$©$ September 21, 2020, Christopher D. Carroll HallJorgenson

The Hall-Jorgenson Model of Investment

Hall and Jorgenson (1967) consider the problem of a ﬁrm that produces output using capital $k$ as its only input,

y = f(k)

(1)

and which obtains its capital $k$ from a market in which a unit of capital can be rented for a unit of time at rate $ϰ t$ .

In period $t$ , the ﬁrm maximizes proﬁt,

max _{k_t} k_t^α − ϰ_tk_t

yielding ﬁrst order conditions

′ f (kt) = ϰt α− 1 αk t = ϰt (y ∕k ) α = ϰ t t t kt = (yt∕ϰt )α.

(2)

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 $r$ , and buying a unit of capital, renting it out at rate $ϰt$ , and then reselling it the next period.

The price at which capital goods can be bought at date $t$ is:

Pt − purchase price of one unit of capital,

(3)

and in continuous time, the rate of change of $Pt$ is $P˙t$ . Assume that capital depreciates geometrically at rate $δ$ . The net proﬁt from the continuous time purchase-and-rent strategy is

ϰt − δPt + P˙t − Income from renting, minus loss from depreciation plus capital gain from the change in price of capital.

Thus, the no-arbitrage condition is

˙ rPt = ϰt − δPt + Pt (r + δ )P = ϰ + ˙P . t t t

(4)

Now to simplify our lives we will assume constant capital goods prices, $P˙t = 0$ . Thus, substituting the value for $ϰt$ from (4) into (2) we have:

k = αy ∕ ϰ t t t = αyt∕ (r + δ)Pt.

(5)

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 )

The net, discounted, after-tax price of capital to the ﬁrm is¹

ˆPt = (1 − ζ)Pt.

(6)

Now let’s rewrite the arbitrage equation (4) taking account of taxes:

ˆ ˆ˙ (r + δ )Pt = (1 − τ)ϰt + Pt.

(7)

If we simplify again by assuming that $P˙ˆt = 0$ , we have

ϰt = (r + δ)Pt (1 − ζ )∕(1 − τ).

(8)

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 $t$ and $t − 1$ . Thus, the Hall-Jorgenson model of gross investment is

i = k − k + δk t− 1 ( t t)− 1 t− 1 yt = Δ --- α + δkt− 1 ϰt

(9)

(where we neglect some minor complications having to do with the distinction between continuous and discrete time).

References

Hall, Robert E., and Dale Jorgenson (1967): “Tax Policy and Investment Behavior,” American Economic Review, 57, Available at http://www.stanford.edu/~rehall/Tax-Policy-AER-June-1967.pdf.