The supply of a particular type of labor to a firm is less elastic than the supply of labor to the ...

False.  The supply of labor to a firm is arguably perfectly elastic whereas the supply of labor to the market is less elastic and positively sloped.  The supply of labor to an individual firm is perfectly elastic because, assuming the basic conditions of employment are essentially the same across employers of the labor type in question, if a given firm offered a wage below the going wage, workers simply would flock to other firms paying the going wage.  Likewise, a given firm has no incentive to pay above the going wage for the labor type, as it is able to attract as many workers as it needs at the going wage rate.  Labor supply to the market is less elastic because occupations differ in terms of their basic conditions of employment and because workers differ in terms of their tastes and preferences with respect to these conditions of employment.  For example, relatively few people would be willing to do construction work on high-rises or skyscrapers if the wage for such work was low because many other jobs would pay comparably with much less risk to safety.  A higher wage for such work, wages in other occupations constant, will attract more into high-rise construction because the additional pay will be enough to compensate some for their fear of heights.