ECON 4721, Summer 2022 Problem Set 1
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Problem Set 1 - Answer Keys
ECON 4721, Summer 2022
Problem 1 (20 points)
Consider an Overlapping generations economy in which Nt young individuals are born each period. Individuals are endowed with y = 15 units of the consumption good when young
and nothing when old. The utility function of one typical agent is a typical time-separable
CRRA: u(c1 , c2 ) = + p . p ∈ (0, 1] is the time discount factor and > 0 is the
inverse of the elasticity of intertemporal substitution. Population of the future generations are determined by Nt+1 = nNt for all t ≥ 1, N0 = 100.
1. What is the equation for the feasible set of this economy?
Ntc1t + Nt−1 c2,t−1 ≤ Nty
OR
c1t + c2,t−1 ≤ y
It is also correct if the constraint is written with equality. Also correct if numerical values are substituted.
2. Portray the feasible set on a graph.
Anything similar to the graph below:
3. Let’s solve the Planner’s Problem. (i) State the Planner’s problem as a constrained maximization problem. (ii) Write down the Lagrangean for this problem. (iii) What are the FOCs? (iv) Assuming a stationary equilibrium, find the optimal allocations as a function of p , and n only.
(i) The planner’s problem is to find the allocations c2,0 , {c1,t, c2,t}0 solving: c2,0 , {ct(1)x2,t − 1 } 0 +p + c2,0
s.t.
Ntc1,t + Nt−1 c2,t−1 ≤ Nty, ∀t ≥ 1
(ii)
L = +p + c2,0 + 入t(Nty − Ntc1,t − Nt−1 c2,t−1)
(iii) The FOCs are:
c t = Nt入t
pc t = Nt入t+1
(iv) In a stationary solution, we must have:
入t Nt+1
=
入t+1 Nt
Hence,
c2 = c1 (p ) 1\ = c1 (pn)1\
Substituting in the feasibility constraint:
n
c1 = y
c2 = n(y − c1)
4. How does consumption when young respond to changes in p? What about ? And n?
c1 clearly decreases with p and . To check wahat happens with n:
?c 1 − 1 (pn)1\
?n (n + (pn)1\)2
Therefore, if > 1, c1 is increasing in n
5. Now, p = 0.5, = 2 and n = 2. Substitute and find 1 and 2 . 2 = 10 1 = 10
6. With arbitrarily drawn indifference curves, illustrate the stationary combination of 1 and 2 that maximizes the utility of future generations.
Anything similar to the graph below with y = 15, ny = 30, 1(*) = 2(*) = 10
7. Now look at a monetary equilibrium. Write down equations that represent the con- straints on first and second-period consumption for a typical individual. Combine these constraints into a lifetime budget constraint.
1t + vtmt ≤ y
2t ≤ vt+1mt
vt
vt+1
8. Suppose the initial old are endowed with a total of M = 400 units of fiat money. What condition represents the clearing of the money market in an arbitrary period t? Use this condition to find the real rate of return of fiat money.
MC:
vtM = Nt(y − 1t
Real Rate of return:
vt+1 Nt+1 (y − 1,t+1)
vt Nt(y − 1t)
9. Let’s Find the competitive equilibrium allocation: (i) Write down the future genera- tions problem for this economy (ii) State the definition of a competitive equilibrium. (iii) Write down the Lagrangean for the future’s generation problem. (iv) What are the FOCs? (v) Find the stationary equilibrium allocation as a function of and p and n only .
(i)
max c 1(1)t(−) +p c 2(1)t(−)
c1,t,c2,t 1 − 1 −
st
vt
vt+1
(ii) Some variation of: A Competitive equilibrium is a sequence of feasible allocations and prices in which every generation solves its problem and market clears. (iii)
L = +p + 入t(y − c1t − c2t)
(iv) FOCs:
c t = 入t
pc t = 入t
( ) − = p
c1 = c2 (p ) −1\
(v) Note that:
vt+1 Nt+1(y − c 1,t+1) Nt+1
vt Nt(y − c 1t) Nt
This implies that the BC:
c1 + c2 = c1 + c2 ≤ y
The BC is exactly equal to the feasibility constraint. Note that the tangency condi- tions are also the same so it must be that the equilibrium allocation is equal to the planner’s problem solution, check in the question below the exact answer. The last step is to check if the allocation is also optimal and feasible for the initial old. We know that it is optimal for them to consume all their money endowment:
c2,0 = c2 = v 1M\N0
c1 + c20 = c1 + v 1 ≤ y
v 1 = y − c1 = c2
So it is feasible. We may also find v 1 ,
=⇒ v 1 = c2 = 10 =
Where the last equality comes from the values above, not asked here. Note that: vt+1 = nvt
=⇒ v2 = nv 1
v3 = nv2 = nnv 1 = n2v 1
Repeating this steps we get that:
vt = nt −1v 1
Thus, all the sequence vt is determined. Ok if left as a function of N0
10. How does the return of money affect consumption when young, when old, and real money holdings (vtmt)? The intuition I was looking for is the income and substitution effect. This is easier to see if we look at the lifetime budget constraint: c1t+c2,t+1 ≤
y . If vt+1\vt increases that means that consumption when young becomes relatively more expensive than when old, hence you want to substitute away from c1 . Also when vt+1 when vt increases you know that the same amount of goods that you give up when young will be more worthy when you are old, hence the real value of money has increased.
