Horizontal mergers in oligopoly: First-price vs. profit-share auction
---Linear example with N=2 and uniform distribution from which graphs in the paper are generated

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

3. Takeover bidding without signaling

Downstream oligopoly subgames

In  Cournot subgames, the equilibrium profits depend  only  on the synergy parameter of the merged firm, denoted by θ. Set N=2 and assume linear demand P(Q):=max{1-Q,0}, Q=q1+q2.

πm[qm_,qn_]:=(1- qm - qn-c+θ)qm
πn[qm_,qn_]:=(1- qm - qn-c)qn

diffl=(∂_qn)πn[qm,qn];diffw=(∂_qm)πm[qm,qn] ;

{qm,qn}={qm,qn}/.Solve[{diffl==0,diffw==0},{qm,qn}][[1]]

πm[θ]=FullSimplify[πm[qm,qn]]

πn[θ]=FullSimplify[πn[qm,qn]]

If  no merger has taken place, the default equilibrium profit is  π0

π0:=((((1-c))^2)/16)

3.1. First-price auction with an entry fee R

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

Πn[y_]:=F[y](πm[θ]-b[y]) +(∫_y^c)πn[z]f[z]\[DifferentialD]z-R

R=F[r](πm[r]-π0);

The equilibrium bidding strategy is:

b[θ_]:=(∫_r^θ) ((F'[x])/(F[θ]))(πm[x]-πn[x])\[DifferentialD]x

Without externality, the equilibrium bid function is:

(OverHat[b])[θ_]:=(∫_r^θ)((F'[x])/(F[θ])) (πm[x]-π0)\[DifferentialD]x

Linear Example with N=2 and uniform distribution, With N=2, G(.)=F(.):

πm[θ_]:=((((1-c+2θ))^2)/9);πn[θ_]:=((((1-c-θ))^2)/9);π0=((((1-c))^2)/16);
F[θ_]:=(θ/c);f[y_]:=(1/c);

Solve[πn[θ]-π0==0,θ][[1]]

Externality: If θ<((1-c)/4), there is positive externality

c=0.49;r=0.001;

FullSimplify[b[θ]]

FullSimplify[(OverHat[b])[θ]]

Solve[πn[θ]-π0==0,θ][[1]]

list={{0.1275,0},{0.1275,0.083}}

figure11=ListPlot[list,PlotJoined->True, PlotStyle->{{DotDashed,Line[{{0,0},{2,1}}]},Thickness[.003]}]

figure7=Plot[b[θ],{θ,0.001,0.49},PlotRange->{0,0.09},PlotStyle->{Thickness[.006]}]

figure8=Plot[(OverHat[b])[θ],{θ,0.001,0.49},PlotRange->{0,0.09},PlotStyle->{Dashing[{.01,.021}],Thickness[.006]}]

Show[{figure7,figure8,figure11},Epilog->{Text["r",{0.012,0.005}],Text["θ",{0.43,0.005}],Text["b(θ)",{0.12,.015}],Text["(OverHat[b])(θ)",{0.1,.03}]},PlotRange->{0,0.09}]

3.2 Profit share auction

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

Πs[y_]:=F[y] (1-s[y]) πm[θ]+(∫_y^c)f[z] πn[z]\[DifferentialD]z-R

R=F[r](πm[r]-π0);

The equiliibrium strategy is:

s[θ_]:=(∫_r^θ)((F'[x])/(F[θ]))(( πm[x]-πn[x])/(πm[x]))\[DifferentialD]x

The effect of externality on bidding in the profit-share auction

Linear example with uniform distribution:

πm[θ_]:=((((1-c+2θ))^2)/9);πn[θ_]:=((((1-c-θ))^2)/9);π0=((((1-c))^2)/16);
F[θ_]:=(θ/c);
c=0.49;r=0.001;

Integrands=FullSimplify[((F'[x])/(F[θ]))(( πm[x]-πn[x])/(πm[x]))]

int=∫(Integrands)\[DifferentialD]x

s1=FullSimplify[FullSimplify[int/.x->θ ]-FullSimplify[int/.x->r]]

