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Example 19 (A simulation model of the energy market)
Source: Will oil markets tighten again? Richard J. Gilbert and Knut Anton Mork;
Equations of energy-macro model; 134 - 141
Production possibility constraint
Long-run unit cost function (translog):
ˆ = a + a ln( w ) + a ln v + a ln p + 1 b (ln( w ))2 + b ln( w )ln v
(1) ln Φ
0
L
K
E
LL
LK
P
2
P
P
+ bLEln(
w
1
1
)ln p + bKK(ln v)2 + bKE(ln v)ln p + bEE(ln p)2
P
2
2
Short-run cost function, approximating putty-clay:
(2) Φ (
w
1−α w
, v, p) = γ 0 [ a1 v + (1 - a1) p ] ( )α
P
P
Combined cost function:
(3) Φ(
w
w
g
w
1−g
, v, p) = [ Φ( , v , p]) , [ Φ( , v , p] )
P
P
P
Labor market
Cost-of-living escalation and employer`s share of the payroll tax for predetermined
wages:
(4) w ∗ = (1 + τ w )[1 + γ
( P− P )
]w
P
Specification of partial wage rigidity (wage aggregation):
ˆ f w∗(1−f)
(5) w ∗ = w
Correction for cyclical variation in labor productivity:
(6) w
= w0(
Employment:
w * h(1-f)
)
,h>0
ˆ
w
(7) Lˆ
=(
w
)Φ w / P( w , v, p) ⋅ Y
ˆ
w
P
2
Supply of labor:
w
w
, v, p) ⋅ Y
∗ )Φ w / P (
w
P
(8) L
=(
(9) L
= f Lˆ + (1 - f) L
(10) Lˆ
= L 0e n , n > 1 given, L0 exogenous;
Unemployment rate: (11) u
= a11,1 + a11,2
Lˆ − L
ˆ
L
Capital market
Capital aggregation:
(12) K
= Kˆ b K1− b
Price aggregation:
(13) v
= vˆ b v∗1−b
Demand for each category of capital:
(14) Kˆ
v
w
= ( )Φ v ( , v , p ⋅)Y
vˆ
P
(15) K
v
w
= ( ∗ )Φ v( , v , p⋅ )Y
P
v
(16) K t
= It + (1− δ)Kt−1 , 0 < δ < i, K0 given
(17) Kˆ t
b
b − bt −1
= ˆI t + (1− δ){ t −1 Kˆ t−1 + t
Kt−1}
bt
bt
Aggregate investment:
(18) I
= bˆI + (1− b)I
Demand for energy:
(19) E
w
= Φ p ( , v , p ⋅) Y
P
Market for goods and services
Consumer behavior:
(20) C
= c ⋅ Lˆ
Trade balance:
(21) X
= (X + X0 )
(22) PV[ X0
C
C0
C
, X exogenous, X0 endogenous
C0
] = PV (p 0M)
Distribution of output:
(23) Y
= C + I + X + G + Z, G exogenous
Distribution of real GNP:
(24) GNP = Y − (p 0 − τM )M × a 24,1 − Z
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Price equation:
[
(25) (1− τs )P = [ Φ(w 0 , P⋅ v , P⋅ p)] Φ( w0 ,P ⋅ v ,P⋅ p)]
a
1−a
, a given
Money market
Money market equilibrium:
(26) ln (
P⋅Y
) = a26,1 + a26,2r, H = H0em, m > 0
H
Arbitrage condition between nominal interest rate and real return to new capital:
(27) rt =
vˆ t
− δ − θ +ln [(1 − d t+1 )Pt+1 ] − ln[(1 − d t )Pt ]
(1− d t )
Domestic energy supply
Marginal cost of extraction:
(28) c(R, S, t; ν) = c ν1 R νe µ + c 2e−ρ S
Total cost of domestic energy supply:
(29) Z = (
1
)cν R ν+ 1e µ 2 + c 2 Re −ρS , p < p
1+ v 1
Arbitrage condition for domestic energy supply:
∂c
(30) (vˆ t − δ)[ pt − τ pt − c(R t , St , t ; vt )] = p t+1 − τ p,t +1 − pt + τ pt −
(R , S , t ; tv)
∂R t t t
∂c
∂c
(Rt+1 - Rt) −
(R t ,S t ,t; vt ) (v t+1 − vt ) − (R t , St , t ; v
t)
∂vt
∂t
Decumulation of resource stock:
(31) St = St-1 - Rt, S0 given
Resource depletion: terminal resource stock, S , determined by:
(32) S =
1
c
ln( 2 )
ρ
p
World oil market
U.S. demand for imported oil:
(33) M = cQ(E - R), if p < p
Aggregate production function for the nonenergy sectors of non-U.S., non-OPEC
countries:
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(34) Y F = [( ηA) a + B a (R F + MF )a
]
1/a
, a = 1-1/σ
Foreign demand for energy (profit maximization):
(35) p 0 − τ M + τ FM = B 1−σ (R F + M F )1/σ
Domestic energy supply in non-U.S., non-OPEC countries:
(36) RF = R F
Total world demand for OPEC oil exports:
(37) Q = M + MF
OPEC price reaction function (for the first five years after the shock):
k −D
(38) p - τM = a 38,1 + a38,2
k−D− Q
Translation from domestic price of oil to overall domestic price of energy:
