RDP 2016-07: The Efficiency of Central Clearing: A Segmented Markets Approach Equation (B1)

L= 𝔼 0 ∑ t=0 ∞ β t { ∫ 0 1 u( c m,t )+ μ m,t [ A m,t + p m,t s m,t−1 +( 1− λ m ) f m,t s m,t−1 + λ m [ M( y m,t ) y m,t +( 1−M( y m,t ) ) f m,t +D( y m,t )max( 0,Z−[ f m,t − y m,t ] ) + ( 1−D( y m,t ) )Z ] s m,t−1 − c m,t − p m,t s m,t − λ m s m,t Z − λ m 1 2 ∫ 0 1 λ n λ ¯ { M( y n,t+1 )( s n,t y n,t+1 + s ˜ n,t f n,t+1 + D ˜ ( y n,t+1 ) s ˜ n,t min( Z, [ y n,t+1 − f n,t+1 ] ) ) +( 1−M( y n,t+1 ) )( s n,t f n,t+1 + s ˜ n,t y n,t+1 −D( y n,t+1 ) s n,t min( Z,[ f n,t+1 − y n,t+1 ] )) }dn ]dm + ∫ 0 1 u( c ˜ m,t )+ μ ˜ m,t [ A ˜ m,t + p m,t s ˜ m,t−1 +( 1− λ m ) f m,t s ˜ m,t−1 + λ m [ M( y m,t ) f m,t +( 1−M( y m,t ) ) y m,t − D ˜ ( y m,t )max( 0,Z−[ y m,t − f m,t ] ) − ( 1− D ˜ ( y m,t ) )Z ] s ˜ m,t−1 − c ˜ m,t − p m,t s ˜ m,t − λ m s ˜ m,t Z − λ m 1 2 ∫ 0 1 λ n λ ¯ { M( y n,t+1 )( s n,t y n,t+1 + s ˜ n,t f n,t+1 + D ˜ ( y n,t+1 ) s ˜ n,t min( Z, [ y n,t+1 − f n,t+1 ] ) ) +( 1−M( y n,t+1 ) )( s n,t f n,t+1 + s ˜ n,t y n,t+1 −D( y n,t+1 ) s n,t min( Z,[ f n,t+1 − y n,t+1 ] )) }dn ]dm } 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