(OverHat[s])[θ_]:=FullSimplify[(∫_r^θ)((F'[x])/(F[θ]))(( πm[x]-π0)/(πm[x]))\[DifferentialD]x]

Integrandshat=FullSimplify[((F'[x])/(F[θ]))(( πm[x]-π0)/(πm[x]))]

inthat=∫Integrandshat\[DifferentialD]x

s1hat=FullSimplify[FullSimplify[inthat/.x->θ]-FullSimplify[inthat/.x->r]]

pse=FullSimplify[s1 πm[θ]]

psne=FullSimplify[s1hat πm[θ]]

figure12=Plot[pse,{θ,0.001,0.49},PlotStyle->{Thickness[.006]}]

figure13=Plot[psne,{θ,0.001,0.49},PlotStyle->{Dashing[{.01,.021}],Thickness[.006]}]

Solve[πn[θ]-π0==0,θ][[1]]

list={{0.1275,0},{0.1275,0.19}}

figure14=ListPlot[list,PlotJoined->True, PlotStyle->{{DotDashed,Line[{{0,0},{2,1}}]},Thickness[.003]}]

Show[{figure12,figure13,figure14},Epilog->{Text["r",{0.015,0.0065}],Text["(Subscript[π, n])(θ)=(Subscript[π, 0])",{0.13,0.2}],Text["s(θ)(Subscript[π, m])(θ)",{0.2,.035}],Text["(OverHat[s])(θ)(Subscript[π, m])(θ)",{0.09,.049}]},AxesLabel->{"θ","s(θ)(Subscript[π, m])(θ), (OverHat[s])(θ)(Subscript[π, m])(θ)"}]

3.3 Superiority of the profit-share auction

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

The expected profit of the target firm is Us:

Us=FullSimplify[Integrate[s[y]πm[y]f1[y],{y,r,c}]+2R(1-F[r])]

R=F[r](πm[r]-π0);
F[θ_]:=(θ/c);f[θ_]:=(1/c);f1[θ_]:=2 F[θ]f[θ];

s[θ_]:=(∫_r^θ)((F'[x])/(F[θ])) ((πm[x]-πn[x])/(πm[x]))\[DifferentialD]x

FullSimplify[s[θ]]

πm[θ_]:=(((1-c+2θ))^2)/9;πn[θ_]:=(((1-c-θ))^2)/9;π0=((((1-c))^2)/16);

integrands=Simplify[((πm[x]-πn[x])/(θ πm[x]))]

Ints=FullSimplify[∫integrands\[DifferentialD]x]

ss=FullSimplify[FullSimplify[Ints/.x->θ]-FullSimplify[Ints/.x->r]]

Integrandus1=FullSimplify[ss πm[θ]f1[θ]]

intus1=FullSimplify[∫Integrandus1\[DifferentialD]θ]

us1=FullSimplify[FullSimplify[intus1/.θ->c]-FullSimplify[intus1/.θ->r]]

US=FullSimplify[us1+((2 (c-r) r (πm[r]-π0))/(c^2))]

Uc=FullSimplify[Integrate[b[θ]f1[θ],{θ,r,c}]+2R(1-F[r])]

b[θ_]:=1/F[θ](∫_r^θ)F'[y] (πm[y]-πn[y])\[DifferentialD]y

Integranduc1=FullSimplify[((2 θ b[θ])/(c^2))]

Intuc1=FullSimplify[∫Integranduc1 \[DifferentialD]θ]

uc1=FullSimplify[FullSimplify[Intuc1/.θ->c]-FullSimplify[Intuc1/.θ->r]]

UC=FullSimplify[uc1+((2 (c-r) r (πm[r]-π0))/(c^2))]

c=0.49;

figure9=Plot[US,{r,0,0.49},PlotStyle->{Thickness[.006]}]

figure10=Plot[UC,{r,0,0.49},PlotStyle->{Dashing[{.01,.021}],Thickness[.006]}]

figure12=Plot[π0,{r,0,0.49},PlotStyle->{{DotDashed,Line[{{0,0},{2,1}}]},Thickness[.006]}]

Show[{figure9,figure10,figure12},Epilog->{Text["r",{0.5,0.005}],Text["(Subscript[U, c])(r)",{0.1,.067}],Text["(Subscript[U, s])(r)",{0.1,.105}],Text["π0",{0.1,.021}]}]

