Uncertainty and Capacity Constraints: Reconsidering the Aggregate Production Function

The CobbDouglas function is today one of the most widely adopted assumptions in economic modeling, yet both its theoretical and empirical bases have long been under question. This paper builds an alternative function on very different (albeit also neoclassical) microfoundations aimed at both addressing those theoretical drawbacks and providing a better empirical fit than the CobbDouglas formula. The new model, unlike the CobbDouglas function, does not portray installed capacity as aggregate capital but as a sunk cost generating economic rents. An analysis of 1949-2008 annual U.S. growth data suggest this alternative model explains nearly 85 percent of GDP fluctuations and is empirically more robust than the CobbDouglas, whilst both contemporary and lagged aggregate capital are statistically rejected as explanatory variables. This lends support to the old Cambridge Critiqueâ€?, according to which using value-weighted capital aggregates to explain production simply makes no sense. At face value, these results not only pose a question on any macroeconomic model assuming a CobbDouglas function but also point towards an alternative interpretation of phenomena such as the way monetary policy impacts productivity.


Introduction
Few assumptions in economics are as widely adopted as the Cobb-Douglas production function. Not only is it taught as one of the first lessons in basic macroeconomics textbooks, but it also stands, in more or less complex forms, at the core of most of today's mainstream Dynamic Stochastic General Equilibrium (DSGE) models. Yet it has also been, almost since it was first put forward in Cobb and Douglas (1928), the object of various well-grounded criticisms questioning both its theoretical rationale and its empirical fit. Hence it makes sense to explore whether an alternative function could be built on a neoclassical foundation such that it is not subject to the same theoretical objections, and then test whether it would provide better empirical results than the Cobb-Douglas. This is the purpose of this paper.
Perhaps the most serious theoretical challenge to the Cobb-Douglas function is the one at the center of the so-called 'Cambridge Capital Critique' that started with Robinson (1953-54). The objection was that, if we define capital as a priceweighted aggregate of heterogeneous goods, it makes no sense to model its return as a marginal product for, in production processes requiring time, capital value is determined by that rate of return acting as a discount rate on the future cash flows it generates -which makes the reasoning circular. 1 As Cohen and Harcourt (2003) note, although capital aggregates continued to be used in modeling practice, the theoretical challenge was never satisfactorily addressed, and hence remains valid today.
The Cobb-Douglas function is also empirically weak. True, estimating GDP growth as a function of capital and labor growth rates through an OLS regression without a time trend (as in Cobb and Douglas 1928) yields results supporting the hypothesis that the regression parameters of the two input variables approximate their shares of output. Yet, when one includes a time trend in the mix (as in Lucas 1970, Romer 1987, Klette and Griliches 1996, Griliches and Mairesse 1998or Felipe and Adams 2005, the regression coefficient associated to aggregate capital usually turns out to be either negative or statistically insignificant, which squarely _________________________ contradicts the model. DGSE models usually sidestep this issue by calibrating capital and labor coefficients a priori, instead of estimating them empirically; yet this of course does not validate the function, since it assumes away what it should actually test. Furthermore, the fact that the regression without a time trend displays such good results does not necessarily constitute real evidence because (as first pointed out by Phelps Brown 1957 and then by Simon and Levy 1963, Fisher 1971, Shaikh 1974, 1980, Samuelson 1979, Felipe and Fisher 2003or Felipe and McCombie 2005, 2009) such a specification is formally identical to the Fundamental Growth Accounting Identity. Thus, as long as the factor shares of output remain stable over time (as is indeed the case), the empirical fit is bound to be good simply because it constitutes a test of the accounting identity itself, not of the underlying production function.
This paper puts forward the hypothesis that observed GDP fluctuations are better modeled by regarding capital, both physical and human, as a "sunk cost" (whose returns are therefore economic rents) than as a variable input whose reward is its marginal product, as the basic Cobb-Douglas function assumes. As capital investment requires a lead time to become productive, in an uncertain world the initial response to random demand changes will be subject to existing capacity constraints. 2 Importantly, since only variable inputs need to be included in the production function, if the time horizon is short enough then only one input is truly variable: labor time. Under these conditions, value-weighted factor aggregates are no longer necessary, which leads to an aggregate production function that is exempt from the Cambridge Critique.
This does not negate the role of installed capital in the production processonly its measurement. At the time an investment is made, of course, the investor expects it to yield at least the market return; yet in an uncertain world these expectations may or may not be fulfilled, so going forward past costs become irrelevant. Hence, the return on capital should not be modeled as a marginal product on a historical investment, but as an economic rent resulting from _________________________ 2 Lead investment time and capacity constraints have been invoked in production function models before (e.g. Jorgenson and Griliches 1967, Kydland and Prescott 1982, Basu and Fernald 2000or Hansen and Prescott 2005, but nearly always as a refinement of the Cobb-Douglas function, as opposed to being the basis to build a different production function altogether.
www.economics-ejournal.org available production capacity vs. output demand: the higher the spare capacity, the lower the ability of asset owners to charge a rent. The structure of this paper is as follows. Section 2 provides an intuitive rationale for the alternative model, which is then developed analytically in Section 3. Section 4 describes the empirical strategy and presents the statistical test results for both the proposed model itself and for the standard Cobb-Douglas function. Finally, Section 5 ends with a summary of findings and conclusions.

