A Real Intertemporal Model with Investment Economics 3307 - Intermediate Macroeconomics Aaron Hedlund Baylor University Fall 2013 Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 1 / 30
A Real Intertemporal Model with Investment
Economics 3307 - Intermediate Macroeconomics
Aaron Hedlund
Baylor University
Fall 2013
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 1 / 30
Introduction
We have looked at static models with production but no savings, andwe have looked at dynamic models with savings but no production.
Now we construct a dynamic model with production to studyinvestment, business cycles, fiscal/monetary policy, etc.
Three economic actors:1 A representative consumer makes consumption/savings decisions and
supplies its labor.
2 A representative firm hires labor and makes production andinvestment decisions.
3 The government engages in government spending using tax revenuesand borrowed funds.
Three markets: labor market, credit market, goods market.
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 2 / 30
The Representative Consumer
The consumer’s budget constraints are given by
C1 + Sp1 = w1(h − l1) + π1 − T1
C2 = w2(h − l2) + π2 − T2 + (1 + r)Sp1
The intertemporal budget constraint is
C1 +C2
1 + r= w1(h − l1) + π1 − T1 +
w2(h − l2) + π2 − T2
1 + r
Consumers solve
maxC1,l1,C2,l2
u(C1, l1) + βu(C2, l2)
subject to
C1 +C2
1 + r= w1(h − l1) + π1 − T1 +
w2(h − l2) + π2 − T2
1 + r
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 3 / 30
The Representative Consumer
Optimality conditions:
Within-period decisions:
{ul (C1,l1)uC (C1,l1) = MRSC1,l1 = w1
ul (C2,l2)uC (C2,l2) = MRSC2,l2 = w2
Intertemporal decisions:uC (C1, l1)
βuC (C2, l2)= MRSC1,C2 = 1 + r
C1 +C2
1 + r= w1(h − l1) + π1 − T1 +
w2(h − l2) + π2 − T2
1 + r
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 4 / 30
Current Labor Supply
Factors affecting current labor supply, Ns1 = h − l1:
1 The current real wage: We will assume that the substitution effectdominates, i.e. ↑ w1 ⇒↓ l1 ⇔↑ Ns
1 .
2 The real interest rate: Intertemporal substitution of leisure just asfor consumption. Note that our existing solution conditions imply
ul(C1, l1)
βul(C2, l2)= MRSl1,l2 =
w1(1 + r)
w2
We will assume that the substitution effect dominates, i.e.↑ r ⇒↓ l1 ⇔↑ Ns
1 . Note that higher future wages ↑ w2 ⇒↑ l1 ⇔↓ Ns1 .
3 Lifetime wealth: Consumption and leisure are normal goods, so higherlifetime wealth (↑ π, ↓ T , etc.) causes reduced labor supply.
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 5 / 30
Labor Supply
NS(rL) NS(rH) NS (r)
NS (r) H L
Higher real interest rates increase labor supply (left) while higherlifetime wealth decreases labor supply (right).
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 6 / 30
Consumption Demand
Cd(r)
Cd(rH )
Cd(rL ) Slope = MPC
The marginal propensity to consume measures how muchconsumption increases when aggregate income Y increases by 1.
Higher real interest rates reduce current consumption (right).
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 7 / 30
The Representative Firm
Output Y1 = z1F (K1,N1) and Y2 = z2F (K2,N2).
The firm invests some of its output in capital accumulation:
K2 = (1− d)K1 + I1
Profits π1 = Y1 − w1N1 − I1 and π2 = Y2 − w2N2 + (1− d)K2.
The representative consumer owns the firm and receives profits asdividend income.
The firm maximizes the present value of dividend income,V = π1 + π2
1+r .
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 8 / 30
The Representative Firm
The firm solves
maxN1,I1,N2
z1F (K1,N1)− w1N1 − I1
+z2F (
K2︷ ︸︸ ︷(1− d)K1 + I1,N2)− w2N2 + (1− d)
K2︷ ︸︸ ︷[(1− d)K1 + I1]
1 + r
Optimality conditions:
Within-period decisions:
{z1FN(K1,N1) = MPN(K1,N1) = w1
z2FN(K2,N2) = MPN(K2,N2) = w2
Investment decision: z2FK (K2,N2)︸ ︷︷ ︸MPK (K2,N2)
−d = r
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 9 / 30
Optimal Investment Schedule
r =
Re
al In
tere
st R
ate
r =
Re
al In
tere
st R
ate
I1 = Investment in New Capital I1 = Investment in New Capital
rL
I1 H
z2FK(K2,N2) – d
rH
I1 L
z2FK(K2,N2) – d
H
z2FK(K2,N2) – d
L
Diminishing MPN and MPK ⇒ downward sloping I d and Nd .
