Hence, for a given q, the LXM firm produces where [[pi].sub.x] [greater than or equal to] 0, implying Ward's  well-known result that the LXM firm produces a smaller output than its PM twin when w [greater than] 0 (and the same output otherwise).
Analogously, the LXM firm achieves its quality and quantity optima of [q.sup.**] = [q.sub.1] and [x.sup.**] = [x.sub.1] at the points labeled [W.sup.1] = ([x.sub.1], [q.sub.1]) where [Lw.sub.q] = 0 and [Lw.sub.x] = 0.
The latter asserts that the slope of the LXM firm's [Lw.sub.x] = 0 function equals the PM firm's [[pi].sub.x] = 0 function plus a nonnegative number whenever [[pi].sub.x] = 0 is positively sloped in the (x, q) plane, or plus a nonpositive number whenever [[pi].sub.x] = 0 is negatively sloped.
As seen in Figure 1, when the [[pi].sub.x] = 0 and [[pi].sub.q] = 0 curves are upward sloping, then [q.sub.1] [less than] [q.sup.*] and [x.sub.1] [less than] [x.sup.*], or the LXM firm produces a smaller quantity and a lower quality product than its PM twin.
The effects of a change in fixed cost on the LXM firm's quality and quantity of output may be determined by totally differentiating the first-order conditions, (1a) and (1b), regarding F to obtain:
Hence, [dq.sup.**]/dF and [w.sub.xq] have the same sign whereas [dx.sup.**]/dF is necessarily positive, or the LXM firm increases output (hence, the size of the membership) when its fixed costs increase.
The LXM firm, therefore, increases quality in response to an increase in fixed cost whenever [[pi].sub.xq] [greater than] 0, or whenever the increase in the price that the consumer is willing to pay for a higher quality product ([p.sub.q]) exceeds the increase in marginal cost required to produce a higher quality product ([c.sub.xq]).
Thus, an upward shift in demand unambiguously reduces the LXM firm's output.
Thus, regarding quality and quantity, the LXM and PM firms react differently to an upward shift in demand.
It is immediately apparent that the simultaneous solution to the LQM firm's problem is directly analogous to the LXM firm's, with quality and quantity exchanging roles and with [[pi].sub.q] [greater than or equal to] 0 because [L.sub.q] [greater than] 0.