Model Rationale
Imagine an economy composed of many production units, each one devoted to transforming inputs into a given set of outputs. For every given output volume, there are multiple productive processes or "techniques" available, each one requiring a given fixed investment in plant capacity in addition to a variable cost per unit produced. We assume each one of these producers is rationally aiming to select the output volume and productive process that maximizes real profit (i.e. the difference between output value and input costs, measured in output units). We also assume that the optimal technology curve that results from their selecting the most profitable technique at each production level displays economies of scale i.e. that, given an increase in production volume, there is always a technique that would allow to reduce the overall cost per unit (always expressed in terms of output units) and therefore increase the real profit. This is represented graphically by curve 'LT' in Figure 1. In this diagram, both the horizontal and vertical axes represent real output, whereas every line in the quadrant represents the cost structure of a given productive process or technique. Therefore, if we select an output demand level on the horizontal axis, its projection on the vertical axis according to the curve representing a given technique indicates how much of its output value would correspond to fixed cost ('R1') vs. variable cost ('VC') vs. "pure" profit ('R2'). For example, curve 'LT´ represents the optimal long-term production cost curve i.e. the lowest production cost possible for every given output level, regardless of how long it would take to deploy the associated production process. Every point along this curve (say, point 'Y') is associated to a given optimal production technique (for example 'B') requiring a fixed upfront cost ('R1') plus a certain www.economics-ejournal.org variable cost per unit (represented by the slope of 'B'), up to a plant capacity equal to 'Y'. Conversely, the straight line 'A' represents the cost profile of a production technique with constant variable costs per unit, no fixed costs and no barriers of entry, which would of course result in the unit price equating the marginal (i.e. variable) cost (hence it forms a 45° angle respective to the axes). Evidently, technique 'A' has a steeper slope than 'B' because its variable costs are higher, and is more inefficient for a given level of production 'Y' because the overall cost per unit for technique 'B', including both fixed and variable costs, is lower than that of technique 'A' -the difference being of course the "pure" profit R2.
In economics, prices are conventionally broken down into marginal costs (i.e. the incremental cost of producing the last output unit) and economic rents (that is, the difference between marginal cost and actual price). Evidently no rational, profit-maximizing producer would willingly sell for less than the marginal cost (i.e. in the case of a production level 'Y', for less than the length of segment 'VC'). Conversely, from a short-term perspective, fixed cost investments www.economics-ejournal.org constitute a "sunk cost" whose historical size is simply irrelevant for the maximization of future profit, and whose return therefore constitutes an economic rent (i.e. in this case the sum of segments 'R1' and 'R2') since the concept of "marginal cost" does not apply to it.
To be sure, if a production unit were considering whether to commit an upfront investment or not, it would require the present value of the expected return to equal or exceed the upfront cost. Hence, in a deterministic world it would make sense to model those future rents as equating marginal costs with marginal returns for that historical investment amount; yet, in an uncertain world, as soon as the conditions change so do those rents, after which no relationship may exist between them and the sunk costs.