Higher expected TFP or lower current capital cause increased I d .
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 10 / 30
Investment and Credit Market Imperfections
Suppose the borrowing interest rate is r + x , where the spread x is thedefault premium from limited commitment/asymmetric information.
Firms that borrow choose investment to satisfy
z2FK (K2,N2)− d = r + x ⇔ r = z2FK (K2,N2)− d − x
r =
Re
al In
tere
st R
ate
I1 = Investment in New Capital
z2FK(K2,N2) – d – xL
z2FK(K2,N2) – d – xH
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 11 / 30
Investment and the Interest Rate Spread
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 12 / 30
Investment and the Interest Rate Spread
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 13 / 30
Competitive Equilibrium
A competitive equilibrium is prices r , w1, w2; household allocationsC1, Ns
1 , C2, Ns2 ; firm allocations K1, Nd
1 , I1, Nd2 ; and allocations for
the government G1, G2, T1, T2 such that:1 C1, Ns
1 , C2, and Ns2 solve the household’s optimization problem.
2 Nd1 , I1, and Nd
2 maximize discounted profits V = π1 + π2
1+r , given K1.
3 The government’s budget constraint is satisfied: G1 + G2
1+r = T1 + T2
1+r .
4 Labor market clearing: Nd1 = Ns
1 and Nd2 = Ns
2 .
5 Credit market clearing: Sp1 + Sg
1 = 0⇔ Sp1 = B1 where
Sp1 = w1N
s1 + π1 − T1 − C1 and B1 = −Sg
1 = G1 − T1.
6 Goods market clearing: C1 + I1 + G1 = z1F (K1,Nd1 ) and
C2 + G2 = z2F (K2,Nd2 ) + (1− d)K2 where K2 = (1− d)K1 + I1.
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 14 / 30
Walras’ Law
Walras’ law states that we have a redundant market clearingcondition, i.e. (1) – (3) + any two of (4) – (6) automatically implythe third market clearing condition.
Here we show (1) – (3), (4), and (6) ⇒ (5). From (6) and (2),
C1 + I1 + G1 = z1F (K1,Nd1 )⇒ C1 = z1F (K1,N
d1 )− I1 − G1
⇒ C1 = w1Nd1 + z1F (K1,N
d1 )− w1N
d1 − I1︸ ︷︷ ︸
π1
−G1
Using Ns1 = Nd
1 from (4), the household budget constraint from (1),and B1 = G1 − T1 from (3) gives
C1 = w1Ns1 + π1 − (B1 + T1)⇒ B1 = w1N
s1 + π1 − T1 − C1︸ ︷︷ ︸
Sp1
⇒ B1 = Sp1
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 15 / 30
Equilibrium Conditions
Household optimality:
Within-period decisions:
ul (C1,h−Ns
1 )
uC (C1,h−Ns1 )
= w1
ul (C2,h−Ns2 )
uC (C2,h−Ns2 )
= w2
Intertemporal decisions:uC (C1, h − Ns
1 )
βuC (C2, h − Ns2 )
= 1 + r
C1 +C2
1 + r= w1N
s1 + π1 − T1 +
w2Ns2 + π2 − T2
1 + r
Profit maximization:
Within-period decisions:
{z1FN(K1,N
d1 ) = w1
z2FN(K2,Nd2 ) = w2
Investment decision: z2FK (K2,Nd2 ) − d = r
Government budget constraint: G1 + G21+r
= T1 + T21+r
.
Market clearing: Nd1 = Ns
1 and Nd2 = Ns
2 (labor); C1 + I1 + G1 = z1F (K1,Nd1 ) and
C2 + G2 = z2F (K2,Nd2 ) + (1 − d)K2 (goods).
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 16 / 30
Labor Market Clearing and Output Supply
Higher r causes Ns1(w1; r) to shift right. Market clearing N1 increases,
causing higher Y1 = z1F (K1,N1). Thus, Y s1 (r) is upward-sloping.
N1
N1
Y1
w1 N1(w1)
d
w1 L
w1 H
N1 L N1 H
N1 L
N1 H
Y1 H
Y1 L
Y1 (r) s
rL
rH
Y1 L Y1
H
r
Y1
N1(w1 ; rL) s
N1(w1 ; rH) s
Y1 = z1F(K1 ,N1)
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 17 / 30
Goods Market Clearing and Output Demand
Higher r reduces the demand for goods because of lower C1(Y1; r)and I1(r). Thus, the goods market clearing Y1 decreases.
