around 0.35.
For a more detailed view of changes in optimal values of θ1 ,θ2, and aggregate
profit moving from the decentralized to centralized regime, Figure 2 shows percentage
changes in each of these variables across the same range of inverse information costs.
The dotted line at the top of Figure 2 shows percentage change in θ1, whose uniformly
positive sign indicates that the centralized solution always calls for Firm 1 to acquire
more information than it chooses on its own. This makes sense, because Firm 1
cannot internalize the benefit it provides to Firm 2 in the decentralized regime. In
contrast, Firm 2 usually acquires less information in the centralized regime, but not
always. The cases where the central planner dictates that both firms acquire more
information correspond to environments in which the cost of information acquisition
is relatively large (σa/c near zero on the x-axis). The range of exogenous parameters
in which both firms acquire more private information in the centralized regime reflects
synergistic complementarity in the two firms’ value of private information, which is
nowhere present when using the exponential cost function specification introduced
earlier. For quadratic acquisition costs and relatively high values of c, the marginal
benefit of Firm 2’s private information increases when the central planner raises Firm
1’s level of private information.4 The solid line in Figure 2 is the percentage change
in aggregate profit, which is always positive because the central planner optimally
utilizes information externalities to achieve greater aggregate profit.
2.5 Discrete choice in acquiring a signal of fixed precision
It is sometimes the case that firms cannot exert continuous control over the precision
of private signals they acquire. For example, a consulting firm might offer a report on
4A related point concerns the nonmonotonicity of Firm 2’s response to a change in θ1. This can be seen analytically
. .....∂β* . .......... ...... . . . . ____
in the indeterminate sign of d^, observed by implicit differentiation of the characteristic equation (27) and noting
that dh(θ2) is positive while dh(θ2) is of indeterminate sign.
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