How to use .fieldname to jq's startswith

In the jq script below I want to print either the concatenation of the first_name and the last_name or just the last_name.

$ echo '{"first_name": "John", "last_name": "Johnson"}' | jq -c '{"name": (if (.last_name | startswith(.first_name))            then .last_name            else .first_name +" " + .last_name            end)}'

The error is

jq: error (at <stdin>:1): Cannot index string with string "first_name"

The jq command fails in the call to startswith - as if startswith only accepted string literals. If I change startswith(.first_name) to startswith("John") then the expression compiles and works as expected.

In the real-world example, there is many input records and many different first_names. Is there a way I could plug in .first_name to startswith?