# A Quick Overview of the “q Theory” of Investment

Tobin’s q is defined as the ratio of market value to replacement value for a firm’s capital.

The market value in the numerator reflects the profitability to the firm of one additional unit of capital.  The replacement cost in the denominator can be thought of either as the cost of acquiring new capital or the price earned by selling existing capital.

If q>1, then the additional profit a firm could expect from one more unit of capital (equipment, buildings, etc) is greater than its replacement/acquisition cost.  The firm should increase its capital stock.

If q<1, then the additional profit would be less than the acquisition cost.  We assume diminishing marginal productivity of capital, so this also means that the last unit of capital the firm acquired is producing less for the firm than its market value.  The firm should reduce its capital stock (sell equipment, etc).

In both cases, diminishing marginal utility means q should tend toward 1.  If q>1 and a firm acquires more capital, average productivity will decrease until the market value of the next machine will equal the acquisition costs.  Firms are indifferent to paying \$1 for \$1 worth of equipment.  If q<1 and the firm sells capital, average productivity will rise until the market value of the remaining capital equals the replacement/sales price.  Firms are also indifferent to selling \$1 worth of equipment for \$1.

In q theory, a firm acts to maximize the present value of its after-tax net receipts.  A firm’s investment level is a function of its marginal q.  For the specifics of the theoretical model, and for the explicit relationship between marginal q and average q, see “Tobin’s Marginal q and Average q:  A Neoclassical Interpretation,” (Hayashi 1982).

Marginal q refers to the market-value-to-replacement-cost ratio of the next unit acquired.  This cannot actually be observed.  What can be observed is the firm’s average q, or the ratio of market value to acquisition cost for the firm’s entire existing capital stock.

# Dynamic Agency and the q Theory of Investment

DeMarzo, Peter M., Michael J. Fishman, Zhiguo He, and Neng Wang, 2012, “Dynamic Agency and the q Theory of Investment,” The Journal of Finance, Vol. 67, No. 6 (2012), 2295-2340.

Purpose:  To introduce an agency problem into the standard q theory of investment; to show that cash flow is not the best predictor of investment.

Motivation:  A large body of literature uses cash flow to predict firm investment levels.  This paper argues that a better proxy is “financial slack,” which is directly related to the agency problem.

Model:

• Productivity is a Brownian motion, and the agent controls the drift but not the volatility.
• w = W/K is the agent’s total expected payoff per unit of capital, and must be high enough to incentivize the agent to maximize productivity.
• The level of w depends on λ, σ, and historical firm profitability.
• λ is a measure of the extent of the agency problem.
• σ is the volatility of firm productivity.
• Past productivity raises or lowers w, and the agent loses his job when w = 0.
• A portion of w is deferred, giving the agent a stake in continued firm success.
• Investor’s expected payoff per unit of capital, p(w), is a function of how much they pay the agent.
• Average q is total firm value per unit of capital stock, or qa = p(w) + w.
• Marginal q, or qm = p(w)wp’(w).
• Firms invest when marginal q is less than 1, so investment is a function of w. It follows that investment depends upon λ, σ, and past firm performance.
• “Financial slack” equals w/λ, and is the largest productivity shock the firm can suffer without changing agents.
• The agent accumulates cash and available credit equal to the firm’s “financial slack,” then distributes excess income to shareholders.

Conclusions:

• Financial slack is a better predictor of investment than cash flow.
• Average q is higher than marginal q because an increase in capital stock K reduces w, and hence reduces the agent’s [historically determined] incentives to maximize productivity.
• Financial slack and profitability are substitutes in determining average q.
• When the firm is profitable, w rises, the agent’s incentives grow, and return on investment increases.
• Investment is serially correlated.
• The cost of incentivizing the agent leads to underinvestment in every state of the world.