11. What is the value of money in period t (vt)? What is the price of the consumption good pt?
We found above that vt+1\vt = 1\n and v 1 = 10\2. We can find vt:
vt+1 = nvt
=⇒ v2 = nv 1
v3 = nv2 = n2v 1 ...
vt = nt −1v 1
2t−1 10
2
Ok if left as a funtion of N0
12. Suppose n increases. What would happen to the rate of return of fiat money and real money holdings of any generation t? What would happen to the value of a unit of fiat money in the initial period and the utility of the initial old? Explain your answers. [Hint: Answer these questions in the order asked.]
This is an exercise on comparative statics. With population grows at rate n,
vt+1 Nt+1
vt Nt
Suppose now n changes from 2 to 8. Then c1 = 12, c2 = 24, vtmt = 3 Since n increases, the rate of return to money would increase. Since c20 = c2 has increased initial old is now better off .
13. Suppose instead that the initial old were endowed with a total of 200 units of fiat money. What would happen to the price level and consumption allocations? Are the initial old better off with less units of fiat money? Explain. pt = 1\vt therefore, pt+1\pt = 1\n and pt = (1\n)t −1p1 . Also remember that
c2nN0 M
M nN0c2
So pt decreases by half. Consumptions allocations will not change and the initial old will consume the same.
14. What is the utility function when 厂 = 1? (HINT: Take limits and use the L’Hôpital’s
rule) L’Hôpital’s rule:
f (x) f \(x)
x →a g(x) x →a g\(x)
Therefore:
c1− elog(c1− ) e(1− )log(c) e(1− )log(c) ( −lOg(c))
→1 1 − 厂 →1 1 − 厂 →1 1 − 厂 →1 −1
So the utility function is: u(c1 , c2) = log(c1) +p log(c2).
Problem 2 (10 points)
Consider an Overlapping generations economy with constant population N = 100 of young individuals are born each period. Individuals are endowed with y1 units of the consumption good when young and y2 when old, the initial old also starts with y2 . The utility function of one typical agent is: u(c1 , c2) = log(c1) + .
1. Consider the case in which y1 = 5 and y2 = 2.
State and solve the Planner’s Problem. The Planner’s Problem is:
∞
∑⃞
t=1
st
Nc1t + Nc2,t−1 ≤ Ny 1 + Ny2
OR
c1t + c2,t−1 ≤ y 1 + y2
FOCs
= 0.5 =⇒ c2 = 0.5c1
c1 + 0.5c1 = y 1 + y2
c1 = (y1 + y2) =
c2 = (y1 + y2) =
(b) Suppose the initial old are endowed with a total of M = 100 units of fiat money. Write down equations that represent the constraints on first and second-period consumption for a typical individual. Combine these constraints into a lifetime budget constraint. BCs are
c2 ≤ vt+1mt + y2 =⇒ mt = (c2 − y2)\vt+1
The lifetime BC is:
c1 + (c2 − y2) ≤ y 1
OR
c1 + c2 ≤ y 1 + y2
(c) Find the Stationary Competitive Equilibrium. FOCs:
= 0.5 =⇒ c2 = 0.5c1
From BC and MC:
vtmt = y 1 − c1 =⇒ vt = N (y1 − c1)\M
= 1 =⇒ c2 = 0.5c1
vt+1
From lifetime BC
c1 + 0.5c1 = y 1 + y2
c1 = (y1 + y2) =
c2 =
From the initial old:
c2 = v 1M\N + 2
= v 1 + 2 =⇒ v 1 = 1\3
=⇒ vt = 1\3, At
2. Consider the case in which y 1 = 2 and y2 = 5.
(a) State and solve the Planner’s Problem. The solution to the Planner’s problem is exactly the same as in 1. (a) with the right values for y 1 and y2
(b) Suppose the initial old are endowed with a total of M = 100 units of fiat money. Write down equations that represent the constraints on first and second-period consumption for a typical individual. Combine these constraints into a lifetime budget constraint. The lifetime BC is exactly the same as in 1.(b) - with the right substitutions
(c) Find the Stationary Competitive Equilibrium. The budget constraint for the initial old is
c20 = v 1M0\N + 5
From the budget constraint it is clear that c20 ≥ 5. Further from the resource constraint for period 1, it follows that
c1 + c20 ≤ 7 =⇒ c1 ≤ 2
From the FOCs, it is clear that the inter generational smoothing would mean c1 > 2, but this is not possible as it would reduce the consumption of initial old below 5. Hence the only possible solution in this case is autarky, where c1 = 2, c2 = 5. Note that Autarky is always one possible equilibrium in the model. Even in part 1. To see that, recall that agents maximizes their utility taking prices as given. When the price of money is 0, autarky is an equilibrium. So in this question, vt = 0, At
Problem 3 (10 points)
Consider two economies, A and B. Both economies have the same population, supply of fiat money, and endowments. In each economy, the number of young people born in each period is constant an N, and the supply of fiat money is constant at M. Furthermore, each person is endowed with y units of the consumption good when young and zero when old. The only difference between the economies is with regard to preferences, in economy A, utility function is UA (c1 , c2) = lOg(c1) + . while in economy B, utility function is given by: UB (c1 , c2) = log(c1) + 2lOg(c2). We will also assume stationarity.
