STOCHASTIC

(1.1) and (1.2) give
SOME EFFECTS OF TAXES ON RISK-TAKING 291
H=^o. ...(1.3)
k1) the tax implies higher risk-taking.
Richter [16] deals with the problem along much the same lines as Tobin. He also
uses von Neumann-Morgenstern utilities and he deals with two parameter distributions.
He gives a fuller treatment of the lump sum tax case and above all he does not base his
analysis on ratios of assets. This is very important in the case of borrowing since the
effect of the tax is shown to be an increase both in money holdings and in risky assets
which will not be seen if the analysis is based on asset ratios.
The analysis is based on quadratic utility functions. 1 The main result is the same
as Tobin's, namely that increased taxation produces greater risk-taking in the society.
Lintner [9] gives an extension of Tobin's work by allowing for borrowing and lending
at a specified rate of interest. As was shown above, Tobin considered the special case
with no return on the riskless asset. Furthermore, Lintner studies the case with explicit
constraints on the non-negativity of assets, and constraints indicating that the holdings
of risk assets must not be greater than the total amount of assets available.
He utilizes the following notation:
where
r" = interest rate on riskless assets,
r = mean return on portfolio,
w = ratio of gross investment in stocks to total net investment (stock plus riskless
assets minus borrowing). 2
The investor's net return per dollar of total net investment will be
^ = (l-iv)r* + wr = r* + H'(r-r*), ...(1.4)
w 1 indicates that the investor borrows.
1 The assumption of quadratic utilities has been criticized by several authors as follows. The range
over which the quadratic utility function is valid is limited by the condition that more return must produce
higher utility. Attempts to widen this range lead to the consequence that the risk aversion represented
by the function is not as strong as the one observed in experiments. Another consequence is that higher
wealth leads to more risk aversion which does not seem to hold empirically. This has motivated the use
of higher-ordered utilities. (See Puu [15].)
2 __ stock
stock —borrowing
stock
stock + lending"
3. EFFECTS OF TAXES ON RISK TAKING 315