By Jack Zanville Sissors; E Reynold Petray
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Step 4: d^ (vH )=dvH > 0 for all vH 0 vH SB ( ; v )) ^ (vH ) is implicitly de…ned by the equation VH (CH H VH (CP ( ; vH )) = 0, whose left- hand-side is a function F ( ; vH ). From the implicit function theorem we have d^ (vH )=dvH = (@[email protected] )=(@[email protected] ). Therefore, d^ (vH ) dvH = = SB ( ; v ))[email protected] @VH (CH H H SB ( ; v ))[email protected] @VH (CH H @VH (CP ( ; vH ))[email protected] @VH (CP ( ; vH ))[email protected] SB ( ; v ))[email protected] @VH (CH @VH (CP ( ; vH ))[email protected] H H > 0, @VH (CP ( ; vH ))[email protected] 30 SB ( ; v ))[email protected] where we use @VH (CH H SB ( ; v ))[email protected] = 0 from Proposition 4 and @VH (CH H H @VH (CP ( ; vH ))[email protected] > 0 and @VH (CP ( ; vH ))[email protected] > 0 from Step 3.
1 as vH ! vH 0 we have C SB ! C F B from Proposition 4. Also we know C F B As vH ! vH H H H Hence, for every SB ( ; v 0 < 1 there exists an " > 0 such that CH H ") H H CP for 0 CP ( ; vH < 1. "), which proves the statement. Taken together, Steps 1, 2, 4 and 5 prove the proposition for the case H L > 1. If this condition is not met, the high types will prefer …nancing of the pooling media product by price rather than advertising whenever values in CP change if 1. In this case, it may thus be that the equilibrium H P gets too low.
Hence, there exists a unique ^ (0) 2 (0; 1) P SB . CH 0 vH Step 2: ^ (vH ) = 1 for all vH This follows immediately from Proposition 4, as the …rst best must constitute a pure strategy 0 . vH Nash equilibrium for every vH SB )[email protected] > @V (C )[email protected] Step 3: @VH (CH H H H P 0 0 and @VH (CP )[email protected] > 0 for all vH < vH Substituting the equilibrium values in (1) for high types and taking derivatives we …nd @VH (CP ( ; vH )) P = vH @vH ' 1 '( SB ( ; v )) @VH (CH H = @vH q and @VH (CP ( ; vH )) = @ H L L) H 2( ' vH for vH 2 [0; v H ) L) H L H L 0 ).
Advertising media planning by Jack Zanville Sissors; E Reynold Petray