Reconciling Hotelling Resource Models with Hotelling's Accounting Method.

AuthorCairns, Robert D.
  1. INTRODUCTION

    Of what does income consist? What generates or is responsible for income? How is income related to value? Does the numeraire matter? The meaning of income has long been a deep question in economics. In recent decades, the question has had a resurgence in resource and environmental economics. It is widely acknowledged that nature's many contributions to human well-being should be recognized in measures of performance such as net income. In a study of environmental (or green) accounting, Nordhaus and Kokkelenberg (1999) note that Hicksian income no. 1 (Hicks 1946) is the concept used in the national accounts and their extension to the environment. Income no. 1 customarily pertains to the present value of flows of benefit. (1)

    Green accountants (e.g., Arrow, Dasgupta, Goulder, Mumford and Oleson 2012; World Bank 2011), harking back to Weitzman (1976), generally argue that net product--and hence net income by an accounting identity--is equal to the current-valued Hamiltonian of an evaluation problem and that capital gains within the problem should be neglected. The neglect follows a tenet of not including receipts due to the "pure passage of time" (e.g., Weitzman 2003), or to what may be called drift. On the other hand, Ferreira and Vincent (2005) remark that the capital gains to a non-renewable resource should act as an offset to depletion in calculating income and product. The inclusion of capital gains in income and product is consistent with Diewert and Lawrence's (2000) critique of the System of National Accounts as being at variance with capital theory (Jorgenson 1963) in not recognizing capital gains as productive.

    Much thinking about green accounting is founded on Hotelling's (1931) model of an exhaustible resource. As Hartwick (2000) and Cairns (2018) point out, Hotelling's optimal path is directed by the prospect of capital gains. We explain below that, if there is a non-zero drift, depreciation in green accounting does not obey a fundamental condition of Hotelling (1925) depreciation, which Schmalensee (1989: 293) notes is also called exact or economic depreciation. An implication is that net product and income should be considered to include capital gains and thus to be equal to the so-called augmented Hamiltonian rather than the usual Hamiltonian. The presence of capital gains depends on whether the dynamic extraction problem satisfies a technical condition called non-autonomy.

    We begin by studying some general fundamentals of accounting. Then we apply them in examining the role of non-autonomy in six Hotelling-style optimization problems. The austerity of the simple models of a competitive market and of a planner allows the role of capital gains to emerge clearly. However, the introduction of a stock effect or of durability of the resource above ground masks that role. Revision of the representations of sectoral aggregation illuminates how capital gains affect properties of the optimal paths and leads us to express a predilection for first accounting at the micro level and then aggregating--as in national-accounting practice.

    A critical feature of the sole model of a firm considered (Gray 1914, Scott 1967) is a non-priced residual that precludes the comprehensiveness of accounting. Non-priced and even unaccountable capital variables, to which Arrow et al. call attention, are a source of non-autonomy. The residual prompts us to broaden our theoretic examination. We glean a perception of capital gains that clears up a superficial difference between the pure passage of time and the more familiar changes in the prices of assets.

  2. SOME ACCOUNTING FUNDAMENTALS

    In an elementary evaluation problem for any sector or activity of an economy, including an extractive sector, at time t a stock S(t) is known and available and is used to produce a quantity q(t) that provides a flow of net benefits f(q(t).t). The flow may or may not be valued at market or observable prices. For example, it may be utility or utility net of some cost (Aronsson, Lofgren and Backlund 2004) or be revenues evaluated at shadow prices (Arrow et al. 2012). It is discounted at the force (instantaneous rate) of interest r(t) to form the (net present) value,

    [Please download the PDF to view the mathematical expression] (1)

    The flow of benefits may or may not maximize V(s(t),t). In general, the flow and the value are determined by what Dasgupta and Maler (2000) call a resource-allocation mechanism (RAM) that allocates the stock to different times according to a transition equation, S(t) = g(s(t),q(t),t) and to rules and conventions that for present purposes can remain unspecified.

    If the value V(S(t),t) depends explicitly on time as written, the problem is called non-autonomous. Non-autonomy occurs through the appearance of time as an argument in one or more of the equations that govern the problem--through the variable s in the flow function f, the variable t in the transition equation g or the variable r in the function for the interest rate r. In contrast, in an autonomous problem, time is not an explicit argument of any of the functions: their partial derivatives with respect to time, and hence that of the value function V. are zero.

