Technological Progress

TheoryGuru applied to Chicago Price Theory

Setup

Load Economicreasoning package only if it is not already loaded

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

Setup

competitivefactormarkets={(Subscript[ΔS, L])-(Subscript[ΔS, k])==ΔW +ΔL-ΔR-Δk,
ΔTFP==ΔY-((Subscript[S, L])ΔL+(Subscript[S, k])Δk),
ΔTFP==(Subscript[S, L])ΔW+(Subscript[S, k])ΔR,
ΔB==ΔW-ΔR-(1/σ)(Δk-ΔL)(* also embeds homothetic production *)};

longrun=ΔR==0;

shortrun={ΔL==0,Δk==0};

unbiased=ΔB==0;

capitalbiased=ΔB<0;

constantshare={(Subscript[ΔS, L])==0,(Subscript[ΔS, k])==0};

Growth Results

Long-run link between TFP and the real wage

TheoryGuru[{competitivefactormarkets[[3]],longrun},ΔW==((ΔTFP)/(Subscript[S, L]))]

Short-run link between TFP and the real wage

TheoryGuru[{competitivefactormarkets[[3;;4]],(Subscript[S, L])+(Subscript[S, k])==1,shortrun},ΔW==ΔTFP+(Subscript[S, k])ΔB]

Sufficient conditions for signing technological bias

TheoryGuru[{competitivefactormarkets[[4]],
ΔW-ΔR<0,Δk-ΔL>0,σ>0},
ΔB<0]

Share changes and factor prices

General case

TheoryGuru[competitivefactormarkets[[{1,-1}]],
(Subscript[ΔS, L])-(Subscript[ΔS, k])==((1/σ)-1)(Δk -ΔL)+ΔB]

Short run

TheoryGuru[{competitivefactormarkets[[1]],shortrun},(Subscript[ΔS, L])-(Subscript[ΔS, k])==ΔW -ΔR]

Long run with constant shares

TheoryGuru[{Most@competitivefactormarkets,constantshare,(Subscript[S, L])+(Subscript[S, k])==1,longrun},
Δk==ΔY&&ΔW==((ΔTFP)/(Subscript[S, L]))]

Wage changes decomposed into TFP, capital deepening, and technological bias

TheoryGuru[{competitivefactormarkets[[3;;4]],(Subscript[S, L])+(Subscript[S, k])==1},
ΔW==ΔTFP+(Subscript[S, k])((1/σ)(Δk-ΔL)+ΔB)]

Long-run capital deepening

TheoryGuru[{competitivefactormarkets[[4]],ΔW>0,σ≥1,unbiased||capitalbiased,longrun},
Δk-ΔL==σ(ΔW-ΔB)>0]

TheoryGuru[{competitivefactormarkets[[3;;4]],ΔW>0,σ≥1,unbiased||capitalbiased,longrun},
Δk-ΔL==σ(((ΔTFP)/(Subscript[S, L]))-ΔB)>0]

Variable interpretations

Capital-tax Results

Additional setup

longruntax={ΔR+Δ[1-τ]==0,ΔTFP==0,ΔB==0};

taxincrease={0<τ<1,Δτ>0,τ Δτ+(1-τ)Δ[1-τ]==0};

sharerestrictions={(Subscript[S, k])+(Subscript[S, L])==1,0<(Subscript[S, L])<1,σ>0};

fixedlabor=ΔL==0;

Results

TheoryGuru[{competitivefactormarkets[[3]],Most@longruntax},
ΔW==((Subscript[S, k])/(Subscript[S, L]))Δ[1-τ]]

TheoryGuru[{taxincrease,sharerestrictions},
((Subscript[S, k])/(Subscript[S, L]))Δ[1-τ]<0∧σ((Δ[1-τ])/(Subscript[S, L])) <0∧((Subscript[S, k])/(Subscript[S, L]))σ Δ[1-τ]<0]

TheoryGuru[{competitivefactormarkets[[3;;4]],(Subscript[S, k])+(Subscript[S, L])==1,longruntax},
Δk-ΔL==σ ((Δ[1-τ])/(Subscript[S, L]))]

TheoryGuru[{competitivefactormarkets[[2;;4]],(Subscript[S, k])+(Subscript[S, L])==1,longruntax},
ΔY-ΔL==((Subscript[S, k])/(Subscript[S, L]))σ Δ[1-τ]]

TheoryGuru[{competitivefactormarkets[[2;;4]],unbiased,sharerestrictions,
taxincrease,longruntax},
ΔW<0&&Δk-ΔL<0&&ΔY-ΔL<0]

TheoryGuru[{competitivefactormarkets[[3;;4]],sharerestrictions,
fixedlabor,taxincrease,longruntax},
τ (Subscript[S, k])(Δτ+ΔR+Δk)+(Subscript[S, L])(ΔW+ΔL)<0]