A note on wealth in a volatile economy

I show that if the capital accumulation dynamics is stochastic a new term, in addition to that given by accounting prices, has to be introduced in order to derive a correct estimate of the genuine wealth of an economy. In a simple model with multiplicative accumulation dynamics I show that: 1) the value function is always a decreasing function of volatility 2) the accounting prices are affected by volatility 3) the new term always gives a negative contribution to wealth changes. I discuss results for models with constant elasticity utility functions. When the elasticity of marginal utility is larger than one, accounting prices increase with volatility whereas when it is less than one accounting prices decrease with volatility. These conclusions are not altered when adopting optimal saving rates. Estimating genuine wealth for an economy has been the subject of much recent interest as it is a key instrument to assess sustainability. In particular Arrow, Dasgupta and Maler [1] – hereafter ADM – have broadly discussed the issue in a recent paper. In brief, ADM show that wealth changes over time can be computed from the knowledge of accounting prices of the various resources in the economy. Their approach is mainly deterministic, apart from the case of a discrete time stochastic model. Here I generalize the treatment to the case of continuous time stochastic processes and show that the effect of volatility cannot be expressed in terms of accounting prices. I discuss a simple model and its application to constant elasticity utility functions in the next section. I discuss both the case of imperfect economies and of efficient ones, where the saving rate is adjusted to its optimal value, as discussed in [2]. Let V (k0, t0) = E ∫ ∞ 0 dτeu(ct0+τ ) (1) be the wealth function, where ct0+τ is the consumption flow at time t0 + τ . We imagine that k(t) follows a stochastic dynamics dk = a(k, t)dt+ b(k, t)dW, k(t0) = k0 (2)