Substitute(4)and(5)into (3)yields ,1-C1-C2 1-1-2C,1-2C11-21-2C 9 (24-1) P 1-C PI-C Here we use CI=C2 in the equilibrium. So Lt< O if u <72 For the left derivative.ie pt< pe we have CPI=C)P2 from(1) Differentiation yields CR+C-u)=CP+C(-p)=2- Together with (5), we have (1-2C)24 P So L:>0. To conclude, if u <12, R=I is TC 3 solution, i. e. Precommitment Solution is equivalent to Time Consistent Solution if price stickiness is sufficiently small. This is a simple si version of Proposition 3. 3 in Albanesi, Chari and Christiano(2003)• For the left derivative L - , i.e. Pf < Pe , we have C1P1 = C2P2 from (1). Differentiation yields • Substitute (4) and (5) into (3) yields ( )             − − − − − − − + − = − − + C P C C C C P P L     1 1 1 1 1 1 1 1 1 1 2 ( ) (2 1) 1 1 1 2 1 1 2 1 1 1 2 1 − − − − =      − − + − − −  = − − +      C C C P C C P C P L • Here we use C1=C2 in the equilibrium. So L + < 0 if μ < ½. ( ) ( ) 1 1 2 2 2 2 2 1 1 1 ' ' ' 1 ' 1 C C C C C P +C −  = C P +C −   = • Together with (5), we have ( ) P C C C C C 2 1 2 ' ' 1 1 2 2 + = − • So L - > 0. To conclude, if μ < ½, R = 1 is TC solution, i.e., Precommitment Solution is equivalent to Time Consistent Solution if price stickiness is sufficiently small. This is a simple version of Proposition 3.3 in Albanesi, Chari and Christiano (2003)