The assumption behind the long-term curve 'LT' is that, when planning for the long run, producers can jump from one technique to the next as their output volumes change, choosing for every level of production the technique with the lowest cost. Conversely, when unexpected shocks hit demand, it is not possible to do this in the short run, for the upfront investments to expand capacity and deploy a more efficient production process cannot be deployed instantly, nor can installed capacity be easily divested, even if such a thing is possible at all. Hence, if demand, for example, drops unexpectedly by a magnitude 'ΔY' (i.e. down to 'Y-ΔY'), the producer will use the same technique to produce at less than full capacity, whereas, if demand increases instead by 'ΔY' (i.e. up to 'Y+ΔY'), the producer will have to resort to the less efficient technique A to produce the supplementary units required. This means that, although the long-term cost function is concave (i.e. has positive returns to scale), the short-term one is convex (that is, displays diseconomies of scale), for, given a planned output level 'Y', actual production follows segment B when it falls below 'Y' but segment 'A2' (parallel to 'A') when it raises above 'Y'.
Let's imagine now that we have many industries in an economy, each producing a different set of goods and services but all subject to a cost function with the same characteristics. Then, if the structure of aggregate demand changes unexpectedly, so that demand for one product increases at the expense of another while the total consumer budget stays the same, the costs of those industries whose demand dropped will go down comparatively less than the costs of the industries with higher demand will go up. This will therefore result in an aggregate loss of productive efficiency: www.economics-ejournal.org 1. The larger the variability of demand (i.e. 'ΔY' in the diagram) and/or 2. The smaller the angle 'α' between segments 'B' and 'A2' In turn, since any increment in the share of economic rents ('R1+R2') over the total revenue 'Y' results in squeezing segment 'VC' and hence flattening segment 'B' and closing angle 'α', we may say that, under demand uncertainty, the higher the ratio Y R2 R1+ , the lower the overall productive efficiency in the short run.
As a result, under uncertainty the link between economic rents and productivity in the short run is exactly opposite to the long run. In the long run what matters is curve 'LT' and, since segment 'B' represents the tangent to this curve, the larger the ratio Y R2 R1 + , the flatter (i.e. the more concave) the curve will be at that point -i.e. the higher will be its economies of scale. Conversely, in the short run, the larger this ratio (i.e. the more concave the 'LT' curve), the more convex the angle 'α' will be. If the world were deterministic, only the long-term function would matter, since rational agents would plan only at one point in the beginning of time with a view in the long run, and never have to revise their expectations again. Yet, in a stochastic world, circumstances change continuously and, as the agents adapt every time to the new conditions, it is the short-term function that determines the initial response.
In sum, the model predicts that the short-term production is a function of variable inputs only (as opposed to both fixed and variable), combined with the percentage of economic rents over total output (which determines the angle 'α' and therefore the degree of convexity) as well as the variability of demand composition (which in essence represents the average shock 'ΔY'). Furthermore, since all variable inputs translate at an aggregate level into capital plus labor, it can be argued that, if the time horizon is sufficiently short, the only "variable" input at a macro level is labor time.
Importantly, since the shape of the curve is directly dependent on the ratio Y R2 R1+ , any policy aimed at changing the weight of economic rents over production would actually have the power to change the shape of the production function, and thus, by implication, to manipulate the rate of productivity growth. The implications are substantial, because the weight of economic rents can actually be modified through public intervention, be it by the central bank (through basic interest rates, which transfer income from borrowers to lenders) or by the government (e.g. through taxes and subsidies, or through legal monopolies and www.economics-ejournal.org other constraints on competition). This means that, even if the market were perfectly rational and efficient, so that money supply were absolutely neutral from a demand perspective, the central bank's discretionary control on monetary supply would have the power to modify the shape of the production function (and hence the rate of productivity growth), accelerating it with low real interest rates and slowing it down with higher ones. An important prediction of this model is therefore that there is a direct, positive correlation between cheap, abundant credit and observed productivity growth.

