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# Economics 4021B Final Exam

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Economics 4021B Final Exam

Instructor: Dana Galizia Carleton University April 2017

PLEASE WRITE YOUR NAME AND STUDENT NUMBER BELOW, AND ON THE TOP OF EACH PAGE IN THE PROVIDED SPACE.

Part A:

Part B: TOTAL:

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_______ / 80 _______ / 100

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A Short Answer (20 marks total)

Answer each of the following questions. Answers will typically range from one word up to 2-3 sentences (you won’t automatically lose marks if yours are longer, but they shouldn’t need to be much longer than that).

1. (1 mark) In the search model of the labour market, what is the “bargaining set”?

The range of wages that are acceptable to both firms and workers.

2. (3 marks) In the two-period endowment model, what would happen to the model if we allowed positive levels of saving in the second period? What if we allowed negative levels of saving (i.e., borrowing) in the second period?

Since there is no period after the second period, allowing positive savings would have no effect, since households would never be able to benefit from those savings but would have to give up current consumption, which can’t be optimal. If we allowed borrowing, however, then since the household would never have to pay this borrowing back, they would try to borrow an infinite amount.

3. (2 marks) Derive the lifetime budget constraint for the two-period endowment model.
The first-period constraint is c1 + s = y1, while the second-period constraint is c2 = (1 + r)s + y2.

Substiting the first into the second and rearranging yields the LBC c1 + 1 c2 = y1 + 1 y2. 1+r 1+r

4. (3 marks) In a dynamic model with no financial frictions, explain (in words) why ρt+1 1/(1 + rt+1) can be interpreted as the price of ct+1 in terms of ct.

Suppose the household reduces ct by the amount ρt+1, increasing their period-t savings by ρt+1 and using the resulting extra resources available in period t + 1 to buy as much ct+1 as possible. Since the household ends up with (1 + rt+1)ρt+1 = 1 units of extra resources in period t + 1, they can buy one more unit of ct+1 in exchange for the ρt+1 units of ct given up; that is, ρt+1 is the price of ct+1 in terms of ct.

5. (3 marks) What is the natural borrowing limit? Explain in words why, under our standard assumptions, a household who faces this borrowing limit will never actually hit it.

Econ 4021B - Winter 2017 Dana Galizia, Carleton University

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Name: _____________________ Econ 4021B - Winter 2017 Student #: __________________ Dana Galizia, Carleton University

The natural borrowing limit is equal to the total present value of all future income. For a household who faces this limit, if they hit it at any point then all subsequent income for the rest of their life will have to go to paying off the debt. But this leaves nothing for consumption, and since having zero consumption in any period is infinitely bad for the household, they would rather consume a little bit less in some prior period in order to avoid hitting the borrowing limit.

1. (3 marks) As in LN5, let Ωj,t+1 be the gross return on asset j from period t to t + 1. Explain in words why we can’t have Ωj,t+1 > Ωl,t+1 in equilibrium for two assets j and l. You don’t need to go through a mathematical proof, just explain in words what would happen if that were the case, and why that can’t happen in equilibrium.

If we had Ωj,t+1 > Ωl,t+1, then households earn a higher return on j than on l. It would then be in the best interest of any investor to sell short l and use the proceeds to buy j, since the return on j next period would be more than enough to cover the cost of covering the short sales of l in that period. With all investors trying to sell short l and buy j, there would be an excess supply of l and an excess demand for j, which can’t happen in equilibrium.

2. (2 marks) For the infinite-horizon endowment economy, write down the transversality condi- tion and interpret it in your own words.

The TVC is limt→∞ R?1st+1 = 0. In words, this says that the present value of savings very far 0,t

into the future should be (close to) zero.

3. (3 marks) Explain briefly what the “comovement problem” refers to in the context of our two-period model with investment. Be sure to make reference to patterns we see in the data.

In the data, consumption and investment tend to move together over the business cycle. The comovement problem refers to the fact that, in response to a change in second-period productivity, consumption and investment move in opposite directions, which is (or may be) inconsistent with the above-stated fact.

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B Problems (80 marks total)

Answer each part of the following questions. NOTE: BLANK ANSWERS IN THIS SECTION WILL RECEIVE AN AUTOMATIC 20% OF THE AVAILABLE MARKS FOR THAT PART OF THE QUESTION.

