InÖnite cake eating problem
Ifα =1 β^1 /σ σλ = 1 β^1 /σthen
1 β^1 /σ σ
=1 β^1 /σ( 1 σ)
+β1 β^1 /σ σ
β(^1 σ)/σ==1 β^1 /σ( 1 σ)
+1 β^1 /σ σ
β^1 /σ==1 β^1 /σ σ
1 β^1 /σ+β^1 /σhence(α,λ)solves the equation. Ifα =1 β^1 /σ σλ = 1 β^1 /σthen
1 β^1 /σ σ
=1 β^1 /σ( 1 σ)
+β1 β^1 /σ σ
β(^1 σ)/σ==1 β^1 /σ( 1 σ)
+1 β^1 /σ σ
β^1 /σ==1 β^1 /σ σ
1 β^1 /σ+β^1 /σhence(α,λ)solves the equation.