Substitute(4)and(5)into (3)yields r=(0-x)-1=+41-C-C2C(1 1-1-2C,1-2C11-21-2C 9 (24-1) P1-C P1-O Here we use C=C, in the equilibrium. So L+< O if u <72 For the left derivative.ie pt< pe we have CPI=C2P2 from(1). Differentiation yields C2P+C(-p)=CP+C(-p)=2= Together with (5), we have 2C P So L>0. To conclude, if H < /2,R=I is 9 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)