Keynesian Investment Theory

Again, relies upon "animal spirits," or expectations. Not empirically successful. 

Neoclassical Investment

Uses the Hall/Jorgenson approach to emphasize the optimal stock of capital. (See handout). Helpful way to think about the capital investments: buying replacement cars as a car rental agency: price of the investment is price of a car, make the decision based on the cash flow that the car would generate, time is how long you own the car, integrating gives you the sum of the rentals (c and s), the car depreciates and is discounted. In general, the first equation on the handout is a continuous time version of the decision whether or not to buy the investment (here, a car) and the theory is that it should be greater or equal to the price of the investment. 
\(p_k (t) = \int \limits ^\infty _t e^{- r (s - t)} c(s) e^{- \delta (s - t)} ds\)
To differentiate with respect to t: count the t's in the expression (4), thus there will be 4 terms in the differentiation. Then you evaluate the integral at the limits:
\(\dot p_k = r p_k + \delta p_k - [e^{-r(t - t)} c(t) e^{- \delta (t - t)}]\)
\(\dot p_k = (r + \delta) p_k - c(t)\) or \(c = p_k (r + \delta ) - p_k\)
Or, we can derive it by logic. What's the definition of profit on a share on Microsoft? It would be the difference of the price plus profits:
\(R = \frac{DIV}{P_M} + \frac{\dot P_M}{P_M}\)
Multiplying on each side by the price of the share:
\(R P_M = DIV + \dot P_M\)
Similar to 
\(c = p_k (r + \delta) - \dot p_k\)
Where we have an expected return equal to a discounted cash flow (dividends) minus the change in price. This last equation is the biggest component of this theory -- all investment decisions should follow this. 
When Hall and Jorgenson test this theory, they use this equation:
\(K^* = \frac{\alpha PY}{c}\)
For the desired stock of capital. Alpha is the same as always -- the share of capital of income -- about one third. 
Refresher: 
Max \(Y(K, L) = pY - rK - wL\)
Subject to \(Y - AK^\alpha L^{1 - \alpha}\)
They just use different variables:
\(pY - cK - wL\)
Now they formulate an investment equation, with a distributed lag. Since they're interested in net investment, depreciation is subtracted and then the optimal level of K is substituted in the equation:
\(N_t = I_t - \delta K_t = \gamma _0 \Delta K_t ^* + \gamma _1 \Delta K_{t - 1} ^* + \gamma _2 \Delta K_{t - 2} ^* + ... - \omega N_{t - 1}\)
This is the famous accelerator revenue formula. Each term is put over the cost of the investment and then there is an omega lag to include all the other lags. 
The problem: this is not a theory of investment, but rather a theory of capital as as stock. Investment, rather, is a flow. It assumes that every time the price of the investment changes (c) then the K* level can change instantly. Doesn't explain how to get to the new state of investment. We need to add a cost to make it more realistic to describe how firms get to the new level of capital. 
One way of adding a cost is by making it internal -- expanding a warehouse or adding more trucks depends solely on the firm's internal investment and time. The bulk of research is here, called q-theory. However, it may be external costs. For example, when all the firms demand more capital (all the car rentals want to add cars) the price will be driven up and now each firm will acquire it more slowly. 

Q-theory

Key: finding the fundamental value of capital (i.e. restaurants, trucks, equipment, etc.) Helps justify the Austrian Business Cycle Theory. Basic intuition: market value to relative/replacement cost. For example, you could either buy all new equipment or repurpose another company's resources. The ratio should be one for full efficiency-- you should sell your company if it is worth more on the market than still running. I.e. if the q is very high of a business, their market value is high and other people will copy it and the replacement costs will increase and push it toward q. The model introduces internal adjustment costs -- an improvement over basic neoclassical model. 
Analogy for profit opportunities: rush hour traffic, the equilibrium car-to-car distance is about 2 ft. If someone isn't paying attention, they will fail to move their car and it will be about 5 cars. Within 5 seconds, someone will see that opportunity and move into the gap, thus filling in the equilibrium again and moving the whole traffic forward. 

Q-theory in Continuous Time

This will feel a lot like the RCK model (with phase diagram). Firm maximizes the present value of profits (requires calculus of variations, Hamiltonians):
Maximize \(\Pi = \int _{t = 0} ^\infty [\pi (K(t)) k(t) - I(t) - C(I)] dt\)
Subject to the constraint \(\frac{dk(t)}{dt} = I(t)\)
What is happening here? Back to the car company analogy. There is a profit rate that depends on the stock of capital: the pi term is the rate of profit (car rental calculates the profit over time with their capital amount) multiplied by whether or not you'll buy one more car (k) and multiplied by the total capital of the industry (K), minus the initial cost of the asset (car), and then minus the internal cost of adjusting your stock. 
Note, the derivative of pi with respect to K is negative -- more cars in the industry, the higher your costs. Constant returns to scale. The constrain says that the relative change of capital is equal to investment. 
(Romer textbook version does the Lagrangian in discrete time then just takes the logical step to get to where the Hamiltonian gets you - two differential equations.)

