An interleukin receptor subunit with specificity for $INTERLEUKIN-$13. It dimerizes with the $INTERLEUKIN-$4 RECEPTOR ALPHA SUBUNIT to form the TYPE II INTERLEUKIN-4 RECEPTOR which has specificity for both $INTERLEUKIN-$4 and $INTERLEUKIN-$13. Signaling of this receptor subunit occurs through the interaction of its cytoplasmic domain with JANUS KINASES such as the TYK2 KINASE.