A=Simplify[y-x Log[x+y]/.y->V]

B=Simplify[y-x Log[x+y]/.y->0]

FullSimplify[A-B]

FullSimplify[((V+x Log[x]-x Log[V+x])/V)]

figure1=Plot[((V+x Log[x]-x Log[V+x])/V)/.x->0.1,{V,0,1},PlotRange->{0,1}]

figure2=Plot[((V+x Log[x]-x Log[V+x])/V)/.x->0.01,{V,0,1},PlotRange->{0,1}]

figure3=Plot[((V+x Log[x]-x Log[V+x])/V)/.x->0.001,{V,0,1},PlotRange->{0,1}]

Show[{figure1,figure2,figure3},PlotRange->{0,1}]

s[V_]:=((V+x Log[x]-x Log[V+x])/V)

Plot[Evaluate[s[V]/.x->{0.1,0.01,0.001},{V,0,1},PlotRange->{0,1},AxesLabel->{"v", "s(v)"}],TextStyle->FontFamily->{"Lucida Bright"},
Epilog->
{Text["x=.1",{0.8,0.6}],Text["x=((.01)^ )",{0.7,0.85}],Text["x=((.001)^ )",{0.4,0.95}]}, PlotStyle->{Thickness[.006]}]

Entry fee?

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

s[v_]:=(1/(G[v]))(∫_0^v)((G'[y]  y)/(y-v))\[DifferentialD]y

((V+x Log[x]-x Log[V+x])/V)/.x->c

figure1=Plot[((V+c Log[c]-c Log[V+c])/V)/.c->-0.1,{V,0,1}]

(1/v)(∫_0^v)((  y)/(y-c))dy

∫((  y)/(y-c))\[DifferentialD]y

Ac=y+c Log[-c+y]/.y->v

Bc=y+c Log[-c+y]/.y->0

Simplify[Ac-Bc]

Simplify[((v-c Log[-c]+c Log[-c+v])/v)]

((v-c Log[-c]+c Log[-c+v])/v)/.c->0.1

Plot[%,{v,0,1}]

Compare profit:

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

G[y_]:=y

G'[y]=1

Clear[s]

s[v_]:=(1/v)(∫_0^v)((  y)/(y+c))\[DifferentialD]y

∫((  y)/(y+c))\[DifferentialD]y

A=y-c Log[c+y]/.y->v

B=y-c Log[c+y]/.y->0

((A-B)/v)

((v+c Log[c]-c Log[c+v])/v)(v+c)2 v

∫2 (c+y) (y+c Log[c]-c Log[c+y])\[DifferentialD]y

AA=2 (((y^3)/3)+(1/2) (c^2) y (1+2 Log[c])+(1/4) c (y^2) (3+2 Log[c])-(1/2) (c^3) Log[c+y]-(1/2) c y (2 c+y) Log[c+y])/.y->1

BB=2 (((y^3)/3)+(1/2) (c^2) y (1+2 Log[c])+(1/4) c (y^2) (3+2 Log[c])-(1/2) (c^3) Log[c+y]-(1/2) c y (2 c+y) Log[c+y])/.y->0

FullSimplify[AA-BB-2c]

ES=(2/3)-(c/2)+(c^2)+c (((1+c))^2) (Log[c]-Log[1+c])

EC=Integrate[2y (1-y),{y,0,1}]

figure4=Plot[{ES},{c,0,.3}]

figure5=Plot[EC,{c,0,.3}, PlotStyle->{Dashing[{.01,.021}]}]

Show[{figure4,figure5}]

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

Merger

Profit-share auction

Π[y_]:=F[y]Πm[θ] (1-s[y])+Integrate[Πn[z]F'[z],{z,y,c}]

FullSimplify[D[Π[y],y]]

s[θ_]:=(1/(G[θ]))(∫_0^θ)((G'[y] (Πm[y]-Πn[y]))/(Πm[y]))\[DifferentialD]y

Πm[θ_]:=((((1-c+2θ))^2)/9)
Πn[θ_]:=((((1-c-θ))^2)/9)
Π0=((((1-c))^2)/16)
G[θ_]:=θ

s[θ_]:=(1/θ)(∫_0^θ)(( Πm[y]-Πn[y])/(Πm[y]))\[DifferentialD]y

FullSimplify[((Πm[θ]-Πn[θ])/(Πm[θ]))]

