Economic Modelling
Lecture 5
Optimal Investment (Micro-foundation)
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1
Profit Maximisation Problem of Firms
Marginal Product of Capital = User Cost of Capital
y
y=f(k)
K


y
1


P
2 k
 P1k k 
Max  
1  r 
1 r
Subject to:
y  f (k )  k 
k
Investor Compare user cost of capital with its productivity
r     
K
1  r     K 
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 1

k
MPK=
2
Investment Decision Analysis
(See Problem 16.3 in Blanchard)
Breaks even point:


r   K  R
V  18000
r 
The value of this investment project:
Cost of the Project (K):
Capital
r
Earning = 18000
100,000
100000
0.05
100000
0.1
100000
0.15
0.08
0.08
0.08
C=(r+d)*K
13000
18000
23000
R (Earnings)
18000
18000
18000
138461.5
100000
78260.87
d
PV
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Analysis of Earnings (R) and Cost (C) from an Investment Project
K = 100000; d = 0.08; R (Earning) =18000
C =(r+d)*K
23,000
C>R
Cost
And
Earning
18,000
Break Even
Earning (R)
C<R
13,000
0.05
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0.1
0.15
r
4


Max   PK L  wL  rK
K
L
Low interest rate induces
producers to substitute out
labour by capital
K
 w

L
 r

  1
 PK L  w  0
L

 1 
 PK L  r  0
K
o
w
r
Producer’s like to maximise profit given factor prices, r and w.
They use more capital relative to labour if wage rate is higher.
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Marginal Productivity Theory of Investment calculations
Y  F K 
Output depends on capital stock:
K t  K t 1 1     I
t
Capital stock depends on investment:
Investment depends on expected profits:

 1
I V   e   I 
e 
t
t 1

 
1 r
1  rt
t



1
1   te  1  ....  
1  r e 

t 


 
 Yt
 t  
 Kt
Expected profits depends on productivity of capital:



Producers invest more until the marginal product of capital equals the user cost of capital:


MPK  r     K P1
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k
6
Derivation of the Marginal Productivity = User Cost of Capital Condition

F K 
1   P2K K
k

P K 
1  r  1
1 r
Producer’s Problem:
Optimality Condition:
Implication:
K


1


P
 F ' K 
2 0

 Pk 
K 1  r  1
1 r
MPK  1  r P1k  1   P2K



MPK  1  r   1    1   K P1


MPK  r     K P1
Assumptions:
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K
P2K
 K 1
P1
k
k
 K  0
7
Role of Investment Tax Credit in Promoting Investment
Why Manufacturers Lobby for a Tax Credit?
MPK
r     
K
1  r     K 
MPK =
0
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K1
K2
k
K
8
 1
Optimal Capital Stock for the Car Company
The user cost of capital :
i  
K
= 6% +3%-3% =6%
F K1   K 
Let
Marginal product of capital:
Optimal Investment condition:
F ' K1   K  1  0.75K 0.751

P.F ' K1   P1k i     k

80000.75K 0.751  20006%  3%  3%
8.0.75K
K
0.25
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0.25
 3 


 0.06 
 26%
6K
K  50
0.25
 26%
4
= 6.25 million
9
Role of Financial Market in Optimal Saving and Investment
Saving
r=i-
Investment
0
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S*, I*
Saving and Investment
10
Elasticity of Substitution is 1 in Cobb-Douglas Function
(Factors are paid according to their marginal productivity)


Yt  At Kt Lt



1
rK

K
L
.
K
 
Y
Y
rK  wL  1
Y
Y


1


K
L
L
wL


Y
Y
K L
K L
K L
K L
 KL 
 K L 1


w r 
K L


1




1




AK
L

AK
L


wr
K L




1




1
AK L
AK
L
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Business School,
 is the elasticity of substitution between capital and labour.
11
References
•
•
•
•
Blanchard (4,5,16)
Bank of England (2001) Financial Stability Review, www. Bankofengland.co.uk.
Cass, David, (1965) Optimum Growth in Aggregative Model of Capital
Accumulation, Review of Economic Studies, 32:233-240.
Lucas, Robert E. (1993) The Determinants of Direct Foreign Investment, World
Development, March 21:3, 391-406.
•
Modiogliani Franco, and Miller, Merton H. (1958) The Cost of Capital,
Corporation Finance and the Theory of Investment, AER, vol. XLVII,
June.
•
Levine, Ross and Sara Zervos (1998) “Stock markets, banks and economic
Growth” American Economic Review, 88, 537-58.
Ramsey, F.P. (1928) “A Mathematical Theory of Saving,” Economic Journal 38,
543-559.
Romer, Paul "Capital Accumulation in the Theory of Long Run Growth" in Barro R.
J. (1989) ed. Modern Business Cycle Theory, Harvard University Press.
•
•
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Business School,
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