1. Will there be a difference in the rates of return of fiat money in the two economies? If so, which economy will have the higher rate of return of fiat money? We have seen in class that in a stationary solution:
vt+1 Nt+1Mt
vt NtMt+1
This does not depend on the utility function!
2. Will there be a difference in the value of money in the two economies? If so, which economy will have the higher value of money? FOCs imply that in each economy:
c 1(A) = 2c2(A)
c 1(B) = 0.5c2(B)
From lifetime BC:
c1(A) + c2(A) = y =⇒ c1(A) = y\3, c2(A) = 2y\3 c1(B) + c2(B) = y =⇒ c1(B) = 2y\3, c2(B) = y\3
From the initial old BC:
c20(A) = c2(A) = v1(A)M =⇒ v 1(A) =
c 20(B) = c2(B) = v1(B)M =⇒ v1(B) =
In economy A, the value of money higher. The intuition is that in economy B agents are more patient tha in economy A meaning that they prefer to consume more when old than people in economy A. So money trades for a higher amount of present consumption.
Problem 4 (10 points)
Consider an overlapping generation economy with the following characteristics: each gen- eration is composed of 1000 individuals. The fiat money supply changes according to
Mt = 2Mt−1 . The initial old own a total of 10000 units of fiat money (M0 = $10000). Each period, the newly printed money is given to the old of that period as a lump-sum transfer (subsidy). Each person is endowed with 20 units of the consumption good when born and nothing when old. Preferences are: U (c1 , c2) = (c1(p) + c2(p))1\p .
1. Find the stationary Competitive Equilibrium in this economy as a function of p .
Initial old problem:
max u(c2,0)
C2,0
st
c2,0 ≤ v 1m0
Future generations problem:
max u(c1t, c2t)
C1t,C2t
st
c2t ≤ vt+1mt + at+1
The lifetime BC is:
c1t + c2t ≤ y + at+1
A Competitive Equilibrium (CE) is a sequence of allocations c2,0 , (c1,t, c2,t, mt)1 and money value (vt)1 such that:
(a) Given v 1 , c2,0 solves the Initial Old problem.
(b) Given vt, vt+1, at , c1,t, c2,t, mt solve the Future Generations problems At ≥ 1.
(c) Allocations are feasible: At ≥ 1, Ntc1,t + Nt−1 c2,t−1 ≤ Nty
(d) Market Clears: Ntmt = Mt , At .
(e) Government budget is balanced: Nt−1at = (1 − 1\z)vtMt
In this exercise, z = 2,Nt = 1000, At , y = 20.
We have found in class that
vt+1 Nt+1 Mt
vt Nt Mt+1
Note that:
(c1 , c2) = (c1(p) + c2(p))1\p−1pc1(p)−1 = U1−p c1(p)−1 (c1 , c2) = (c1(p) + c2(p))1\p−1pc2(p)−1 = U1−p c2(p)−1
c1(p)−1 = c2(p)−1 =⇒
c1(p)−1 = c2(p)−1
c1 = c241\(1−p)
Substituting in lifetime BC:
c241\(1−p) + c2 = 20
c2 =
c1 = 41\(1−p)
2. If p = −1. What is the gross real rate of return on fiat money? How many goods does an individual receive as a subsidy? What is the price of the consumption good in period 1, p 1 , in dollars? Real rate of return was found above. In this case:
c1 = 2c2 =⇒ c2 = c1 \2
From lifetime BC:
c1 + 2c2 = y + 2a =⇒ c1 + c1 = y + 2a
c1 = (y + 2a)
From the definition of a:
a = (1 − 1 )vtMt = (1 − 1\z)(y − c1)
Where in the last equality we used the money market clearing condition:
vtMt = vtNtmt = Nt(y − ct)
Substituting c1 into a:
a = (1 − 1\z)(y − (y + 2a)) = (1 − 1\2)(y\2 − a)
a = y\4 − a\2
a = y\4
y
6
So the old individual receives 20/6 units. Substituting in consumption:
c1 = y = 20
c2 = y = 20
From the initial old BC:
c20 = v 1M0
y 20 1
3. Find the social Planner’s optimal allocation for p = −1. In the SPP solution with constant population, it must be that:
?u ?u
=
?c 1 ?c2
=⇒ c = c
=⇒ c1 = c22 =⇒ c1 = √2c2
From Feasibility:
c1 + c2 = y
c2 (1 + √2) = y
c2 = y = 20
c1 = y = 20
2022-07-23