The Individual Production Function
Consider an economic system with given basket of consumable goods and services selected as numéraire (and whose unit price we conventionally represent as ' ') i.e.: On this basis, we define the economic production function Note that this economic production function differs from the technical one in its using output prices to weight the various (and otherwise heterogeneous) output quantities into a scalar metric, the real output flow. As this paper aims to develop an expression linking a set of inputs to a measure of GDP (i.e. ' '), it is the economic production function we will focus on going forward.
We now define the profit (' ') of the production unit j as the difference between its output flow at current prices and its total production cost (' When this profit is expressed in terms of the basket of goods and services we have selected as real output numéraire, we refer to it as "real" profit (' ') i.e.: At this point we introduce four assumptions that will be central to this section:

Profit Maximization:
Producers select their output quantities as well as their corresponding demand for inputs to maximize their real profit flow .

Existence of an Optimum:
There is at least one finite maximum point for every real profit function .

Homogeneous Production Function:
The economic production function For convenience, we also introduce the following two definitions: • We define the combined input (' ') as: represents the marginal cost of an input '1' selected as aggregation unit (and which, incidentally, does not need to coincide necessarily with the basket of goods we selected as output numéraire).
ρ ') as the output value premium over real marginal costs per unit of input i.e.: The four assumptions above suffice to build the following general expression: PROPOSITION 1: Under Assumptions 1 to 4, the economic production function of a production unit may be expressed as: www.economics-ejournal.org Where both the integration coefficient (which we will refer to as "productivity coefficient") and the rent ratio

PROOF: See Appendix 1
This is an important expression, for it represents the analytical equivalent of the broken line B-A2 in Figure 1 (only, now under the homogeneity condition imposed in Assumption 4, which the broken line B-A2 would obviously not fulfill due to its non-differentiability at the inflexion point). As one would expect on the basis of the reasoning in Section 2, the individual production function, represented here by the analytical expression (11), will display diseconomies of scale as long as the rent coefficient t j, ρ is positive. 3

The Path of Output Growth
The purpose of this subsection is to determine the path of output growth over time under the conditions established in Subsection 3.1. We now introduce the following three additional assumptions:

Neutral Price Breakdown:
The way prices are broken down between marginal costs and economic rents (as represented by the rent ratio t j, ρ ) has no impact, other things being equal, on

Random Walk Perturbation on Technology Shocks:
Technology shocks follow an Itô stochastic diffusion process subject to a Wiener perturbation (that is, a linear, continuous, normally-distributed random-walk process otherwise known as "Brownian motion") i.e.: Where represents the productivity coefficient in γ and represent functions whose value is known at time t, and is a Wiener process i.e. a stochastic process such that and t j s ,

Random Walk Perturbation on Output Demand:
The growth rate of output demand also follows an Itô stochastic process subject to a Wiener perturbation, i.e.: σ is a function whose value is known at time t, the operator indicates the expected value according to the information available at instant t, and is a Wiener process i.e. a serially-uncorrelated, normally-distributed standardized white noise just as . For simplicity, we will also assume that and are independent random processes so that Assumption 5 is fairly intuitive: demand may be impacted by changes in output prices, but what portion of that price is devoted to paying for marginal costs as opposed to economic rents has no relevance for consumers, and should therefore have no impact on their demand -which we are implicitly assuming to always equal supply. Note that the condition The other two assumptions introduce a stochastic element into the model. The assumption that technology progress follows a random walk (Assumption 6) is consistent with the standard literature (e.g. in DSGE models), except perhaps for its being expressed in continuous instead of discrete time. Similarly, Assumption 7, which postulates that output growth perturbations are the sum of technology and non-technology shocks, should also be fairly intuitive (except perhaps for the fact that, for the sake of analytical simplicity, we assume these two sources of uncertainty to be independent from each other), although the second source of uncertainty is sometimes assumed to be nil in standard DGSE models.
Under these conditions, if we differentiate expression (7) respective to time (see the detailed derivation in Appendix 3) we obtain the following output growth function: PROPOSITION 2: Under Assumptions 1 to 7, the output growth rate of a production unit may be expressed as: There are two key takeaways from this expression (11). One is obvious: the output fluctuations that are not matched by technology change (as represented by ) must be reflected on input change (represented by ). The other, however, is more important: the higher the non-technology variance (' ') and/or the rent ρ '), the slower will output growth be, even under the same levels of input www.economics-ejournal.org growth and technology progress. This is exactly the conclusion anticipated in Section 2.

Aggregation
To extend expression (11) to a macroeconomic scale, we still need to aggregate output across all the production units In this context, we introduce only one additional assumption:

Single Marginal Cost for the Reference Input:
The marginal cost of the input '1' selected as aggregation unit is the same for all producers i.e.
This is actually quite a weak assumption, as it only requires to find one commodity that displays the same marginal cost for all producers (for example a currency trading in a highly liquid, competitive market), and then take it as the accounting unit of .  In this context there is however an important distinction to be made between the overall non-technology variance β represent the relative shares of, respectively, output and input of the various productive units i.e.
. Needless to say, these two magnitudes would www.economics-ejournal.org necessarily be equivalent for an individual production unit, but not for an aggregate. Under these conditions the following proposition and its corollary hold: PROPOSITION 3: Under Assumptions 1 to 8, the aggregate output growth rate may be expressed as: COROLLARY 1: Under Assumptions 1 to 8, aggregate output may be expressed as: Where the aggregate productivity coefficient t A is a martingale such that: Importantly, notice that expression (15) is NOT just a weighted aggregate of the individual productivity growth expression (9), as one might expect. This is because the aggregate productivity coefficient t A reflects not just the impact of technology changes but also that of changes in the share of output of different _________________________ units whose individual production functions, as depicted graphically in Section 2 and analytically in Subsection 3.1, are not necessarily linear.
The question remaining now is how to measure t X : after all, this metric is defined as the sum of the individual producers' input quantities weighted by their own marginal costs, which may be different from one produce to the next. The definition of "variable input", however, depends on the time horizon of analysis: the shorter it is, the fewer inputs can be regarded as variable. Hence, there must be a period short enough for only one input to be genuinely regarded as variable.
Which input would this be? At an aggregate level all inputs are a combination of labor time and a stock of capital resulting from investments made (and sunk) at some earlier point in time (be it under the form of physical fixed assets, human capital or any other). Hence, if we define the time horizon as short enough, we may assume that:

