Comparative statics

In this section we study how the prices changes as the mar-ket’s power levels change. In analysing the effect of a change in the bargaining power on a certain price, we first consider only changes in the parameters that will make the conditions under which the price is observed to still hold also after the changes.

In other words, if the price Pyis observed, it is because a certain parameter x that enters its expression respects the condition to be on path y, for example x < ¯x; if we want to study what is the effect of a change in x on Py, that is, ^dP_dx^y, we need to have that x + dx < ¯x, otherwise we could incur in a change of the path in the game modeled in this section and, hence, in the occurrence of a price different from Py.

Assuming that the condition we just mentioned holds, one can demonstrate that all the equilibrium prices of each step and in each possible path are increasing in the parameters of bargain-ing power that appear in their expressions.

Observing the closed-form expressions of the various prices (see Appendix A), one can see that their formulas vary depend-ing on how many steps there are in the path to which the price is referred and what is its position in this path. Considering the number of steps in the path, we can distinguish among three different paths:

• Paths with only one step (ah)

• Paths with two steps (aih, ago, and agh)

• Paths with three steps (aigh and aigo)

Another distinction is based on the position of the considered price on the path in which it is observed, that is:

1. Prices in a one-step path (that is, P_ah^ahin the ah path) 2. Prices in a two-step path with a further step in front (that

is, Paih^ai in the aih path, and Pago^ag and P_agh^ag in the ago and aghpaths)

3. Prices in a two-step path with no further steps in front (that is, P_aih^ih in the aih path, and Pago^go and P_agh^gh in the ago and agh paths)

4. Prices in a three-step path with two further steps in front (that is, P_aigh^ai in the aigh path, and Paigo^ai in the aigo path) 5. Prices in a three-step path with a step behind and a fur-ther step in front (that is P_aigh^ig in the aigh path, and P_aigo^ig in the aigo path)

6. Prices in a three-step path with two steps behind (that is P_aigh^gh in the aigh path, and Paigo^go in the aigo path)

From a first analysis of all these prices, we can easily see that all of them have a similar structure, that is:

Px= (1 − ξ)Pz+ ξPy

where ξ is the bargaining power of the considered agent, Pzis either his purchasing price (if the agent is a gallery or an in-sider) or his reserve price (if he is an artist), and Py is either the reserve price of the outsider (if the seller is the gallery, or the artist when facing the auction house, or the insider when facing the auction house, by assumption (2.3)) or simply the price that will be obtained in the following step (if the seller is the artist when facing the gallery or the insider, or the insider when facing the gallery). Given how we built the model, we

have that Py> Pz, and hence Pxis increasing in ξ.

So, all the prices that appears in Section 2.3 are increasing in the parameter that appears in them, but this implies that their closed form, as they result in Appendix A, are increasing in all the parameters that appear in them as well.

The explanation of this result is as follows: since both the lower and the upper bound of the interval over which the bargaining of each of the prices is made are increasing in the parameters that appear in them, as well as the price itself is increasing in the parameter it has inside it, an increase in each of the pa-rameters that appears in each price formula have an increasing effect on it, assumed that these changes in the parameters do not change the validity of the condition under which the price is observed.

However, a change in a parameter could also make one or more than one of the conditions to not hold anymore, that is, it could change the path the artwork will follow; this is due to the effect a parameter have on the equilibrium price of each of the path, and hence could change the choice of the agents. For ex-ample, assuming conditions (2.30), (2.32), and (2.38) hold, that is, assuming the artwork is in the ah path, a reduction in η such that η > _{1−β(1−ν)}^βν does not hold anymore, but η > _{1−α(1−µ)}^αµ still holds (that is,_{1−β(1−ν)}^βν > _{1−α(1−µ)}^αµ ) will imply that the art-work will pass to the agh path; if instead η changes such that

βν

1−β(1−ν) still hold, while _{1−α(1−µ)}^αµ does not hold anymore, the artwork will pass to the aih path.

If we assume that the artwork is in the ago path, that is, that conditions (2.31), (2.35), and (2.46) hold, a reduction in β such that β > 1−α(1−γ(2−δ))−γ(1−δ)^αγ does not hold anymore will make the artwork pass through path aigo instead.

These examples we presented depict how a particular agent could emerge in the path the artwork will follow, due to the effect this agent has on the equilibrium price of this path: an artist will find profitable to sell her artwork to the insider as her market power towards the auction house decreases, while her market power against the gallery is not high enough; the emergence of the insider within the ago path, transforming it in aigo, can be explained with a similar argument.

It is worth noting that all the analysis carried through in this subsection assume the ceteris paribus condition, that is, a change in one of the parameter is not associated to the change of any of the other parameters.