1. (9 marks total) Consider the two-period model with investment. Recall that we could reduce that model to the two equations

c1 + k2 ? (1 ? δ) k ? = z1f ??1, k ??? u(c1) = βu(z2f (1, k2)) z2fk (1, k2)

in two endogenous variables c1 and k2. From a solution to these equations, we can then get c2,iandρasc2 =z2f(1,k2),i=k2?(1?δ)k?,andρ=βu(c2)/u(c1).

(a) (5 marks) Suppose initially that c1 = c2. Show what happens to c1 and k2 if z1 increases by a small amount (i.e., solve for dc1/dz1 and dk2/dz1). For each of c1, c2, k2, and i, say whether it increases, decreases, stays the same, or whether the effect is ambiguous.

Totally differentiating the first two equations, we get

dc1 +dk2 =f??1,k ???

dk2 ?? ?? 2 z2fkk?? ???1 ?? ??? dz=β(z2fk)+σ +1 f1,k

1

2 z2fkk???? ?? 2 z2fkk?? ???1 dz =β (z2fk) + σ β (z2fk) + σ +1

dc1
where σ u′′/u< 0. We can then solve this to obtain

and thus

dc1 ?? 1

?? ??? f 1,k

dz1 11

dz1 dz=β(z2fk)+σ dz

??

2 z2fkk ?? dk2

Sincefk >0,andfkk,σ<0,wehavedk2/dz1 >0anddc1/dz1 >0. Sincec2 andiare both increasing in k2, we also have dc2/dz1 > 0 and di/dz1 > 0.

(b) (4 marks) Recall that the manager’s objective in period 1 is to maximize the share price, which in turn is given by p1 = d1 + ρd2. Explain how the manager “knows” how to

Econ 4021B - Winter 2017 Dana Galizia, Carleton University

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Name: _____________________ Econ 4021B - Winter 2017 Student #: __________________ Dana Galizia, Carleton University

change i in response to an increase in z1. (HINT: An increase in z1 must increase y1. Suppose all of that increase in output went to increasing c1, keeping i the same. How would this affect the manager’s optimization problem, and how would it respond?)

If all of the increase in y1 went to increasing c1, with i kept constant, then k2 wouldn’t change, and therefore y2 and c2 wouldn’t change either. Since a rise in c1 leads to a fall in u(c1), and c2 doesn’t change, from the given expression for ρ, we see that ρ must rise. From the manager’s period-1 objective function, this rise in ρ puts more weight on d2 (relative to d1), and thus the manager will respond by trying to increase d2. The only way to increase d2 is to increase y2, which in turn requires an increase in k2, and therefore an increase in i. Thus, the manager will increase i (financed by a reduction in d1).

2. (34 marks total) Consider a modification of the search and matching model of the labour market discussed in class. First, we assume that there is no unemployment insurance (i.e., b = 0). Second, we assume that g(h) = 0 for every household h. Finally, we assume that M(V,H) = VμH1?μ, where μ is a parameter satisfying 0 < μ < 1. In every other respect, the model set-up is the same (including the assumption that wages are determined by Nash bargaining with the worker’s share given by β).

1. (a) ?(4 marks) Consider a matched firm-worker pair. Assuming the two come to an agreement over a wage w, write down the firm’s surplus, the worker’s surplus, and the total surplus. Using these expressions, find the Nash wage.

The firm’s surplus is sf = z?w, the worker’s is sw = w, and the total surplus is s = sf + sw = z. Since the worker gets share β of the total, we have sw = βz, which combined with the fact that sw = w, implies that the Nash wage is w = βz.

2. (b) ?(4 marks) Using the functional form for M given above, find expressions for the proba- bility that a worker finds a match, p(θ), and the probability that a firm finds a match, q(θ).

We have p(θ) = κM(θ,1) = κθμ and q(θ) = κM(1?1) = κθ?(1?μ).

3. (c) ?(6 marks) Given the Nash wage you determined in (a) and the expression for q(θ) you found in (b), write down an expression for the firm’s expected profit from searching for a worker, R, as a function of θ and exogenous variables only. Use this expression to solve for the equilibrium level of tightness, θ?, as an explicit function of exogenous variables. Use this expression to argue that we must have θ? > 0.

The firm’s expected profit from searching is given by

R(θ) = q(θ)(z?w)?k
= κθ?(1?μ)(1?β)z?k

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Econ 4021B - Winter 2017 Dana Galizia, Carleton University

In equilibrium, we must have R (θ?) = κθ??(1?μ)(1 ? β)z ? k = 0, which can be solved to

obtain

θ=k Since,κ,z,k>0andβ<1,wemusthaveθ? >0.