Labor Hours are the only Variable Input within the Target Time Horizon:
According to this assumption, if we represent the total sum of labor hours as t H , then obviously expression (14) may be rewritten as: Where t H represents a non-weighted sum of work hours of all persons (as opposed to the usual Tornquist labor input aggregate, where input time is weighted by the share of output of each type of worker). In short: this model's main prediction is that GDP growth is better explained by taking non-weighted labor time and an estimate of the rent ratio as explanatory variables than by a Cobb-Douglas function depending on value-weighted capital and labor aggregates. This is the hypothesis that will be tested in Section 4.

Comparison with the Cobb-Douglas Function
For reference, it may now make sense to establish what sort of assumptions would derive a Cobb-Douglas production function within the same analytical framework. This requires introducing two additional assumptions. The first is the following:

Market Prices Equal Marginal Costs:
www.economics-ejournal.org The marginal costs of all inputs equal their market prices i.e.
This assumption implies that: • The rent ratios are always zero (i.e. 0 , = t i ρ ), as there are no economic rents • Assumption 9 becomes indefensible, as all inputs are now instantly variable Under this assumption, the aggregate production function obviously takes the form: Where the aggregate productivity coefficient t A is a martingale such that: And where t X can now be calculated as a sum of inputs multiplied by their market prices i.e.

∑∑
The impact of Assumption 10 is therefore substantial: not only does the function become linear (i.e. exhibits constant returns of scale), but also, as a result, the growth rate of the productivity coefficient t A becomes a direct aggregation of the individual expression (9), which, per Assumption 6, is only determined by technology factors.
The second key assumption is related to aggregate output elasticity. Specifically, given the usual Tornquist aggregates for labor (' t L ') and capital (' t K '), it assumes:

Constant Elasticity of Aggregate Capital vs. Output:
The elasticity Since, according to expression (17), this is a homogeneous function of degree one , which means that the production function may be rewritten as: Which is of course the constant-returns Cobb-Douglas function.
Note that, as shown in Appendix 2, by dropping Assumption 1 (i.e. profit maximization) this result can be extended to non-constant returns i.e.

4
Empirical Strategy and Testing

Empirical Model
Following the traditional approach to testing macroeconomic production functions empirically, this paper resorts primarily to an OLS regression against a GDP growth series. This requires rearranging expression (14) in logarithmic form: Where, since we know from expression (15) that t A follows an Itô diffusion process: And therefore we can rewrite expression (21) as: Which, taking discrete time increments, becomes: ln Under these assumptions we can rewrite expression (24) as: is compatible with the theoretical model, and therefore the regression results must be such that this hypothesis cannot be rejected.
In addition, if the regression is performed against the data of an economy large enough for the foreign sector to be proportionally small (e.g. the USA), one should also expect the value of 1 β to be lower than zero. The reason is that, although in principle the definition of 2 2 1 δ σ β − ≡ is compatible with any sign, in a closed or nearly-closed economic system, where the total number of labor hours available can be taken as a given, increases in the demand of hours by some producers can only be fulfilled though reallocation from other producers, so the variance

