A

As in Figure 1, the demands ei of a PT and a NPT with the same initial endowments ai are identical at s=μ−σ2rai. In order to show this, the demand (2) of a PT is set equal to the demand (3) of a NPT with ai. Inserting λ via (5) yields

B

In order to show that the absolute value of the equilibrium transaction size of a NPT increases in the number of PTs P, the squared equilibrium transaction size of a NPT εn−an2 is considered. Its derivative with respect to P is given by

The derivations of ε and π with respect to P amount to

Inserting these expressions into (14) yields

In particular, this expression is positive when the signs of qL−1 and ε−an are identical. This means qL−1⋅ε−an>0. Inserting the initial endowment of a representative NPT an=qˉ=1−Pq/N and ε=qh+1h+L into this condition yields

which is always true as h⩾0. Thus, dεn−an2dP>0 and the absolute value of the equilibrium transaction size of a NPT increases in the number of PTs P.

C

Before trading, the initial certainty equivalent of an investor i is given by

The certainty equivalent after trading has taken place amounts to

Inserting the equilibrium values of s and en or ep, respectively, into (15) yields the certainty equivalent of a PT or NPT, respectively:

Thus, the increases in welfare ΔΦi=Φi−Φiinitial are given by

Given P PTs, a PT p profits more (less) from trading than a NPT n does if ΔΦp>(<)ΔΦn. Because of (16) and (17), this is the case if

where

By substituting π=PP+h, (18) can be written as

Inserting εn−an=πε−an from (9) yields

with

D

Inserting ap=q, an=1−Pq/N, α, h and ε into (10) yields

with δ=P/L.

The representative PT p profits more from trading when the left-hand side (LHS) exceeds the right-hand side (RHS). Note that this requires a positive LHS, L>2δ/1−δ. Both sides of (19) are squared and rearranged to

(20) can only be satisfied when δ<0.5. Thus, (20) can be restated to

It can be shown that the RHS of (21) exceeds 2δ/1−δ within the relevant range. Therefore, (21) together with δ<0.5 are necessary and sufficient conditions for ΔΦp>ΔΦn.

E

A PT p would profit more if he were a NPT when ΦPap>ΦN+1ap−c . If he deviates to non-price-taking, the other N NPTs calculate with the “old”  λ because his deviation is unobservable. The PT who becomes a NPT calculates with a different price impact λN+1. It is derived analogously to the basic model. The resulting equilibrium share price sN+1 is given by

The resulting equilibrium demand of the PT who becomes a NPT amounts to

where ε=1+qhL+h. Inserting sN+1 and eN+1 and P=hh+L−2 into ΦPap>ΦN+1ap−c yields

Thus, a PT would not profit more if he were a NPT when his trading potential is sufficiently small: ε−ap<u, with u=2rσ2L+hL+h−2L+h−12−1⋅c.

Analogously, a NPT would not profit more if he were a PT when  Φn−c>ΦP+1. As his deviation is not observable, the remaining NPTs calculate with the “old” λ . The resulting share price sP+1 amounts to

The equilibrium demand of the “new” PT is given by

Inserting these expression into ΦP+1ap<Φnap−c yields

with l=2rσ2⋅c⋅L+h−12L+h−2L+h.

Thus, a NPT would not profit more if he were a PT when his trading potential is sufficiently large. When c=0, every investor with positive trading potential would profit more as NPT.

F

The PT’s increase in certainty equivalent is given by

ΔΦp=σ22rqh+1h+L−ap2=σ22rε−ap2. If the PT deviates to non-price-taking behavior, the initial endowment a PT holds on average q changes to q−=Pq−aP−1. The price-impact is given by λN+1=σ2rP−1h−, with h−=h−Δ and Δ only being dependent on L and P. As a result the trading potential changes to ε−=q−h−Δ+1h−Δ+L, which leads to ΔΦN+1=σ22r1+2h−ΔP−11+h−ΔP−12q−h−Δ+1h−Δ+L−an2. Inserting the expressions for ΔΦp and ΔΦN+1 into ΦPa>(<)ΦN+1a  yields

ΔΦN+1=σ22r1+2h−ΔP−11+h−ΔP−12q−h−Δ+1h−Δ+L−an2. Inserting the expressions for ΔΦp and ΔΦN+1 into ΦPa>(<)ΦN+1a yields

with Q=ε−aq−a. Three cases have to be distinguished. First, h+L+h−PΔP−1Q<0 is considered. In this case ΦPa>(<) ΦN+1a leads to

which can be rewritten to

In the second and third case, h+L+h−PΔP−1Q>0. If Q>0, ΦPa>(<) ΦN+1a yields

If Q<0,

In all of the cases, the PT would not profit more as a NPT if the absolute value of Q is sufficiently high, which implies his trading potential has to be sufficiently high.

G

In the following the condition for depds=rpσ2<dends=1λρ+σ2rn is derived. Inserting λρ into this inequality yields

Inserting hρ and squaring both sides leads to

Restating yields

This means that the price-demand function of a NPT sen can be flatter than that of a PT if ρ=rprn is sufficiently small.

H

λρ is given by

where hρ=ρ−1δL+L2−4L−δL−1−ρ−1δL−L−2. Its derivative with respect to δ amounts to

This expression is positive (negative) if

where A≡ρ−1δL+L2−4L−δL−1. Squaring both sides yields

Restating leads to

I

In order to derive eq. (13), recall the eqs (3) and (4): ∑p=1Pep=∑prpσ2μ−s and ens=1λn+σ2rμ−s+λnan. With Rp denoting ∑prp the equilibrium share price is given by

Differentiating with respect to en yields

Consequently, the price effect of a NPT i≠n is given by 1λi=Rpσ2+∑j≠i1λj+σ2rj. Subtracting 1λi from 1λn yields

Solving for λi delivers

which finally yields

Inserting this expression into 1λn=Rpσ2+∑j≠n1λj+σ2rj leads to eq. (13).

J

In the case of N=2λ1 is given by

with h1=12−1+1+4r1R1−r1R2−r2R2. Analogously, λ2=σ2r2⋅h2. Inserting λ1 and λ2 into the market clearing condition 1=∑p=1Pep+∑n=1Nen and solving for s yields

with γA=1+h1h2∑pap+h11−a1+h21−a21+h1h2RpR+h11−r1R+h21−r2r.

K

Investor i’s increase in certainty equivalent is given by

Inserting the PT’s equilibrium demand ep∗=γAR⋅rp yields

Analogously, inserting en∗=11+hnγAR⋅rn+hn⋅an delivers

With the help of these two expressions, Proposition 1 A follows:

with αA≡1+2hn1+hn2<1. In order to obtain Proposition 2 A, the investors’ transaction sizes are considered. ΔΦp> (<)ΔΦn can be restated to

which can be rewritten to