Concave and quasiconcave production functions

TheoryGuru applications

Load Economicreasoning package only if it is not already loaded

If[Length@Names["PLTools`*"]<10,
Get["http://economicreasoning.com"]]

SetOptions[TheoryGuru,method->"Parallel"];

Load other tools by clicking on extras and/or evaluating below

If[Not@MemberQ[$ContextPath,"OtherTools`"],
Get["http://othertools.economicreasoning.com"]]

Derivatives at a Point

Remove[x,y,z,m]

2 inputs

fobeyseulersthm=x((∂f[x,y])/(∂x))+y((∂f[x,y])/(∂y))==f[x,y];

TheoryGuru[{QuasiConcaveFunctionQ[f[x,y],strictly->True],
x>0,y>0,"((∂f[x, y])/(∂x))">0,"((∂f[x, y])/(∂y))">0,
"\!\(\*FractionBox[\(∂\\ fobeyseulersthm\), \(∂x\)]\)","\!\(\*FractionBox[\(∂\\ fobeyseulersthm\), \(∂y\)]\)"(* hod 1 *)},
ConcaveFunctionQ@f[x,y]]

Variable interpretations

3 inputs

fobeyseulersthm3d=x((∂f[x,y,z])/(∂x))+y((∂f[x,y,z])/(∂y))+z((∂f[x,y,z])/(∂z))==f[x,y,z];

TheoryGuru[{QuasiConcaveFunctionQ[f[x,y,z],strictly->True],
x>0,y>0,z>0,"((∂f[x, y, z])/(∂x))">0,"((∂f[x, y, z])/(∂y))">0,"((∂f[x, y, z])/(∂z))">0,
"\!\(\*FractionBox[\(∂\\ fobeyseulersthm3d\), \(∂x\)]\)","\!\(\*FractionBox[\(∂\\ fobeyseulersthm3d\), \(∂y\)]\)","\!\(\*FractionBox[\(∂\\ fobeyseulersthm3d\), \(∂z\)]\)"(* hod 1 *)},
ConcaveFunctionQ@f[x,y,z]]

Variable interpretations

4 inputs

fobeyseulersthm4d=x((∂f[x,y,z,m])/(∂x))+y((∂f[x,y,z,m])/(∂y))+z((∂f[x,y,z,m])/(∂z))+m((∂f[x,y,z,m])/(∂m))==f[x,y,z,m];

TheoryGuru[{QuasiConcaveFunctionQ[f[x,y,z,m],strictly->True],
x>0,y>0,z>0,m>0,"((∂f[x, y, z, m])/(∂x))">0,"((∂f[x, y, z, m])/(∂y))">0,"((∂f[x, y, z, m])/(∂z))">0,"((∂f[x, y, z, m])/(∂m))">0,
"\!\(\*FractionBox[\(∂\\ fobeyseulersthm4d\), \(∂x\)]\)","\!\(\*FractionBox[\(∂\\ fobeyseulersthm4d\), \(∂y\)]\)","\!\(\*FractionBox[\(∂\\ fobeyseulersthm4d\), \(∂z\)]\)","\!\(\*FractionBox[\(∂\\ fobeyseulersthm4d\), \(∂m\)]\)"(* hod 1 *)},
ConcaveFunctionQ@f[x,y,z,m]]

Variable interpretations

Revealed preference derivation (8 points)

a, b, v, w are vectors

Define: a==((μ v)/(f[μ v])),b==(((1-μ) w)/(f[(1-μ) w])),λ==((f[μ v])/(f[μ v]+f[(1-μ) w]))

λ a==((μ v)/(f[μ v]+f[(1-μ) w])),(1-λ)b==(((1-μ)w)/(f[μ v]+f[(1-μ) w]))

definitions={λ==((f[μ v])/(f[μ v]+f[(1-μ) w]))};

positivef={f[μ v]>0,f[(1-μ) w]>0};

homogeneousf ={f[a]==f[b]==1,f[μ v+(1-μ)w]==(f[μ v]+f[(1-μ) w])f[λ a+(1-λ)b],
μ>0\[Implies]f[μ v]==μ f[v],μ<1\[Implies]f[(1-μ) w]==(1-μ) f[w]};

quasiconcavef={(f[a]==f[b]&&0<λ<1)\[Implies]f[λ a+(1-λ)b]≥1};

concavef={0<μ<1\[Implies]f[μ v+(1-μ)w]≥μ f[v]+(1-μ)f[w]};

TheoryGuru[{definitions,positivef,homogeneousf,quasiconcavef},concavef]

Variable interpretations