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THE JOURNAL OF ECONOMIC SCIENCES: THEORY AND PRACTICE, V.82, # 2, 2025, pp. 96-116
A.4 Linearization outline and steady state (compact)
Linearized IS (consumption Euler):
= − ( − − ). (A14)
+1
+1
Linearized Phillips curve:
= + . (A15)
+1
Steady-state relations (symbolic):
1 + ‾ = 1 + ‾ − , ‾ = ( ‾ + ‾), (A16)
and the static substitution equation:
‾ − ‾ = ‾ + ‾ ( ‾ − ‾ ). (A17)
0
1
A.5 Welfare functional and planner FOC
Representative-agent welfare:
2
= ∑ ∞ [ − ]. (A18)
0
=0
2
Planner’s marginal condition (linear approximation):
= (1 − ), (A19)
1
2
which pins down the interior optimal SDC remuneration qualitatively (higher
allows a higher ).
Appendix B - Sensitivity snapshot and Welfare Approximation
B.1 Sensitivity snapshot
Holding = −10%:
● = 0.0: = 0.8 × 1 × (−0.10) = −8.0%.
● = 0.8: = 0.8 × 0.2 × (−0.10) = −1.6%.
● = 0.6, = 0.6: = 0.6 × 0.4 × (−0.10) = −2.4%.
B.2 Consumption-equivalent welfare approximation (method)
For small welfare perturbations, compute consumption-equivalent solving:
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