Output demand curve Y d1 (r) is downward-sloping.
De
man
d f
or
Per
iod
1 G
oo
ds
r =
Re
al In
tere
st R
ate
Y1 = Period 1 Income Y1 = Period 1 Income
Y1 (r) d
Y1 H
Y1 L
C1(Y1 ; rH) + I1(rH) + G1
C1(Y1 ; rL) + I1(rL) + G1
rH
rL
Y1 H
Y1 L
Y1 H
Y1 L
45° line
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 18 / 30
Equilibrium
w
1 =
Pe
rio
d 1
Re
al W
age
r =
Re
al In
tere
st R
ate
N1 = Period 1 Employment Y1 = Period 1 Output
r
Y1 (r) d
N1(w1 ;r) s
N1 Y1
w1
Y1(r) s N1(w1)
d
By construction, Nd1 = Ns
1 everywhere on Y s1 and C1 + I1 + G1 = Y1
everywhere on Y d1 .
Thus, all markets clear when r adjusts to cause Y s1 (r) = Y d
1 (r).
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 19 / 30
Effects of Higher Government Spending – Output Demand
Higher G1 ⇒ higher T1 and/or T2. Suppose Y1 ⇒ Y1 + ∆Y .
Net change in household disposable income ∆Y −∆T = ∆Y −∆G .
C1(Y1; r) + MPC (∆Y −∆G )︸ ︷︷ ︸C1(Y1+∆Y−∆G ;r)
+I1(r) + G1 + ∆G = Y1 + ∆Y ⇒ ∆Y = ∆G
De
man
d f
or
Per
iod
1 G
oo
ds
r =
Re
al In
tere
st R
ate
Y1 = Period 1 Income Y1 = Period 1 Income
Y1 (r) d
Y1 Y1 + ΔY
C1(Y1 ; r) + I1(r) + G1
C1(Y1 ; r) + MPC(ΔY – ΔG) + I1(r) + G1 + ΔG
Y1
45° line
Y1 + ΔY
Y1 (r) = Y1 (r) + ΔG d ~ d
C1(Y1 + ΔY – ΔG; r)
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 20 / 30
Effects of Higher Government Spending – Output Supply
The negative wealth effect of ∆T causes Ns1 to shift to the right.
N1
N1
Y1
w1 N1(w1)
d
w1
N1
N1
Y1
Y1 (r) s
r
Y1
r
Y1
N1(w1 ; r) s
Y1 = z1F(K1 ,N1)
Y1 ~
N1 ~
N1 ~
Y1 ~
w1 ~
N1(w1 ; r) s ~
Y1 (r) s ~
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 21 / 30
Effects of Higher Government Spending – Equilibrium
w1 =
Pe
rio
d 1
Re
al W
age
r =
Re
al In
tere
st R
ate
N1 = Period 1 Employment Y1 = Period 1 Output
r
Y1 (r) d
N1(w1 ;r) s
N1 Y1
w1
Y1(r) s N1(w1)
d
Y1 (r) s ~
Y1 (r) = Y1 (r) + ΔG d ~ d
Y1 + ΔG
N1(w1 ;r) s ~
N1(w1 ;r ) s ~ ~
w1 ~
r ~
Y1 ~ N1
~
Small negative wealth effect on Ns1 ⇒ small relative increase in Y s
1 .
Output Y1 increases, but ∆Y < ∆G because higher r crowds out C1
and I1. Summary: ↑ r , ↑ Y1, ↓ C1, ↓ I1, ↑ N1.
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 22 / 30
Effects of Lower Initial Capital – Output Demand
Lower K1 increases MPK next period, z2FK ((1− d)K1 + I1,N2) ↑.
Firms increase I1, driving MPK back down until MPK − d = r . Thus,Y d
1 shifts to the right.
De
man
d f
or
Per
iod
1 G
oo
ds
r =
Re
al In
tere
st R
ate
Y1 = Period 1 Income Y1 = Period 1 Income
Y1 (r) d
Y1
C1(Y1 ; r) + I1(r) + G1
C1(Y1 ; r) + I1(r) + G1
Y1
45° line
Y1 (r) d ~
~
Y1 ~
Y1 ~
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 23 / 30
Effects of Lower Initial Capital – Output Supply
Lower K1 decreases MPN this period, z1FN(K1,N1) ↑.
Labor demand Nd1 shifts to the left, causing Y s
1 to shift to the left.
