconomic effects of a collapse in consumer demand in the New Keynesian model. There are a continuum of identical households. The representative household makes consumption (C) and labor supply (N) decisions to maximize lifetime expected utility: E0 X t=0 t C 1 t 1 N 1+ t 1 + ! Zt (1) subject to their budget constraint: Ct + Bt = wtNt + (1 + it1) Pt1 Pt Bt1 + Dt (2) where wt is the real wage, Nt is hours worked, Bt are real bond holdings at the end of period t, it1 is the nominal interest rate paid between t 1 and t, Pt is the price of the final consumption good and Dt are real profits from firms that are distributed lump sum. As usual, 0 < < 1, > 0 and > 0. Zt is a household preference shock, which is a way of generating shocks to demand. The production side of the model is the standard New Keynesian environment. Monopolistically competitive intermediate goods firms produce an intermediate good using labor. Intermediate goods firms face a probability that they cannot adjust their price each period (the Calvo pricing mechanism). Intermediate goods are aggregated into a final (homogeneous) consumption good by final goods firms. The production side of the economy, when aggregated and linearized, can be described by the following set of linearized equilibrium conditions (the production function, the optimal hiring condition for labor and the dynamic evolution of prices): yt = nt (3) wt = mct (4) t = Et(t+1) + mc t (5) The resource constraint is: yt = ct (6) Monetary policy follows a simple Taylor Rule: it = t (7) The (linearized) preference shock follows an AR(1) process: zt = zt1 + et (8) et is i.i.d. In percentage deviations from steady state: mct is real marginal cost, ct is consumption, wt is the real wage, nt is hours worked, yt is output. In deviations from 3 steady state: it is the nominal interest rate, t is inflation. is a function of model parameters, including the degree of price stickiness.2 Assume that > 1, 0 < < 1. a) First consider the representative household's problem. Write down the household's problem in recursive form and derive the household's first order conditions. b) Show that the linearized first order condition for labor supply from part (a) is: wt = ct + nt (9) and show that under flexible prices demand shocks have no effect on real GDP. Hints: you will need to use equations (3), (4), (6) and (9). c) This model can be reduced to three equations: Etyt+1 yt = 1 (it Ett+1) + 1 Et(zt+1 zt) (10) t = Et(t+1) + yt (11) it = t (12) where zt follows the process in equation (8) and = ( + ). yt = yt y n t is the output gap and y n t is the natural rate of output. Using the method of undetermined coefficients, find the response of the output gap and inflation to a collapse in consumer demand when prices are sticky and monetary policy follows the Taylor Rule above. To do this, guess that the solution for each variable is a linear function of the shock zt . d) Discuss how, and why, a fall in demand affects the natural rate of output, the output gap and inflation in this model. Briefly comment on how a decrease in zt relates to typical recessions we see in the data. e) Instead of following the Taylor Rule above, policy is now set optimally. Derive the optimal monetary policy rule under discretionary policy. (Hint: As in class, assume that the loss function has quadratic terms for the output gap and inflation, with a relative weight on the output gap. For simplicity, assume the steady state is efficient). Using this rule and your knowledge of the model, what is the optimal path for the output gap and inflation in response to a demand shock? Would your answer change if the central bank followed an optimal policy rule under commitment? Explain. (You do not need to derive anything for these last two discussion questions).

Consider the following economy. Time is discrete and runs forever, t = 0, 1, 2, . . . The economy is populated by two types of agents (a measure one of each): farmers and workers. Farmers own a piece of land that pays a stochastic income yt every period. We assume that yt is i.i.d. across farmers and time, and that yt N(y, 2 ). Farmers use all their time working their land. Their preferences are given by u(c) = exp(c) for some > 0. For simplicity, we assume that consumption of farmers can be negative, that is c R. Moreover, farmers can save (or borrow) in a non-state contingent and non-defaultable asset a, which has a rate of return r. Farmers face the following "NoPonzi" condition on assets lim t at (1 + r) t 0. Moreover, farmers can produce and hold capital, k, which is rented to the representative firm in competitive markets (1 unit of final consumption good produces 1 unit of capital). Let r K be the rental rate of capital and the depreciation rate. Unlike farmers, workers don't own land, and they use their available time to work in the representative firm. Assume each worker is endowed with one unit of time. They cannot trade the asset a, but they can produce and hold capital, k (with the same technology as farmers). Their per-period utility is given by u(c), with u 0 (c) > 0, u 00(c) < 0, limc0 u 0 (c) = , limc u 0 (c) = 0. Finally, there is a representative firm that combines capital and labor to produce final good according to f(K, L), where K is the capital they operate, and L the hours/workers they hire. Assume f() satisfies the standard Inada conditions. Thus, the only financial market available in this economy is the market for assets a, where only farmers can trade. Moreover, there is a market for k, where all agents can trade. Both type of agents have the same discount factor (0, 1). a) Given a constant path for r and r K, state the problem of a farmer. Argue that the equilibrium price of capital is equal to 1. Characterize the farmer's problem with the necessary FOCs. Show that in any equilibrium with positive capital it must hold that r = r K . b) Let c F (a, k, y) denote the consumption policy function of the farmers, which depends on asset holdings a, capital holdings k, and land income y. Prove that: c(a, k, y) = c + r(a + k) + r 1 + r y, 5 where c is a constant. Find an expression for c in terms of the parameters of the model. Hint: If x N(, 2 ), then E[e x ] = e + 1 2 2 . c) Let C F and KF denote the aggregate consumption and capital holdings of farmers. Show that in an equilibrium with constant C F and KF , it must hold that (1 + r) < 1. Hint: Remember to check the budget constraint. d) Let's turn to the workers. State their problem. Hint: Remember that workers cannot save in asset a; they can only hold capital. e) Let c W and k W denote the consumption and capital holdings of an individual worker. Show that in an equilibrium with constant c W , it must be that k W = 0. Conclude that an equilibrium with constant C F , KF , c W , and k W = 0 exists Hint: Use the result that in equilibrium (1 + r) < 1. f) Show that in such an equilibrium, an increase in 2 increases the welfare of workers. Explain.