Data and Approach
The variables of the regression are obtained on the basis of the following dataset: • www.economics-ejournal.org that is used for macroeconomic Cobb-Douglas estimations. In truth, both variables are so highly correlated (R 2 =97.77% between the two in this particular sample) that it would matter little which one we used in a modelbut the distinction is important from a conceptual viewpoint.
• As for the rent ratio t ρ , there is unfortunately no single, all-encompassing indicator of the portion of economic rents embedded in the overall price of productive factors. Nevertheless, as in this model all capital investment is regarded as "sunk" costs, any return it generates must be a rent flow. Hence, a way to measure at least part of this flow would be to look into the process of purchasing power transfer between lenders and borrowers in the money market. In this context, the risk-free interest rate represents a flow of purchasing power from debtors to creditors, whilst any additional credit funds represent a flow of purchasing power in the opposite direction. 6 The balance of interests generated less additional credit provided is thus equivalent to a rent paid by producers, under the form of a transfer of purchasing power, for their use of capital invested. In addition, since expression (25) takes as an input its value at instant t instead of its increment between instants t and t t Δ + , we need to take the value of this indicator at the start of the period -which, as we deal with annual data, means at the start of the year. Of course this is a fairly crude measure, as there is nothing inherently "special" in the rent ratio in the month of January, or in the calendar year as a time horizon, but this should allow us to test if the rent ratio today does indeed help to predict the rate of output growth tomorrow. As a result of all this, we take as an estimator of the rent ratio t www.economics-ejournal.org Under these conditions, the model is subjected to four empirical tests: • TEST 1: First, the proposed model is estimated through an OLS regression in order to analyze its empirical fit and its consistency with the theory.
• TEST 2: Next, a Cobb-Douglas production function is estimated under the same conditions, and its empirical fit compared with the previous one. 8 • TEST 3: The two models are then compared through a Davidson-MacKinnon test (following Davidson and MacKinnon 1981) to assess whether one should be preferred to the other, or if they should be regarded as complementary.
• TEST 4: Finally, the regression in test 1 is repeated, only now adding capital input changes (both contemporary and lagged) as an explanatory variable to test whether aggregate capital is better or worse an indicator of installed capacity than financial rents, or if the two are complementary.

Test 1
Here we estimate the parameters of the model as formulated in expression (26) by means of an OLS regression. 9 The results appear in Table A.
The R 2 value suggests that this expression can explain nearly 85 percent of the total variability of the series. The results are also consistent with the model's predictions regarding parameters 1 β and 2 β . The t-test suggests both parameters are different to zero at over 99% confidence, the estimated value of 1 β is negative _________________________ 8 The source data come from http://stats.bls.gov/mfp/ and have been reproduced in Appendix 5. 9 In this and the following tables, the asterisks beside the estimated values indicate their significance according to Student's t-test: one asterisk indicates significance at 90% confidence or more; two asterisks, significance at 95% confidence or more; three asterisks, significance at 99% confidence or more. For further clarity, standard deviations are always represented between brackets.
www.economics-ejournal.org Moreover, the robustness of this result is confirmed by both the Durbin-Watson and the Breusch-Godfrey tests: in both cases, the result indicates that there is no first-order autocorrelation in the residuals.

Test 2
Next we follow the same procedure to test a Cobb-Douglas function as expressed in (20) i.e.  Table B. The statistical fit is clearly much worse than in Table A. Not only is the R 2 lower in Table B, but the F-statistic is also smaller, and all the information criteria (Akaike, Schwarz and Hannan-Quinn), which signal higher likelihood the smaller they are, take much larger values. Furthermore, unlike in Table A, the Durbin-Watson test takes a value which suggests that there might be an autocorrelation issue: indeed, the value of this test falls right between the Durbin-Watson lower and upper critical values for 5% significance (1.51442 and 1.65184, respectively), and therefore its result is inconclusive. A more powerful tool, the Breusch-Godfrey test, allows to reject the autocorrelation hypothesis at 5% significancebut only just, because at 10% it would not. These inconclusive results suggest a possibility that t-ratio values might be somewhat overestimated, thus reducing their explanatory value even further.
Much more importantly, the regression results simply do not support the Cobb-Douglas specification. In particular, the estimated regression coefficient associated to capital input (' 1 β ') is negative (i.e. it portrays capital is a factor of value destruction) which is contradictory to the model, although with such a large standard deviation it is not even possible to reject the hypothesis that the actual coefficient be 0 1 = β (which would still contradict the Cobb-Douglas framework). Nevertheless, the estimated value is so far away from the theoretical one (i.e. from capital's share of output, which is approximately 32 . 0 1 = β ) that we can reject it as a null hypothesis, for its t-ratio would be -2.40541 0.195456 32 . 0 0.150151 − = − 2 , which allows to reject it with a 95% confidence margin. The conclusion for the parameter (' β ') associated to labor is similar to that of capital (although, in this case, its regression coefficient is statistically different from zero): its t-ratio respective to the null hypothesis that it be equal to labor's share of output (i.e. approximately
, which allows to reject the hypothesis with over 99% confidence.
As discussed in the introduction, this is by no means a novel finding. If it is not highlighted more often is probably because most mainstream models are designed to work around it, be it by performing the regression without a constant time trend factor (i.e., in terms of Table B, by imposing a priori that = α ) and then www.economics-ejournal.org estimating parameters ' ' and ' ', be it by calibrating the theoretical weights of capital and labor (i.e. by defining a priori the values of ' 1 β 2 β 1 β ' and ' 2 β ') and then estimating only 'α ' as a residual growth rate. These restrictions may well be useful for other purposes, but do not help to estimate the production function, for they assume away what should actually be proven. The results in Table B are clear-cut: neither can parameter 'α ' be equated to zero (because the t-statistic rejects such hypothesis with a confidence margin of over 99%) nor, once this time trend element is introduced, can the hypothesis of parameters ' 1 β ' and ' 2 β ' representing factor shares be sustained.

