The definition and treatment of capital is an important issue that arises quickly when discussing neoclassical approaches to the business cycle. In particular, one can rapidly fall down the rabbit hole of the Cambridge Capital Controversies. However, if the objective is to focus on what is important for understanding the business cycle, we see that capital is hard to easily model, and this is related to the opacity of business cycle analysis.

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Heterodox economists put great weight on the Cambridge Capital Controversies. They argue that neoclassicals do understand the implications, which call into question the internal consistency of neoclassical theory. For example, see "What Even Famous Mainstream Economists Miss About the Cambridge Capital Controversies," by  Marc Lavoie and Mario Seccareccia. My argument is that the debate is complex because it is in reference to the internal consistency of neoclassical macroeconomics. However, if one takes the viewpoint that neoclassical economics is only useful if it has applications in the real world, there is no need to be derailed by worrying about its internal consistency. The problem with the neoclassical approach is that it is starting from a bad place to make the theory understandable.

What is Capital?

Capital is a word that has a variety of meanings. In accounting, it is related to equity, such as its usual in bank capital regulations. However, the meaning in economics has flipped to the other side of the balance sheet, being synonymous with non-land fixed assets.

The operational definition of capital is that it is a variable in the production function, that appears along side labour (and possibly other variables). It has a few key properties.

1. It is permanent, although it might depreciate.
2. It can be added to via investment that uses up produced goods. (This is why land is ruled out. If we want to add land to the production function, it is another variable.)
3. It is meant to be tangible, which rules out some things like intellectual property. 

This corresponds to non-land fixed assets on business balance sheets, but importantly, it is a  physical measure. E.g., what matters is the number of drill presses, not how they are valued on the balance sheet. The easiest way to understand this if you play strategy video games -- capital refers to the various production units that you control, which then have associated production mechanics (the production function).

The reason we care about this definition of capital is that fixed investment is a major driver of the business cycle.

Ignores Intangible Factors

The working assumption in economic models is that capital is somewhat homogeneous. Although production functions in neoclassical models are simplistic with typically only a single productive good, we could imagine a more complex model with many different real productive assets being tracked. The homogeneity assumption implies that adding a copy of an existing factor of production would give the same output.

This works for small commodity producers, or perhaps for a large firm that is adding to existing capacity. But it breaks down once we consider firms that have differentiated output. A firm could copy the physical capital stock of an existing firm, but there is no guarantee that it can produce output that potential customers view as equivalent to that of the existing firm.

The implication is that even if we modelled all various types of physical capital that exist, output will still not be uniform.

The Price of Capital

The more important issue is: what price or value do we assign to the capital stock of a firm? There are a number of different answers.

1. Book value The accounting book value of the firm's fixed assets.
2. Replacement Value The cost to buy newly produced physical capital that corresponds to the firm's productive capacity.
3. Market Value I The enterprise value of the firm, which is the market value of publicly traded equity and debt. Note that this value is supposed to match the value of the firm's assets, which includes intangible assets, financial asset holdings, as well as real estate.
4. Market Value II The enterprise value of the firm as assessed by a potential acquiring entity that has knowledge of the industry.

The first answer -- accounting book value -- is probably the wrong answer for questions of economic theory. That said, accounting book value is often the only data to work with. The problem with book value is that it is backward-looking, but it should be noted that a great deal of effort is made in accounting standards to ensure that the book value does resemble the other potential valuation methods. An inflationary environment is the most problematic for book value.

The latter two values are two versions of "market value of the firm" and are supposed to be somewhat close to each other. The reality is that equity valuations can jump around for various silly reasons. However, if equity valuations get too far out of line with a sober assessment of valuations, take-over events will occur. If the firm in question in under-valued, it can either go private (e.g., leveraged buyout) or be taken over by another firm. If the firm is over-valued, it can use its expensive equity to take over other firms, and perhaps economic reality might align with stock market valuations. In any event, we are largely stuck with equity market valuations as a proxy for these two concepts, since it is the only available data.

Once we put aside book value, we are left with two main approaches: replacement value of assets versus market value of the firm, with the market value most likely corresponding to the value of publicly traded financial assets. From the perspective of basic financial analysis, it is completely unsurprising that those two valuations diverge.

How Economics Gets Into a Muddle

There are two main concerns within economic theory with respect to capital.

1. The division of firm revenue between the factors of production -- labour and capital.
2. The role of fixed investment within the business cycle.

The first concern is of immense ideological importance to mainstream economics, as they needed to come up with a response to Marxist criticisms of capitalism. Their theory suggests that capital and labour earn their just desserts, making capitalism the optimal form of social organisation. I am not going to touch that debate with a ten foot pole. That leaves us with the discussion of the flows that create capital -- fixed investment.

Economic models are meant to create internally consistent stories about the world, albeit with important simplifications. A standard simplification is that the time period in a model is long, and the transactions take place at the average price. By the definition of the average price, we can analyse transactions as if they took place at the average when calculating changes to stocks and flows. However, the economic analysis is based on entities using those average prices to make decisions. In reality, not all transactions will occur at the average, and each entity will have a different basis to make decisions.

Compounding this simplification, mainstream economics is built around trying to force the notion of equilibrium into economic theory. Prices are supposedly to magically settle at a price utilised by all entities, and all prices are internally coherent with the economics rules of the model. In this case, there are (typically) three assumed properties of interest.

1. Markets are efficient, and the expected return on equity is the same the expected rate of return on bonds. Economic models are simple, and ignore issues like risk and term premia. As such, the nominal interest rate for one period debt instruments (set by the central bank in the model) is the stand-in for those returns.
2. Since credit risk would be hard to model, the business sector does not use leverage. This means that the return on assets is equal to the returns on equity. This explains why economists have allowed "capital" to jump from one side of the balance sheet (equity) to the other (assets). Since capital in the production function only makes sense in terms of physical facilities means that it has to be different than equity -- which is a financial claim on assets. (The asset side balance sheet entry is a monetary value, which we can relate to the physical assets via heroic assumptions.)
3. The replacement value of assets is equal to market value of assets. Otherwise, there is a simple "economic arbitrage": buy investment goods and use them in production (i.e., undertake fixed investment), since those investment goods are undervalued versus their value in financial markets.

This set of assumptions allows mainstream economic theory to effectively treat Treasury bills and fixed assets as being identical objects. This has obvious defects. 

Given the insistence of mainstream economists to only consider equilibrium models, we can see why the Cambridge Capital Controversy was so mystifying. The premises for analysis are incorrect, so any discussion using those premises are going to be hard to follow.

Two Price Models

Hyman Minsky favoured two price models: a price for the replacement value of capital assets (determined by current production), and one for the market value of financial assets. Bubbles in the financial markets can lead to over-production of investment goods, setting up the conditions for a recession -- where fixed investment is slashed to be in line with more realistic economic growth estimates.

My expectation is that it would be possible to invent a neoclassical model to capture this effect, but it is safe bet that the model would then be missing other critical components. 

Concluding Remarks

 Most industry and financial market professionals understand that we face fundamental uncertainty about future economic outcomes. We simply do not know what future profits will be associated with an act of fixed investment. This uncertainty is incompatible with models that are based on notions of equilibrium, which helps explain why such models have a hard time developing convincing business cycle dynamics.