A DEPARTING B737 IS ISSUED AN ARRIVING B737 AS TFC AND ADVISED TO MAINTAIN VISUAL SEPARATION AND ISSUED A CLB.

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TFC WAS CALLED OUT TO US 1-2 O'CLOCK; 18 MI. WE GAINED VISUAL AT APPROX 10 MI. AT THIS TIME WE WERE TOLD TO MAINTAIN VISUAL SEPARATION AND CLB FROM 10000 FT TO 13000 FT MSL WHILE THE TFC WAS CLRED TO DSND FROM 13000 FT TO 11000 FT MSL. THE TFC WAS AT OUR 2 O'CLOCK DSNDING AND CLOSING. I WAS THE PF AND INITIATED THE CLB. THE CAPT SAID HE WAS NOT COMFORTABLE WITH THE SEPARATION AND SAID HE WANTED ME TO DSND BACK DOWN TO 10000 FT; WHICH I COMPLIED WITH. AS I BEGUN THE DSCNT WE GOT A TCASII RA TO INCREASE OUR VERT DSCNT RATE WHICH I COMPLIED WITH. WE NEVER GOT ABOVE ABOUT 10500 FT MSL AND THE OTHER ACFT WAS ONLY CLRED TO 11000 FT MSL. AS WE WERE DSNDING THE OTHER ACFT CHANGED COURSE; PROBABLY MORE TO KEEP US IN SIGHT AS WE DSNDED UNDER HIS NOSE AS OPPOSED TO EVASIVE ACTION. CONTRIBUTING FACTORS WERE THE CTLRS SEEMING INABILITY TO PERCEIVE THE POTENTIAL CONFLICT PRIOR TO ISSUING THE CLRNC (HE STILL DIDN'T UNDERSTAND WHEN WE EXPLAINED OUR DEV DUE TO THE TCASII RA). ALSO; THE CAPT WAS INDECISIVE IN HIS DECISION MAKING PROCESS -- CONTINUE THE CLB OR DSND? FINALLY; ALTHOUGH I WAS HAND FLYING THE AIRPLANE; THE CLB RATE COMMANDED BY THE FMC WAS INADEQUATE TO PROVIDE PROPER SEPARATION WITH THE CLOSURE RATE BTWN THE 2 ACFT. BOTH OF US SHOULD HAVE RECOGNIZED THIS SOONER AND CLBED STEEPER OR REFUSED THE CLRNC UNTIL THE OTHER ACFT WAS CLR OF OUR FLT PATH.