1. (d) ?(6 marks) Given the Nash wage and the expression for p(θ) you found above, write down an expression for a household’s expected benefit of searching, J, as a function of θ and exogenous variables only. Using this expression, argue that, in equilibrium, every household will search for work (i.e., we will have H? = N).

The household’s expected benefit from searching is given by

J(θ) = p(θ)w = κθμβz

Since equilibrium tightness θ? is strictly positive as argued in (c), and also κ,β,z > 0, we must have J(θ?) > 0; that is, the expected benefit from searching in equilibrium will be strictly positive. Since the opportunity cost of searching is g(h) = 0 regardless of h, in equilibrium every household will find it optimal to search. That is, we will have H? = N.

2. (e) ?(14 marks total) Suppose a social planner (SP) wishes to maximize total output net of firms’ search costs (that is, total output of the whole economy net of total search costs incurred by all firms). Assume throughout part (e) that, as above, every household will search for work (i.e., H = N).

1. (4 marks) Let f denote total output net of firms’ search costs. Find an expression for f as a function of V and exogenous variables only.

Total output net of firms’ search costs is given by mz ? V k. Substituting in m = κV μH1?μ = κV μN1?μ, we obtain the objective function

f (V ) = κN1?μzV μ ? V k

2. (5 marks) Suppose the SP can choose V . Find the value of V it would choose and denote it VSP . Under what condition(s) would VSP = V ? (i.e., when would the de-centralized equilibrium V you found in (d) coincide with the optimal V )?

The SP will choose V to maximize f. The first-order condition for this problem is f(VSP ) = 0, i.e., κμN1?μzV μ?1 ? k = 0, which can be solved to obtain

????1 κμz 1?μ

SP

1 ? ??κ(1?β)z??1?μ

VSP =N
Comparing this with the expression for V we found in (d), we see that the two are equal

k

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Name: _____________________ Econ 4021B - Winter 2017 Student #: __________________ Dana Galizia, Carleton University

if and only if μ = 1 ? β.

iii. (5 marks) Suppose a government (without dictatorial powers) would like to imple- ment VSP as the de-centralized equilibrium outcome. The only tool it has at its disposal is a tax on firm search. That is, letting τ be the amount of the tax, the firm’s search cost is (1 + τ )k instead of k. What should the government set τ equal to? (HINT: You should be able to find the resulting equilibrium V without needing to fully re-derive the equilibrium with the tax in place.)

When the search cost is (1 + τ )k, the equilibrium level of vacancies is given by 1

τ=1?β?1 μ

3. (24 marks total) Consider the infinite-horizon endowment model of Lecture Note 4. Recall that ct is consumption in period t, st+1 is savings from period t to period t + 1, rt+1 is the interest rate on those savings, and yt is the endowment income in period t.

(a) (3 marks) Write down the period-t budget constraint, and explain in words what it means.

The period-t budget constraint is ct + st+1 = (1 + rt)st + yt. The right-hand side is the household’s total available resources, given by its income in that period, yt, plus its accumu- lated savings, (1 + rt)st. The left-hand side is the total amount of resources used, given by the amount of consumption purchased, ct, plus the amount of savings from t to t + 1, st+1. The budget constraint says these two things must be equal.

(b) (5 marks) Suppose the household puts one unit of savings in the bank at period t0 (i.e., st0+1 = 1) and henceforth leaves it there to accumulate interest. Let Rt0,t1 denote the accumulated value that savings in period t1. Derive an expression for Rt0,t1. Make sure you show how you obtained this expression (e.g., establish a pattern). What is the economic interpretation of R?1 ?

The accumulated value at t0 + 1 is simply (1 + rt0+1)st0+1 = 1 + rt0+1. The accumulated value at t0 + 2 is then (1 + rt0+1)(1 + rt0+2), at t0 + 3 is (1 + rt0+1)(1 + rt0+2)(1 + rt0+3), and so on. Thus, the accumulated value at t1 is

t1
Rt0,t1 = ?? (1+rj) j=t0+1

??κ(1?β)z??1?μ (1+τ)k

N
Setting this equal to VSP and solving for τ, we obtain the optimal choice of τ as

t0 ,t1

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Econ 4021B - Winter 2017 Dana Galizia, Carleton University

Since R is the accumulated value at t of one unit of savings made in period t , R?1 is t0,t1 1 0 t0,t1

the amount you would have to put in the bank at t0 in order to have one unit of accumulated savings at t . Thus, we can interpret R?1 as the present value at t of one unit of resources

1 t0,t1 0

at t1.
(c) (3 marks) Write down the No-Ponzi condition, and interpret it in words.