∆p
∆p0
(39)
= a 39,1 ( 0 ) , p, p 0 exogenous
p
p
Price of energy after six years:
(40) p
= p 0eµ 3 ,p ≤ p
µ 3 = ln(a40,1), a40,1 > 1
List of variables
C
= aggregate consumption (in billion 1972 dollars)
D
= disruption of OPEC capacity (in mmbd)
d
= investment tax credit (rate)
E
= domestic demand for energy (in 1972 billion dollars)
G
= government demand for goods and services (in 1972 billion dollars)
GNP = real gross national product (in 1972 billion dollars)
H
= money supply (in billion dollars)
I , ˆI = level of investment for categories of capital whose levels are determined
before, after the shock (in 1972 billion dollars)
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I
= Aggregate investment (in 1972 billion dollars)
K , Kˆ = level of capital stock for categories of capital whose levels are determined
before, after the shock (in 1972 billion dollars)
K
= level of aggregate capital stock (in 1972 billion dollars)
k
= OPEC capacity (in mmbd)
L, Lˆ
= Actual, normal employment (in millions of man-years)
L
= employment corresponding to predetermined component of wages ( w ∗ )
(in millions of man-years)
M
= U.S. oil-import demand (in mmbd)
MF
= oil-imported demand for non-U.S. non-OPEC countries (in mmbd)
P
= overall price level in the absence of a shock ( same units 1972)
P
= expected overall price level in the absence of a shock ( same units as Pt)
p
= real domestic price of energy (index, equal to 1 for 1972)
p
= expected real domestic price of energy in the absence of a shock
(index, equal to 1 for 1972)
p
= backstop price of energy (index, equal to 1 for 1972)
p0
= domestic price of oil (in 1980 dollars per barrel)
PV(z) = present value of z
Q
= demand for OPEC oil by non-OPEC countries (in mmbd)
R
= U.S. domestic energy supply (in 1972 billion dollars)
RF
= domestic energy supply for non-U.S., non-OPEC countries
(in mmbd oil equivalents)
r
= nominal interest rate (rate p.a.)
S
= U.S. domestic energy resource stock (in 1972 billion dollars)
t
= time (years)
T
= date after which backstop energy technology is used in the model
u
= unemployment rate
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v∗
= real rental rate (shadow price) for categories of capital whose levels are
determined before the shock
vˆ
= real rental rate for categories of capital whose levels are determined after
the shock
v
= aggregate real rental rate of capital
v
= expected real rental rate of capital in the absence of a shock
w
= predetermined nominal wage rate (in thousands of dollars per man-year)
w∗
= predetermined nominal wage rate adjusted for employers´share of the
payroll tax and cost-of-living increases
thousands of dollars per man-year)
(in
w0
= aggregate nominal wage rate, for use in the price equation
(in thousands of dollars per man-year)
w
= aggregate nominal wage rate, adjusted for cyclical variation in labor
productivity, for use in factor-demand equations
thousands of dollars per man-year)
X
(in
= net U.S. exports of nonenergy goods (in 1972 dollars, deflated by Pt)
Y
= gross output of goods and services in the nonenergy sector of the U.S.
economy (in 1972 billion dollars)
YF
= gross output of goods and services in non-U.S., non-OPEC countries
1972 billion dollars)
Z
= gross real cost of U.S. energy production (in1972 billion dollars)
τM
= U.S. tariff on oil (1980 dollars per barrel)
τ FM
= foreign tariff on oil (1980 dollars per barrel)
τw
= payroll tax (rate of)
τs
= rate of sales tax for goods and services in the United States
τp
= tax on U.S. energy production
θ
= tax rate on capital in production
(in