Mathing Up "Why Bubbles Are Great for the Economy" (It Has Been Done Before Department)
Hoisted from Comments: Robert Waldmann asks, apropos of Daniel Gross's book Pop!: Why Bubbles Are Great for the Economy:
Grasping Reality with Both Hands: Brad DeLong's Semi-Daily Journal: A Review of Daniel Gross's Book "Pop": Yes, but do you know an academic economist who has formalized this argument [that bubbles are good for the economy by] writing a [formal] model in which irrationality is necessary for growth? It would not be hard. And have you asked yourself "If not me, who? If not now, when?" Posted by: Robert Waldmann | May 13, 2007 at 11:55 PM.
It was done two decades ago, Robert: a model in which the introduction of investors subject to irrational exuberance and panic can either raise or lower the economy's productive capital stock and hence enhance or diminish the economic welfare of others in the economy. The model doesn't have all the channels that Gross discusses, but it has some of them--and enough to make the point.
Unfortunately, the authors were chasing the case in which bubbles and panics were socially harmful--not the case when bubbles are beneficial to the rest of society. But that case is there in the model, if the parameter ρ∗ is big enough and the shock variance ratio (ση2)/(σε2) is small enough.
Here's an excerpt from the core of the argument:
Size and Incidence: There are two reasons why the capital stock [in the absence of bubbles and panics] is different.... If... misperceptions... are on average bullish [i.e., prone to bubbles, investors]... on average demand [more stock].... [I]f noise traders are on average bearish [i.e., prone to panics], the equilibrium capital stock is lower.... [I]nvestors’ demands [also] depend on the risk borne.... The θ2ση2 term in the denominator of [equation] (15) captures the reduction in the [economy's] capital stock that arises from aversion to noise trader-generated price risk.... The second term dominates, and the capital stock is lower in the presence of noise traders, if:
(17) ρ∗/(δ -r) < (θ/((1+r)2))((ση2)/(σε2))
For ρ∗≤0 [i.e., a market at least as prone to panics as bubbles], it is always the case that the presence of noise traders reduces the capital stock.
Even if ρ∗ is positive, only if both the noise trader wealth share θ is small and if noise traders’ opinions are not volatile relative to dividend risk (that is, ((ση2)/(σε2)) relatively small) is the ratio of productive capital to wealth increased because of noise traders.
A lower capital stock implies a lower average level of consumption. Since capital gains and losses on stockholdings simply redistribute wealth from one generation to another, the average level of consumption of a generation is simply:
(18) (1+r)W + K∗(δ-r)
which is an increasing function of the capital stock.
J. Bradford DeLong, Andrei Shleifer, Lawrence H. Summers, and Robert J. Waldmann (1989), "The Size and Incidence of Losses from Noise Trading," Journal of Finance 44: 3 (July), pp. 681-696 http://www.j-bradford-delong.net/pdf_files/Noise_Traders_Incidence.pdf.
Note: Robert's comment--that the source of utility gains to the sophisticated in DSSW (1990) is different from the source in Gross (2007)--is completely correct. Gross argues that volatility of opinions is good. DSSW argue implicitly that irrational exuberance can be good when such exuberance is not very volatile.