No, a shortage is the discrepancy between the quantity supplied and quantity demanded at a given price.
When wages offered are too low, there is a shortage of labour supply. Increase those wages and that shortage disappears.
>If we have a shortage of carrots, we dont have enough carrots at ANY price.
This is simply not true.
>If people are unwilling to pay 5$/LB for carrots but thats what they cost in the store, that is not a shortage.
The shortage of carrot occurs at every price below $5/lb. You cannot talk about shortages in the economic sense without prices. Period.
I'm not sure why people are insistent on arguing the semantics of the word "shortage", rather than the far more interesting supply/demand dynamics underpinning the markets being discussed.