((3 θ (2-2 c+θ))/(((-1+c-2 θ))^2))/.c->0.2

∫((3 θ (1.6`+θ))/(((-0.8`-2 θ))^2))\[DifferentialD]θ

A=3 (0.1`+0.25` θ+((0.24000000000000005`)/(0.8`+2.` θ))+0.20000000000000004` Log[0.8`+2.` θ])

B=3 (0.1`+0.25` θ+((0.24000000000000005`)/(0.8`+2.` θ))+0.20000000000000004` Log[0.8`+2.` θ])/.θ->0

H=Simplify[A-B]

(H/θ)

Plot[%,{θ,0,.2}]

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

Π[y_]:=G[y]Πm[θ] (1-s[y])+(1-G[y])Πn[θ]

s[θ_]:=(1/(G[θ]))(∫_0^θ)((G'[y] (Πm[y]-Πn[y]))/(Πm[y]))\[DifferentialD]y

FullSimplify[Π[θ]]

(∫_0^θ)(((Πm[y]-Πn[y]) G'[y])/(Πm[y]))\[DifferentialD]y=(((Πm[y]-Πn[y]) G'[y])/(Πm[y]))/.y->θ-(∫_0^θ)(∂_y) (((Πm[y]-Πn[y])/(Πm[y])))G[y]\[DifferentialD]y

(((Πm[y]-Πn[y]) G'[y])/(Πm[y]))/.y->θ

Πstar=FullSimplify[(-(((Πm[θ]-Πn[θ]) G'[θ])/(Πm[θ]))+(∫_0^θ)(∂_y) (((Πm[y]-Πn[y])/(Πm[y])))G[y]\[DifferentialD]y)Πm[θ]+G[θ] (Πm[θ]-Πn[θ])+Πn[θ]]

FullSimplify[Πstar-Πn[θ]]

First-Price auction with reserve price

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

Π[y_]:=F[y](Πm[θ]-β[y]) +(∫_y^θ)Πn[z]f[z]\[DifferentialD]z

D[Π[y],y]/.F'[y]->f[y]

Profit-share auction

β[θ_]:=(1/(F[θ]))(∫_r^θ)F'[y] (Πm[y]-Πn[y])\[DifferentialD]y

β0[θ_]:=(1/(F[θ]))(∫_r^θ)F'[y] (Πm[y]-Π0)\[DifferentialD]y

β0[θ]

Πm[θ_]:=((((1-c+2θ))^2)/9)
Πn[θ_]:=((((1-c-θ))^2)/9)
Π0=((((1-c))^2)/16);
F[θ_]:=(θ/c)

F'[θ]

FullSimplify[β[θ]]

FullSimplify[β0[θ]]

c=0.49;r=0.001;

B=FullSimplify[β[θ]]

B0=FullSimplify[β0[θ]]

figure7=Plot[B,{θ,0.001,0.49},PlotRange->{0,0.09},PlotStyle->{Thickness[.006]}]

figure8=Plot[B0,{θ,0.001,0.49},PlotRange->{0,0.09},PlotStyle->{Dashing[{.01,.021}],Thickness[.006]}]

Show[{figure7,figure8},Epilog->{Text["r",{0.012,0.005}],Text["β(θ)",{0.12,.015}],Text["(OverHat[β])(θ)",{0.1,.03}]},PlotRange->{0,0.09}]

Epilog->{Text["(Subscript[r, min])",{0.206,.0752}],Text["(Subscript[G, n])(r)",{0.25,.0778}],Text["(Subscript[G, r])(r)",{0.26,.0785}] }

Profit comparison in Merger

Off[General::"spell1"];
Off[Remove::"rmnsm"];
Remove["Global`*"];

Δ=Integrate[(s[y]Πm[y]-β[y])f1[y],{y,r,c}]

f1[y_]:=2 F[y]f[y]

F[y_]:=(y/c)