Test 3
Although all the criteria we have reviewed so far support the regression in Table A (i.e. the model postulated in this paper) against the one in Table B (the standard Cobb-Douglas function), this comparison alone does not suffice to reject the latter, for there is still a possibility that both models be complementary and a "hybrid" one combining the explanatory variables of the two offer an even better result.
To distinguish between these two possibilities we resort to a Davidson/MacKinnon test (following Davidson and MacKinnon 1981). This test requires to re-run each one of the two original regression models, only now including as explanatory variables, in addition to the original ones, the fitted values of the alternative model. A rejection of one of the two models will take place when its fitted values come out as statistically insignificant (on the basis of a t-test) when included in the alternative model, and at the same time the fitted values of the alternative model come out as significant when included as an independent variable in the first one. As a result, this test does not "force" the identification of a single winner: it may happen that none of the models is rejected, in which case we would probably need to consider a "hybrid" model, or that both are, which might point towards a more fundamental specification issue. Hence, if we do end up finding a clear winner, this will arguably provide a very strong piece of evidence in favor of adopting it at the expense of the other model.
As shown in Tables C and D, the outcome of this test is clear. Table C shows the result of performing the same regression as in Table A but including now the fitted values from the one in Table B as an explanatory variable (represented by ' '): t B www.economics-ejournal.org This result proves that, under the conditions of this test, the fitted values from the regression in Table B (i.e. from the Cobb-Douglas model) lose their statistical significance, whereas all the explanatory variables of the model postulated in this paper remain significant.

Coefficient (' ')
-0.163373 ***   Table D displays the results of the opposite exercise, namely, performing the same regression as in Table B but including the fitted values from the one in Table A (represented by variable ' t ') as an input variable, and proves that, under these conditions, only the fitted values from that regression Table A retain their statistical significance as an explanatory variable.In short, these results reinforce the conclusions from Tests 1 and 2, i.e. the model in this paper constitutes a better-fitting specification than the traditional Cobb-Douglas production function and, furthermore, that a hybrid combining the explanatory variables of both models would not provide any additional explanatory power.

Test 4
Although the results of Tests 1, 2 and 3 are statistically robust and clear-cut, one could still wonder whether they should simply be interpreted as rejection of the Cobb-Douglas specification or, more fundamentally, of the significance of the variables it is built upon. The question is almost a moot point in the case of Tornquist aggregate labor (' t ') because it is so highly correlated to the nonweighted sum of work hours (' L t ') -although the latter is statistically more closely correlated to GDP, H 2 11 just as the model in this paper would predict. In the case of aggregate capital, conversely, the question of whether it is a better or worse measure of installed capacity than economic rents is relevant for two reasons: • The production function proposed in this paper takes production capacity as a constraint, but lacks a mechanism to explain how this capacity evolves in the long run. To the extent capacity is built up through investment, one would expect today's net investments (i.e. changes in aggregate capital) to determine future capacity expansions. • On the flip side, if (as the Cambridge Critique would predict) it makes no sense to measure capital input as a value-weighted capital aggregate, then one would expect to find that net investment plays no role in capacity expansion.
To test these alternative hypotheses, we perform a regression on an encompassing model (following the method proposed by Mizon and Richard 1986) including both the production function in (26) and the change in aggregate capital, lagged n periods. To cover a wide range of possibilities, the regression is repeated for aggregate capital lags ranging from zero (i.e. contemporary) to nine years, as shown in Table E.
The outcome is unambiguous. Both the rent ratio and the labor input variables retain their significance in every single case, while none of the regressions with lagged net investment provides any clear improvement in explanatory power (as measured by the R and F statistics). Conversely, for net investments with up to seven years' lag not only is the coefficient negative in most of the cases, but in none of them is it possible to reject the null hypothesis of it being zero… and for www.economics-ejournal.org lags of eight or nine years, bizarrely, the test indicates statistical significance for a negative coefficient: in other words, it suggests that investments made eight or nine years ago have over 95% probability of resulting in a reduced productive capacity today (!). Two straightforward interpretations spring to mind: either there is really a mechanism by which net investment is not only irrelevant but even detrimental to production (which seems contrary to common sense) or using aggregate capital as a measure of non-labor input simply makes no sense, as the Cambridge Critique authors defended so long ago. Faced with this choice, one must let Occam's razor rule and conclude with the Cambridge (England) school that value-weighted aggregate capital metrics play no measurable role in the macroeconomic production function.