    If the discount rate is constant in equation (1), so that the discount factor is written [Please download the PDF to view the mathematical expression] it may be that the problem is autonomous except for time discounting at that constant rate. If the initial conditions do not depend on the initial time t, the program for given initial values is the same no matter when it starts (no matter the value of t). Economists have adapted the term "autonomous" to apply to this case as well (e.g., Kamien and Schwartz 1983: 159ff). Values are discounted back to t rather than to 0.

    Our aim is to evaluate various accounting magnitudes related to the postulated sector or activity, especially net product, net income and depreciation. We adopt Hicksian income no. 1 (Hicks 1946) as the concept of income. It is the level of net flow that can be consumed while leaving the decision maker "equally well-off' in value terms: dV / dt=0. This equality leaves the capital value V in the sector or activity "intact" over any instant of time for which it holds.

    When value is expressed as the integral of a discounted flow as in equation (1), it is natural (cf. Dasgupta and Maler 2000: 78) to differentiate both sides of equation (1) to find that (2)

    [Please download the PDF to view the mathematical expression] (2)

    In studies based on Hicksian income no. 1, the interest rate appears in income accounting in a suggestive and simple way: the force of interest r(t) is applied to the capital value, [V.sup.3.sub.r]. (3)

    Whether the dynamic path of the economy is optimal or not, equation (2) is the foundation of green or comprehensive accounting. Prices and interest rates are interpreted in terms of the RAM that is being implemented by the economy and are expressed in terms of the numeraire assumed in the model. Interest rates may vary but are exogenous to decisions concerning the activity, which is viewed as being embedded in a wider economy. The transition equation S = g(S,q,t) holds under the RAM but does not directly affect the qualitative nature of accounting.

    Let [Please download the PDF to view the mathematical expression] on the path determined by the RAM. Even though the RAM may not be to maximize value, a (current-valued) Hamiltonian function can be defined:

    [Please download the PDF to view the mathematical expression] (3)

    The term [lambda]g(S,q,t) is traditionally interpreted as net investment. If there is more than one state variable, [lambda](t)g(S,q,t) can be interpreted as the inner product of vectors X and S.

    The Hamiltonian H, the sum of net benefits and net investments evaluated at the shadow prices of the RAM, is traditionally considered to be the net product of the activity being studied (cf. Weitzman 2003). The notion of net product is often applied to macroeconomic magnitudes. It is also applicable to microeconomic magnitudes because of the equality of the circular flows of income and product for each component activity of an economy. The sums of these flows are the macro flows of income and product.

    Mathematically equivalent results can be obtained using the fundamental equation of dynamic programming under the RAM. Over a small interval of time dt,

    [Please download the PDF to view the mathematical expression] (4)

    Simplifying to first order yields that, as in equation (2),

    [Please download the PDF to view the mathematical expression] (5)

    Interest on the capital value of the resource at any time, rV, is the total return to the stock, notionally what would have been earned if V had been realized in the market and loaned out at r for a duration dt.

    Rearranging and expanding equation (5) yield that

    [Please download the PDF to view the mathematical expression] (6)

    Equation (6) expresses the dynamic no-arbitrage condition and hence how the activity is integrated into the capital market of the wider economy. The total return on the "stock" of value, rV, is equal to the sum of (a) the net dividend from extraction, [Please download the PDF to view the mathematical expression] and (b) a further term, [partial derivative]V / [partial derivative]t, which we interpret as the "drift" of the program, as a "pure price effect" attributable to the "pure passage of time".

    In a macro model, Arrow et al. (2012: 323-4) also introduce a drift term, interpreting it as the shadow price of an aggregate of exogenous, unexplained, real factors that vary through time, that influence productivity and that can be aggregated to an asset summarized by the variable t. In their equation (8), comprehensive wealth is defined to be the sum of t([partial derivative]V / [partial derivative]t) (a "stock" t multiplied by its shadow price) and the values of the capital stocks, evaluated at the shadow prices of the RAM. The drift is viewed as...

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