6. (10) Consider the discrete time monetary-search model we saw in class. As in the baseline model, in the day time trade takes place in a decentralized market characterized by anonymity and bilateral meetings (call it the DM), and at night trade takes place in a Walrasian or centralized market (call it the CM). There are two types of agents, buyers and sellers, and the measure of both is normalized to the unit. The per period utility is u(q) +U(X) H, for buyers, and q +U(X) H, for sellers; q is consumption of the DM good, X is consumption of the CM good (the numeraire), and H is hours worked in the CM. In the CM, one hour of work delivers one unit of the numeraire. The functions u; U satisfy standard properties. What is important here is that there exists X > 0 such that U 0 (X ) = 1. Goods are non storable, but there exits a storable and recognizable object, called at money, that can serve as a means of payment. The supply of money, controlled by the monetary authority, follows the process Mt+1 = (1 + )Mt , and new money is introduced via lump-sum transfers to buyers in the CM. So far, this is just a description of the model we saw in class. What is dierent here is that only a fraction of buyers turn out to have a desire to consume the DM good in the current period; let us refer to these buyers as C-types (for consumption) and to the remaining 1 buyers as N-types (for no-consumption). The shock that determines each buyers type in every period is iid. A buyer learns her type after all CM trade has concluded but before the DM opens. To make things interesting we will assume that between the CM and the DM there is a third market, where C-types and N-types can meet and trade liquidity", i.e., money. Let us refer to this market as the loan market (LM).4 The LM is a bilateral market for loans, where N-types, who may carry some money that they do not need, meet C-types, who may need additional liquidity. A CRS matching function f(; 1 ) brings the two types together. Importantly, the LM is not anonymous, so that agents can make credible (and enforceable) promises. Hence, when an N-type and a C-type meet, they mutually benet from a contract specifying that the N-type will give l units of money to the C-type right away, and the C-type will repay d (for debt) units of the numeraire good in the forthcoming CM. After the LM trades have concluded (for the agents who matched with someone), C-types proceed to the DM, where they use money to purchase goods from sellers. Assume that all C-type buyers match with a seller. Notice that I have not said anything about the splitting of the various surpluses (i.e., bargaining), because this information will not be necessary for what I am asking here. Let W(:) be the CM value function of a buyer, and V (:) the DM value function of a C-type buyer (since only these buyers visit the DM). Also, let i (:) be the LM value function of a type-i buyer, i 2 fC; Ng. Your task in this question is to describe these value functions. I am not asking you to analyze them. I recommend that you draw a graph summarizing the timing of the model. (a) Describe the function W(:), and show that it is linear in all its arguments/state variables (what these arguments are, however, is for you to determine). (b) Let (q; p) be the quantity of good and the units of money exchanged in a typical DM meeting. Let (l; d) be the size of the loan (in dollars) and the promised repayment (in terms of the numeraire) specied in a typical LM meeting. What variables do the terms q; p; l; d depend on? Hint: Provide quick answers of the form q is a function of the money holdings of the (C-type) buyer". (c) Describe the function V (:), where, again, determining the state variables is your task. (d) Describe the functions i (m), i 2 fC; Ng, for a buyer who enters the LM with m units of money. Hint: Recall that some buyers (of either type) will match in the LM and some will not, and the outcome of the matching process will critically aect a buyers continuation value. Make sure that this is reected in the expression you prov

MCAT PROJECT DEFINE PHASE TECHNICAL REPORT

. Create a professionally formatted technical repot (format up to you) to your supervisor communicating the results of the . . . Design Phase of Projectile LLC's process improvement project.

. Your report should contain the following.

1. Executive summary of the problem.

2. Results of your "Process Walk" (interviews, initial discussions). What were some of

the difficulties with data collection that you observed? Are there any process improvement opportunities that jump out of you from your "Go see" process walk / baseline analysis?

3. VOC analysis.

4. SIPOC chart and discussion.

5. Discuss your plants quality levels, ability to deliver and overall cost of (poor)

quality. What are the opportunities that you can pursue? What is the potential cost

savings to be realized?

6. Complete the Catapult QC Report (baseline) and include it in your memo as a figure.

Complete the MCAT Cost of Quality Analysis. Include the full analysis in an appendix

and a SUMMARY of your cost of quality analysis (don't just cut and paste the spreasheet! Make your own summary table with the most important, bottom line information.)

7. Complete the Project Charter and upload it with your tech memo as a separate document. Though do please reference it in your memo.