Summary and Conclusions
Already more than four decades ago, Fisher (1965Fisher ( , 1968Fisher ( , 1969 and Gorman (1968) proved that a macroeconomic production function cannot be built on the basis of just any set of individual production functions, even if behind them are rational, profit-maximizing agents. One can always add up individual production functions into an aggregation, to be sure, but then its aggregate input results from adding up individual input quantities weighted by their marginal costs, and these may have different productivities depending on which individual producer uses them. Additional assumptions are therefore required to develop a usable function. One such assumption is perfect competition, i.e. the postulate that all marginal costs equal market prices. This postulate is central to the Cobb-Douglas function, and is often justified by supposing that, in the long run, most inputs are variable and so economic rents may be discounted as a mere form of short-term friction. The Cobb-Douglas function, however, poses a number of theoretical and empirical issues which suggest there is room for exploring other specifications. This paper has put forward an alternative based on the exact opposite assumption: that it is the "short term" that matters most to understand output fluctuations, and that, within this time horizon, only one major aggregate input (labor time) may be regarded as variable. From a theoretical perspective, this approach presents a number of advantages, among them not being subject to the Cambridge Critique.
Nevertheless, one could equally think of some strong theoretical points to support the Cobb-Douglas function, so this alone cannot drive the choice: to select a model instead of the other we need to resort to empirical data.
We perform this empirical analysis in Section 4, with the following results: • The model proposed in this paper is empirically robust, and can explain nearly 85 percent of the U.S. GDP fluctuation from 1949 to 2008. • Against the same data set, the Cobb-Douglas function proves to be a lot less robust, with substantially lower fit and likelihood metrics, and rejecting the hypothesis that the capital and labor elasticity parameters are equal to their shares of GDP with a confidence margin of over 95%. • A non-nested model comparison (specifically, a Davidson/MacKinnon test) concludes that the model put forward in this paper represents a better specification than the Cobb-Douglas function and, furthermore, that a hybrid of the two would not add any meaningful explanatory power either. • Last but not least, the estimated impact of capital, both contemporary and lagged, on production is nearly always negative (although the variance around it does not allow to exclude its simply being nil), which suggests that, as the Cambridge Critique postulated long ago, aggregate capital makes no sense as a measure of non-labor productive inputs.
Why does it matter which model we select? It matters because their implications are very different. A Cobb-Douglas world where output depends on the quantities of capital and labor and their marginal productivities, which also happen to be their prices, is very different from one where every income above the marginal product of basic unqualified labor constitutes a rent, and where these rents in turn determine how overall productivity behaves. In this latter world, for example, central bank policy would not require any sort of irrationality or monetary illusion to be effective as long as the balance of interest rates and credit flows acts as an economic rent and therefore drives the behavior of the aggregate production function. Conversely, in such a world it might not be licit to model the choice between investment and consumption as most DSGE papers do, for aggregate capital (i.e. accumulated net investment) could not be represented as a productive input… It would go well beyond the scope of this paper to analyze each one of these implications; but it is probably fair to say here that the predictions www.economics-ejournal.org from models based on the production function put forward in this paper would probably diverge substantially from the standard ones. This is the exactly expression we introduced as Proposition 1 in Subsection 3.1.

Appendix 2
The purpose of this appendix is to show the development of a widely-used derivation of a production function's degree of homogeneity (see for example Varian 1986 for a textbook example, or Fernald 2000 and2002 for empirical applications), and so to highlight where its assumptions differ from those in Subsection 3.1.
We define the marginal cost respective to the output quantity  Thus, if we also adopt Assumption 4 (i.e. homogeneity) in terms of expression (5) and combine it with the above we obtain:  π Which allow to rewrite expression (2.1) in the more familiar form: Note, however, that, up to this point, we have resorted to Assumptions 3 and 4 (and also implicitly to Assumption 2, at least to the extent that, if no optimal point www.economics-ejournal.org existed, the whole exercise would be pointless) -yet Assumption 1 (i.e. maximization of the real profit function ,t j Π ) has played no role so far. If we now formally introduce this assumption, then evidently: