The Hall-Jorgenson Model of Investment
Hall & Jorgenson (1967) k \kap k
y = f ( k ) \begin{gathered}\begin{aligned}
\inc & = \fFunc(\kap)
\end{aligned}\end{gathered} y = f ( k ) and which obtains its capital k \kap k ϰ t \kapRent_{t} ϰ t
In period t t t
max k t k t α − ϰ t k t \max_{\kap_{t}} ~ \kap_{t}^{\kapShare}-\kapRent_{t}\kap_{t} k t max k t α − ϰ t k t yielding first order conditions
f ′ ( k t ) = ϰ t α k t α − 1 = ϰ t ( y t / k t ) α = ϰ t k t = ( y t / ϰ t ) α . \begin{gathered}\begin{aligned}
\fFunc^{\prime}(k_{t}) & = \kapRent_{t}
\\ \kapShare \kap_{t}^{\kapShare-1} & = \kapRent_{t} \\
(\inc_{t}/\kap_{t}) \kapShare & = \kapRent_{t} \\
\kap_{t} & = (\inc_{t}/\kapRent_{t})\kapShare .
\end{aligned}\end{gathered} f ′ ( k t ) α k t α − 1 ( y t / k t ) α k t = ϰ t = ϰ t = ϰ t = ( y t / ϰ t ) α . This equation says the level of capital is always instantly adjusted to y t \inc_{t} y t ϰ t \kapRent_{t} ϰ t
What determines the cost of capital? In the simple case with no taxes and no capital market frictions of any kind, an investor must be indifferent between putting his money in the bank and earning interest at rate r \rfree r ϰ t \kapRent_{t} ϰ t
The purchase price at which capital goods can be bought at date t t t
P t − purchase price of one unit of capital , \begin{gathered}\begin{aligned}
\Price_{t} & - & \text{purchase price of one unit of capital},
\end{aligned}\end{gathered} P t − purchase price of one unit of capital , and in continuous time, the rate of change of P t \Price_{t} P t P ˙ t \dot{\Price}_{t} P ˙ t δ \depr δ continuous time purchase-and-rent strategy is
ϰ t − δ P t + P ˙ t − Income from renting, minus loss from depreciation plus capital gain from the change in price of capital . \begin{gathered}\begin{aligned}
\kapRent_{t}-\depr \Price_{t}+\dot{\Price}_{t} & - & \text{Income from renting, minus loss from depreciation} \\
& \text{plus capital gain from the change in price of capital}.
\end{aligned}\end{gathered} ϰ t − δ P t + P ˙ t − plus capital gain from the change in price of capital . Income from renting, minus loss from depreciation Thus, the no-arbitrage condition is
r P t = ϰ t − δ P t + P ˙ t ( r + δ ) P t = ϰ t + P ˙ t . \begin{gathered}\begin{aligned}
\rfree \Price_{t} & = \kapRent_{t} - \depr \Price_{t} + \dot{\Price}_{t}
\\ (\rfree+\depr) \Price_{t} & = \kapRent_{t}+\dot{\Price}_{t} .
\end{aligned}\end{gathered} r P t ( r + δ ) P t = ϰ t − δ P t + P ˙ t = ϰ t + P ˙ t . where the left-hand side is the return from putting money in the bank at rate r \rfree r
Now to simplify our lives we will assume constant capital goods prices, P ˙ t = 0 \dot{\Price}_{t}=0 P ˙ t = 0 ϰ t \kapRent_{t} ϰ t (6) (3)
k t = α y t / ϰ t = α y t / ( r + δ ) P t . \begin{gathered}\begin{aligned}
\kap_{t} & = \kapShare \inc_{t}/\kapRent_{t} \\
& = \kapShare \inc_{t}/(\rfree+\depr)\Price_{t}.
\end{aligned}\end{gathered} k t = α y t / ϰ t = α y t / ( r + δ ) P t . Now let’s introduce taxes, defined as follows:
τ − corporate tax rate ( ≈ 0.34 in US) ζ − investment tax credit (sometimes 10 percent, sometimes 0) \begin{gathered}\begin{aligned}
\taxCorp & - & \text{corporate tax rate ($\approx 0.34$ in US) }
\\ \itc & - & \text{investment tax credit (sometimes 10 percent, sometimes 0) }
\end{aligned}\end{gathered} τ ζ − − corporate tax rate ( ≈ 0.34 in US) investment tax credit (sometimes 10 percent, sometimes 0) The net, discounted, after-tax price of capital to the firm is[1]
P ^ t = ( 1 − ζ ) P t . \begin{gathered}\begin{aligned}
\hat{\Price}_{t} & = (1-\itc) \Price_{t}.
\end{aligned}\end{gathered} P ^ t = ( 1 − ζ ) P t . Now let’s rewrite the arbitrage equation (6) ϰ t \kapRent_{t} ϰ t ( 1 − τ ) (1-\taxCorp) ( 1 − τ )
( r + δ ) P ^ t = ( 1 − τ ) ϰ t + P ^ ˙ t . \begin{gathered}\begin{aligned}
(\rfree+\depr) \hat{\Price}_{t} & = (1-\taxCorp) \kapRent_{t}+\dot{\hat{\Price}}_{t}.
\end{aligned}\end{gathered} ( r + δ ) P ^ t = ( 1 − τ ) ϰ t + P ^ ˙ t . If we simplify again by assuming that P ^ ˙ t = 0 \dot{\hat{\Price}}_{t}=0 P ^ ˙ t = 0
ϰ t = ( r + δ ) P t ( 1 − ζ ) / ( 1 − τ ) . \begin{gathered}\begin{aligned}
\kapRent_{t} & = (\rfree+\depr)\Price_{t}(1-\itc)/(1-\taxCorp)
.
\end{aligned}\end{gathered} ϰ t = ( r + δ ) P t ( 1 − ζ ) / ( 1 − τ ) . Note that so far we have not derived a formula for investment - we have derived a formula for the level of the capital stock. But net investment is just the difference between the capital stock in periods t t t t − 1 t-1 t − 1
i t − 1 = k t − k t − 1 + δ k t − 1 = ( Δ y t ϰ t ) α + δ k t − 1 \begin{gathered}\begin{aligned}
\inv_{t-1} & = \kap_{t}-\kap_{t-1}+\depr \kap_{t-1} \\
& = \left(\Delta \frac{\inc_{t}}{\kapRent_{t}}\right)\kapShare+\depr \kap_{t-1}
\end{aligned}\end{gathered} i t − 1 = k t − k t − 1 + δ k t − 1 = ( Δ ϰ t y t ) α + δ k t − 1 (where we neglect some minor complications having to do with the distinction between continuous and discrete time).
Hall, R. E., & Jorgenson, D. (1967). Tax Policy and Investment Behavior. American Economic Review , 57 .