N1
N1
Y1
w1
N1(w1) d
w1
N1
N1
Y1
Y1 (r) s
r
Y1
r
Y1
N1(w1 ; r) s
Y1 = z1F(K1 ,N1)
Y1 ~
N1 ~
N1 ~
Y1 ~
w1 ~
Y1 (r) s ~
N1(w1) d ~
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 24 / 30
Equilibrium Effects of Lower Initial Capital: Case 1
w1 =
Pe
rio
d 1
Re
al W
age
r =
Re
al In
tere
st R
ate
N1 = Period 1 Employment Y1 = Period 1 Output
r
Y1 (r) d
N1(w1 ;r) s
N1 Y1
w1
N1(w1) d
Y1 (r) s
Y1(r) s ~
Y1 (r) d ~
r ~
Y1 ~
N1(w1 ;r ) s ~ N1(w1)
d ~
N1 ~
w1 ~
Summary: ↑ r , ↑ I1, ambiguous Y1, N1, and C1.
Labor supply shifts to the right in the new equilibrium because of thehigher interest rate, i.e. Ns
1(w1; r̃) > Ns1(w1; r).
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 25 / 30
Equilibrium Effects of Lower Initial Capital: Case 2
w1 =
Pe
rio
d 1
Re
al W
age
r =
Re
al In
tere
st R
ate
N1 = Period 1 Employment Y1 = Period 1 Output
r
Y1 (r) d
N1(w1 ;r) s
N1 Y1
w1
N1(w1) d
Y1 (r) s Y1(r)
s ~
Y1 (r) d ~
r ~
Y1 ~
N1(w1 ;r ) s ~
N1(w1) d ~
N1 ~
w1 ~
Output increases above because the demand effect from higherinvestment outweighs the supply effect from lower labor demand.
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 26 / 30
Effects of Higher Present TFP z1 – Output Supply
Higher z1 increases MPN this period.
Labor demand Nd1 increases, causing Y s
1 to shift to the right.
N1
N1
Y1
w1
N1(w1) d
N1
Y1 (r) s
r
Y1
r
Y1
N1(w1 ; r) s
Y1 = zLF(K1 ,N1)
Y1 ~
N1 N1 ~
N1 ~
Y1
Y1 ~
w1
w1 ~
Y1 (r) s ~
N1(w1) d
~
Y1 = zHF(K1 ,N1)
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 27 / 30
Effects of Higher Present TFP z1 – Equilibrium
w1 =
Pe
rio
d 1
Re
al W
age
r =
Re
al In
tere
st R
ate
N1 = Period 1 Employment Y1 = Period 1 Output
r
Y1 (r) d
N1(w1 ;r) s
N1 Y1
w1
N1(w1) d Y1 (r)
s
Y1(r) s ~
r ~
Y1 ~
N1(w1 ;r ) s ~
N1(w1) d ~
N1 ~
w1 ~
No effect on C1(Y1; r), I1(r), or G1, and thus no change in Y d1 .
Summary: ↓ r , ↑ Y1, ↑ C1, ↑ I1, ↑ N1. Labor supply shifts to the left,i.e. Ns
1(w1; r̃) < Ns1(w1; r).
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 28 / 30
Effects of Higher Future TFP z2 – Output Demand
Higher z2 increases MPK and MPN next period.
Firms increase I1 because of higher MPK . Also, higher MPN increasesw2, causing C1 to increase (C smoothing). Thus, Y d
1 shifts right.
De
man
d f
or
Per
iod
1 G
oo
ds
r =
Re
al In
tere
st R
ate
Y1 = Period 1 Income Y1 = Period 1 Income
Y1 (r) d
Y1
C1(Y1 ; r) + I1(r) + G1
C1(Y1 ; r) + I1(r) + G1
Y1
45° line
Y1 (r) d ~
~
Y1 ~
Y1 ~
~
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 29 / 30
Effects of Higher Future TFP z2 – Equilibrium
w1 =
Pe
rio
d 1
Re
al W
age
r =
Re
al In
tere
st R
ate
N1 = Period 1 Employment Y1 = Period 1 Output
r
Y1 (r) d
N1(w1 ;r) s
N1 Y1
w1
N1(w1) d
Y1 (r) s
Y1 (r) d ~
r ~
Y1 ~
N1(w1 ;r ) s ~
N1 ~
w1 ~
No effect on Nd1 or Ns
1 , and thus no change in Y s1 .
Summary: ↑ r , ↑ Y1, ↑ N1, ↑ I1, ambiguous C1 because higher w2 andY1 (+) and higher r (−). Labor supply shifts to the right.
Econ 3307 (Baylor University) A Real Intertemporal Model with Investment Fall 2013 30 / 30