The No-Ponzi condition is limt→∞ R?1st+1 0. In words, this condition states that the 0,t

present value of savings very far into the future should be either (close to) zero or positive.

1. (d) ?(4 marks) Set up the household problem using the sequence-of-budget-constraints ap- proach (but don’t solve it yet). That is, write down the objective function, indicate which variables the household gets to choose, and state any relevant constraints. (You can ignore any non-negativity constraints here).

The household maximizes the objective function

??βtu(ct) t=0

by choice of ct and st+1, t = 0, 1, 2, . . ., subject to the sequence of budget constraints ct + st+1 = (1 + rt) st + yt

2. (e) ?(6 marks) Set up the Lagrangian for the problem in (e), and obtain first-order condi- tions for ct, ct+1, and st+1. Combined these conditions into one optimality condition by eliminating any Lagrange multipliers, and interpret this condition in words.

The Lagrangian is

The FOC’s are

∞∞

??βtu(ct)+??λt [(1+rt)st +yt ?ct ?st+1] t=0 t=0

βtu(ct) = λt βt+1u(ct+1) = λt+1 λt =(1+rt+1)λt+1

Combining these by eliminating λt and λt+1, we get βu(ct+1) = 1

u(ct) 1 + rt+1
The left-hand side of this condition is the MRS of ct for ct+1, which is how much ct the

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Name: _____________________ Econ 4021B - Winter 2017 Student #: __________________ Dana Galizia, Carleton University

household is willing to give up in order to get one more unit of ct+1. The right-hand side, meanwhile, is how much ct the household has to give up in order to get one more unit of ct+1. At an optimum, these two things should be equal.

(f) (3 marks) Write down the goods-market equilibrium condition for period t, and find the equilibrium interest rate rt+1.

The goods-market equilibrium condition for period t is that ct = yt. From part (f), then, the equilibrium interest rate is given by

rt+1= u(ct) ?1 βu(ct+1)

4. (13 marks total) Consider the two-period model of limited commitment discussed in LN7. In particular, suppose borrowers are born in period 1 with some fixed quantity of illiquid assets L, which cannot be bought or sold in period 1. In period 2, however, these assets can be bought/sold at exogenous price v. Households are allowed to borrow from the bank in period 1 at interest rate r, and to pledge any or all of their illiquid assets as collateral. In period 2, these households can either repay their debt, or decide to walk away from it (i.e., default). If they walk away, they must give to the bank the portion of their illiquid assets that they pledged as collateral, but the bank receives nothing else.

1. (a) ?(4 marks) Suppose the household pledges fraction φ of their illiquid assets as collateral for a loan of size a in the first period. Under what conditions will the household choose to defaul? Explain your answer.

If the household pays back its loan in period 2, it will cost them (1+r)a. On the other hand, if they walk away, they give up φL units of their illiquid assets, which are worth v each, for a total cost of φvL. The household will choose to default if the cost of paying back the loan exceeds the cost of surrendering their collateral, i.e., if (1 + r)a > φvL.

2. (b) ?(3 marks) Given your answer to part (a), assuming the household pledges fraction φ of their assets as collateral, what is the maximum amount the bank would be willing to lend them? Explain your answer.

The bank knows that if (1 + r)a > φvL then the household will default, and thus it will receive only φvL. Thus, the most the bank would be willing to lend is the amount that would yield them exactly φvL in period 2, since any amount above this would end up being effectively confiscated by the household. Thus, the maximum loan size is φvL/(1 + r).

3. (c) ?(3 marks) Suppose the household wishes to obtain a loan of size a. Given your answer to part (b), what is the minimum fraction of their illiquid assets they will have to pledge as collateral? Explain your answer.

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Econ 4021B - Winter 2017 Dana Galizia, Carleton University

Since the bank will lend at most φvL/(1 + r) when the household pledges fraction φ, the minimum fraction it will need to pledge to obtain a loan of a is the value that solves a = φvL/(1 + r). Thus, in order for the household to be able to borrow a it must pledge at least (1 + r)a/(vL).

(d) (3 marks) Let ψ be the maximum amount the household can borrow in this model. Solve for ψ.

The maximum amount the household can borrow is equal to the maximum amount it can borrow when it pledges all of its illiquid assets as collateral, i.e., when it sets φ = 1. The maximum loan it can get then is ψ = vL/(1 + r).

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