samlet årgang - Økonomisk Institut - Københavns Universitet

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NATIONALØKONOMISK TIDSSKRIFT 2005. NR. 3
may be inserted into (28) and (29) to find the short run equilibrium values of q1 and q2 (and hence the values of r e 1 e 1 /q1 and re 2 e2 /q2 ). If an increase in the tax rate t
reduces the value of qj , it will drive up re j , thereby reducing the real investment activity
of unquoted companies over time. Using (24) through (30) and the formula for the coefficient
of correlation cij = ij /ij , one can show that
dr e 1 2 1 re 1 2
= v 1 (c 2 13 –1)+v 2 (c 13 c 23 – c 12 ) (31)
dt i 1
dr e 2 2 2 re 2 1
= v 2 (c 2 23 –1)+v 1 (c 13 c 23 – c 12 ) (32)
dt i 2
Since the coefficient of correlation cij cannot exceed 1 numerically, the terms (c2 ij – 1)
in (31) and (32) are generally negative, whereas the term (c13c23 – c12 ) cannot be signed
a priori. However, if all share returns are positively correlated, a sufficient (but not
necessary) condition for negativity of the derivatives in (31) and (32) is that
c 12 ≥ c 13 or c 12 ≥ c 23 (33)
In other words, if the returns to the two unquoted shares are at least as correlated as
the returns to quoted and (one of the) unquoted shares, we can be sure that a symmetric
tax on the equity premium will reduce the cost of capital for unquoted companies
when marginal investors are not fully diversified. Again, this is due to the insurance
effect of the tax which reduces the riskiness of investment from the viewpoint of private
investors.
Note that (31) and (32) are consistent with our earlier neutrality result: if the market
value of the assets invested in unquoted firms is small relative to aggregate investor
wealth so that v1 and v2 are close to zero, dre j /dt will also be close to zero, and the shareholder
income tax will be approximately neutral towards real investment decisions.
The tendency for a symmetric tax on the equity premium (with full loss offsets) to
stimulate risk-taking is well known from the literature; see, e.g., Sandmo (1977 and
1989). This non-neutrality may well be socially desirable, since one would expect
that the owners of small companies are inclined to take too little risk from a social
perspective when they have failed to diversify their portfolios. However, to make reliable
welfare judgements on this issue, one should explicitly model the factors (such