Industry Equilibrium

TheoryGuru applications

Load Economicreasoning package only if it is not already loaded

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

Clear symbols that are about to be used

Remove[f,L,k,p,d,c,Y,w,r,x,SL,Sk]

Setup

Homogeneous cost function

c[w_,r_,Y_]:=Y w ch[r/w];

Industry model

Pricing=p==c[w,r,1] (* average cost = marginal cost *);

IndustryDemand=Y==d[p];

FactorDemands={L==Y ((∂c(w,r,1))/(∂w)),k==Y ((∂c(w,r,1))/(∂r))};

Production=Y==f[L,k];

Definitions

DefineShares={SL==((w L)/(p Y)),Sk==((r k)/(p Y))};

DefinePctPriceChanges={Δp==(((\[DifferentialD]p)/(\[DifferentialD]x))/p),Δw==(((\[DifferentialD]w)/(\[DifferentialD]x))/w),Δr==(((\[DifferentialD]r)/(\[DifferentialD]x))/r)};

DefinePctQuantityChanges={ΔY==(((\[DifferentialD]Y)/(\[DifferentialD]x))/Y),ΔL==(((\[DifferentialD]L)/(\[DifferentialD]x))/L),Δk==(((\[DifferentialD]k)/(\[DifferentialD]x))/k)};(* x indicates the thing that is changing, e.g., time, or a comparison between two markets *)

DefineElasticities={ϵd==(p/(d[p]))d'[p],σ==((ΔL-Δk)/(Δr-Δw))};

iqpositive={p>0,Y>0,L>0,k>0,w>0,r>0,0<SL<1,Derivative[1,0][f][L,k]>0,Derivative[0,1][f][L,k]>0};

Results

Marginal cost equals average cost, holding factor prices constant

((∂c(w,r,Y))/(∂Y))==((c[w,r,Y] )/Y)

Output price change is the revenue-share-weighted input price change

TheoryGuru[{FactorDemands,((\[DifferentialD]Pricing)/(\[DifferentialD]x)),
DefineShares,DefinePctPriceChanges},
Δp==SL Δw+Sk Δr]

Scale effect of a wage change depends on labor’s share and the price-elasticity of industry demand

TheoryGuru[{ Δr==0,IndustryDemand,FactorDemands,
((\[DifferentialD]Pricing)/(\[DifferentialD]x)),((\[DifferentialD]IndustryDemand)/(\[DifferentialD]x)),
DefineShares,DefinePctPriceChanges,DefinePctQuantityChanges,First@DefineElasticities},
ΔY==ϵd SL Δw]

Factor shares add to one

TheoryGuru[{Pricing,FactorDemands,DefineShares},
SL==1-Sk]

TheoryGuru[{Δr==0,Pricing,FactorDemands,
((\[DifferentialD]FactorDemands)/(\[DifferentialD]x)),((\[DifferentialD]Production)/(\[DifferentialD]x)),
Last@DefineShares,DefinePctPriceChanges,DefinePctQuantityChanges},
ΔY==ΔL+Sk (Δk-ΔL)]

Two of Marshall’s Laws

TheoryGuru[{Δr==0,Pricing,IndustryDemand,FactorDemands,
((\[DifferentialD]Pricing)/(\[DifferentialD]x)),((\[DifferentialD]IndustryDemand)/(\[DifferentialD]x)),((\[DifferentialD]FactorDemands)/(\[DifferentialD]x)),((\[DifferentialD]Production)/(\[DifferentialD]x)),
DefineShares,DefinePctPriceChanges,DefinePctQuantityChanges,DefineElasticities},
ΔL==(ϵd SL-Sk σ) Δw]

LR and SR demand for the other factor k

LR: k adjusts

TheoryGuru[{ΔY==ϵd SL Δw,ΔY==ΔL+Sk (Δk-ΔL),ΔL==(ϵd SL-Sk σ) Δw,SL+Sk==1,iqpositive},
Δk==(ϵd + σ)SL Δw,
keepall->True]

SR: k is fixed

TheoryGuru[{Δk==0,SL+Sk==1,IndustryDemand,FactorDemands,
((\[DifferentialD]Pricing)/(\[DifferentialD]x)),((\[DifferentialD]IndustryDemand)/(\[DifferentialD]x)),((\[DifferentialD]FactorDemands)/(\[DifferentialD]x)),((\[DifferentialD]Production)/(\[DifferentialD]x)),DefineShares,DefinePctPriceChanges,DefinePctQuantityChanges,DefineElasticities},
Δr(SL σ-ϵd Sk)==(ϵd + σ)SL Δw]

Variable interpretations