Goods Market Equilibrium
(Closed Economy)
Output Supply:
Past Investment  K 

Knowledge, etc.  A   Y
Labor Market  N 
Output Demand:
Consumptio n : Cd


d
d
d
Investment : I

C

I
G

Government Spending : G 
Supply  Demand
Y  Cd  Id  G
Y  Cd  G  Id
Sd  I d
Desired Savings  Desired Investment
Supply of L.F.  Demand for L.F.
Equilibrating Factor: Real Interest Rate
Savings Incentive: Real Interest Rate
Real Interest Rate   Sd 
Investment Incentive: Real Interest Rate
Real Interest Rate   Id 
Real Interest Rate
Sd
r
Id
SI
S, I
Full Characterization of Equilibrium
Y : Labor Market, Production Function
I : Sd  Id
G : Government Decision
C  Y -I-G
Savings Choice
Sdpvt

PDI : Y  (T  TR  INT) 
 Cd


 d

S
Sgov : T  TR  INT  G

Sdpvt
Factors Affecting Sd
Movements Along Sd :Real Interest Rate
Shifts in Sd
Variable
Change
Current Income
Y
Expected Future Income
Yf 
Permanent Income
Y  Y f
Wealth
Wealth
Government Spending
Current
G
Expected Future
Gf 
Permanent
G, G f 
Taxes (Traditional)
T
Taxes (Ricardian Equiv.)
T
Shift
Right
Left
None
Left
Left
Right
None
Right
None
Real Interest Rate
d
S
Sd
Shifts in Sd
Income: Y, Yf
Permanent Income
Wealth
Government Spending: G, Gf, Permanent
Taxes
Firms' Investment Choice
 N : Labor Demand Choice
Factors of Production 
 K : Best Stock of Capital
Capital Stock Choice
Marginal Benefit of Capital: MPK f
Tax Adjusted Marg. Benefit: 1   MPK f
Marginal Cost Considerations
p K : real price of a machine
d: physical rate of depreciation
r: real interest rate
User Cost of Capital: r  d p K
Net of Tax Marginal Benefit  User Cost
1  MPK f  r  d p K
(r  d)p K
 MPK 
(1  )
f
(r  d)p K
(1   )
Reduce K
MPK f
Increase K
f
K
Tax Adj. User Cost, MPKf
Tax Adj. User Cost
(r  d)p K
(1   )
K*
MPK f
Kf
Optimal Investment Choice
Capital, K, is changed through choice of I.
K f  K
Net Investment: 
 I  dK
I  dK  K f  K
I  K f  (1  d)K
Now Set K f  K* .
I d  K *  (1  d)K
Movement Along I d : r  K*  Id 
Shifts in I d : Other Changed Circumstances
Variable
Effective Tax Rate
Expected MPK
Future Productivity
Current Capital Stock
Change

MPK f 
Af ↑
K
Shift
Left
Right
Right
Left
Real Interest Rate
Id
Id
Shifts in Id
Effective Tax Rate
Expected MPKf
Future Productivity
Current Capital Stock
Curve Shifters
ND
NS
Wealth
(-)
wf
(-)
Demographics
(?)
A
(+)
K
(+)
Sd

Y


(

)


Yf
(-)
Wealth
(-)
G
(-)
Gf
(+)
T
(?)
Id
Af
(+)
K
()
MPKf
(+)

(-)
Numerical Problem 4.7
MPK f  20  0.02 K f : K f  Future Capital
d  20%
K  900 : Current Capital
pK  1
  50%
C d  100  0.5Y  200r
G  200
Y  1,000
a.)
Suppose r  10%  0.10
uc
(r  d ) pK (0.10  0.20) 1


 0.6
(1   )
(1   )
(1  0.50)
Set MPKf  Tax-Adj. UC of Capital  K *
20  0.02  K *  0.6  K *  970
Net Investment Definition: K f  K  I  dK
Set K f  K *  970
I d  K f  (1  d ) K  970  (1  0.2)900  250
Therefore, I d  250
b.)
For Arbitrary r:
(r  d ) pK
MPK 
(1   )
(r  d ) pK (r  0.2) 1

 2r  0.4
(1   )
(1  0.5)
f
20  0.02 K *  2r  0.4
0.02 K *  20  2r  0.4
K *  980  100r
Now we equate K f  K *
We also recall: K f  K  I  dK  I  K f  (1  d ) K
Desired gross investment, I d , is the amount of investment
necessary to make K f  K *.
Therefore:
I d  K f  (1  0.2) K
 980  100r  (0.8)(900)
I d  260  100r
Next, we know that:
C d  100  (0.5)Y  200r
Set Y equal to full employment, 1000.
C d  100  500  200r
Therefore: C d  600  200r
By definition, S d  Y  C d  G
Therefore: S d  1000  600  200r   200
S d  200  200r
Finally, equate I d and S d
I d  260  100r  200  200r  S d
260  200  300r
60  300r
60
r
 20%
300
The equilibrium real interest rate; r  20% .
Lastly, we can calculate equilibrium I and C.
I  260  100r
 260  20
I  240
C  600  200r
 600  40
C  560
Note that C  I  G  560  240  200  1,000.
Recall that Y  1,000 .
Therefore Y  C  I  G
w
NS
w1
ND
N
N1
r
Sd
r1
Id
S1  I 1
S, I