Q-theory in Discrete Time

Maximize \(\Pi = \sum \limits ^\infty _{ t = 0} \frac{1}{(1 + r)^t} [\pi (K_t) k_t - I_t - C(I_t)]\)
Subject to \(k_t = k_{t - 1} + I_t\)
Thus the Lagrangian setup will be:
\(\mathcal{L} = \sum \limits ^\infty _{ t = 0} \frac{1}{(1 + r)^t} [\pi (K_t) k_t - I_t - C(I_t)] + \sum \limits _{t = 0} ^\infty \lambda _t (k_{t - 1} - k_t + I_t)\)
To full put the constraint into the maximization, have to multiply it by the real rate of return. Here, 
\(q_t = (1 + r)^t \lambda _t\)
Recall: What is lambda? Marginal benefit for relaxing your constraint by one unit (marginal benefit of adding one more car to the fleet).   
Now we will maximize with respect to capital and investment -- the two choice variables. First, take the derivative (FOC) with respect to I(shows up 3 times so should have 3 terms):
\(1 + C' (I_t) = q_t\)
This function is convex (like a parabola arising from 0 on both ends of neg. and pos. investment). The interpretation is that investing and disinvesting a ton is exponentially more expensive than investing/disinvesting a little. Notice, if there is no investment, the q should be equal to one. In other words, the q is the price relative to replacement costs. 
Now remember, I is equal to the change in capital for the firm:
\(q_t = 1 + C' (\dot k_t)\)
Now, recall that the aggregate investment is simply the per firm amount times the number of firms (n)
\(q_t = 1 + C' (n \dot k_t)\)
Thus, this implies that there is a relationship between the total industry investment and q:
\(\dot K_t = C' ^- (q_t - 1)\)
Here we took the inverse of the price function (c) to understand the investment across the industry. A way to understand this is by thinking about the cars again, where you would be paid back the cost of the car. That was the FOC with respect to I. Now to do the same for k (our other choice variable). CRUCIAL to recall that within continuous time, there will be another kt in the next time period in the q term, which is why we will multiply it by r:
\(\pi (K_t) = \frac{1}{1 + r} (rq_t - \delta q_t)\)
Here, if we hold q constant, the profit per firm will go down. There is a negative relationship between q and K (important for phase diagram). Now realize how close this equation is to the handout equation:
\(c = p_k (r) - \dot p_k\)
If we put the q model in discrete time, it will be the same as the above. This model just adds an internal adjustment cost to Hall and Jorgenson's original model. If this derivation is unappealing, there are two others: Sommer's (less calc) and the Hamiltonian. 

Summary of Key Results:

If q is less than one, capital should be shrinking. If greater than one, increasing. To get the phase diagram, visualize what happens to the other variables here when either q or K increases:
  \(\pi (K_t) = rq_t - \dot q_t\)
Just like RCK, the relationship will explode if it's off the stable arm.

Effect of Output Movements

Suppose there is a permanent increase in output:
  1. The \(\dot q = 0\) line would shift out, intersecting the \(\dot k = 0\) line at a higher point
  2. There will be a new stable arm
Recall, q is the jumpy variable (like consumption in RCK) and the slow variable to adjust is K (like RCK model). 
What happens to stock prices?
They will boom, since investment was increased (q increased, above 1, and their market values relative to their replacement costs increased, signaling that it is a good time to get into the industry). For example, either the car companies will expand or new companies will enter the industry and add K and lower the general profit. Therefore, the permanent increase in output led to a temporary increase in stock prices and per firm profit. 
What happens to capital stock?
K, total capital in the industry, increases because the market values rose and many firms want to grow or enter. Now the profits will fall slowly and come back down to equilibrium until the investment is growing at 0. The new equilibrium level is a permanent increase in capital stock. 
Another question to consider: when will the industry stop expanding? when q goes back to equaling 1, so that the return on capital is equal to its market value (there are no more unexploited opportunities). 
What if the increase in output is temporary? How do the results change?
Here, the q jumps up but it falls downward and back on the stable arm. Now right at the moment it hits, output goes back to the pre-temporary-shift level. So output is low, high, then low again. The stock prices boom, but the stable arm carries it down and turns it back to the original equilibrium. 

Effect of Interest Rate Movements

A decline in interest rates tends to increase stock prices and stimulate investment spending. Temporary vs. permanent cases may make an important difference. Changes in long-term rates also affect stock prices and investment through the term structure. 
Since r shows up in the q-dot line, the slope and entire line will change. It gets steeper and shifts out. And stays at a higher level of capital once it goes back to the equilibrium.  If it is temporary, then it will rise up but again, fall on the original stable arm and go back to the original equilibrium (no level change of K). So output and capital increased temporarily to exploit the game then shrink to bring it back to the original equilibrium. 
This is where the model has implications for ABCT. If there is a credit-expansion, industry capital will increase though firms know it is only temporary, so they will try to be on the right level for it to come back to original equilibrium when the rate decreases again. Thus, there is a cyclical nature to this analysis -- an up and down even though profit is maximized (do not need to hypothesize malinvestment or stupidity). This shows the rationality behind q-theory and ABCT. 

Effects of taxes

Imagine an investment tax credit. This is where firms get rebates from the price of capital (each car that they buy). Therefore, the net cost is 
\(1 + C'(I) - rebate\)
Thus this reduces the cost of investment and the idea behind it is to stimulate investment spending, and hence, growth. 
But here, since the lower price of capital causes the k dot line to drop, we see the stock market and output drop. This is an artificial reduction of price, so since the demand for the firms' services have not increased, the value of each capital product is less productive. There is a higher industry capital stock, but the level of market value has dropped.
Therefore, remember that the fundamentals of the market have not changed, thus this is an artificial adjustment of prices since high capital but no higher output. You flooded the market (increase in supply). When the tax credit is taken back, the industry will shrink and the market will go back to the original valuation. Another example of this is the encouragement of home ownership -- artificial incentive that isn't based upon the fundamentals of the market.

Empirical Implications

Basic results is that the effect of q on investment is statistically significant but small. However, there are huge econometric problems since we need a marginal q not average q, and there are simultaneity effects causing q to be downward biased (coefficient is lower than it should be). 

Some Tests