time saving

Author: tecunningham

posts/2025-12-17-llm-time-saving-demand-theory-substitution.llm.qmd

@@ -126,6 +126,81 @@ Many good:
 
 Two-good case.
 
+$$\frac{1}{(1-t_2)+t_2/A'}\;\le\;\frac{v'}{v}\;\le\;(1-t_2')+A't_2'$$
+
+```mermaid
+graph TD
+  %% Assumptions
+  A1["A1: Tasks & time endowment"]
+  A2["A2: Productivities; z_i = A_i t_i"]
+  A3["A3: y is monotone, concave, homogeneous"]
+  A4["A4: Differentiability (Shephard/envelope)"]
+  C1["C1: CES aggregator"]
+  T1["T1: Activation costs (discrete activation)"]
+
+  %% Propositions / corollaries
+  P1["Prop 1: Time allocation ⇔ time-price formulation"]
+  P2["Prop 2: V(A)=1/P(1/A)"]
+  P3["Prop 3: V(A')/V(A)=P(p)/P(p')"]
+  P4["Prop 4: Laspeyres/Paasche-type bounds"]
+  P5["Prop 5: EV/CV in time units"]
+  P6["Prop 6: Time shares = Hicksian shares"]
+  P7["Prop 7: d ln V = Σ t_i d ln A_i"]
+  C71["Cor 7.1: First-order baseline-share approximation"]
+  P8["Prop 8: Exact integral for large changes"]
+  C81["Cor 8.1: Single-component integral"]
+  P9["Prop 9: Törnqvist/Divisia trapezoid approximation"]
+  P10["Prop 10: CES price index and shares"]
+  P11["Prop 11: Two-good CES gain closed form"]
+  P12["Prop 12: CES share response / elasticity identification"]
+  C121["Cor 12.1: CES limiting cases"]
+  P13["Prop 13: Activation ⇒ possible non-differentiability"]
+
+  %% Dependencies (edges indicate proof dependence)
+  A1 --> P1
+  A2 --> P1
+
+  A3 --> P2
+  P1 --> P2
+
+  P2 --> P3
+
+  A3 --> P6
+  P1 --> P6
+  P2 --> P6
+
+  A3 --> P5
+  P2 --> P5
+
+  A3 --> P4
+  P3 --> P4
+  P6 --> P4
+
+  A3 --> P7
+  A4 --> P7
+  P2 --> P7
+  P6 --> P7
+
+  P7 --> C71
+  P7 --> P8
+  P8 --> C81
+  P8 --> P9
+
+  C1 --> P10
+  A4 --> P10
+
+  P3 --> P11
+  P10 --> P11
+  P6 --> P11
+
+  P10 --> P12
+  P6 --> P12
+
+  P11 --> C121
+
+  T1 --> P13
+  ```
+
 
 
 #        Setup