The mass function dN \propto m ^ { - \beta _ { 0 } } dm of molecular clouds and clumps is shallower than the mass function dN \propto m ^ { - \beta _ { \star } } dm of young star clusters , gas-embedded and gas-free alike , as their respective mass function indices are \beta _ { 0 } \simeq 1.7 and \beta _ { \star } \simeq 2 .
We demonstrate that such a difference can arise from different mass-radius relations for the embedded-clusters and the molecular clouds ( clumps ) hosting them .
In particular , the formation of star clusters with a constant mean volume density in the central regions of molecular clouds of constant mean surface density steepens the mass function from clouds to embedded-clusters .
This model is observationally supported since the mean surface density of molecular clouds is approximately constant , while there is a growing body of evidence , in both Galactic and extragalactic environments , that efficient star-formation requires a hydrogen molecule number density threshold of n _ { th } \simeq 10 ^ { 4 - 5 } cm ^ { -3 } .

Adopting power-law volume density profiles of index p for spherically symmetric molecular clouds ( clumps ) , we define two zones within each cloud ( clump ) : a central cluster-forming region , actively forming stars by virtue of a local number density higher than n _ { th } , and an outer envelope inert in terms of star formation .
We map how much the slope of the cluster-forming region mass function differs from that of their host-clouds ( clumps ) as a function of their respective mass-radius relations and of the cloud ( clump ) density index .
We find that for constant surface density clouds with density index p \simeq 1.9 , a cloud mass function of index \beta _ { 0 } = 1.7 gives rise to a cluster-forming region mass function of index \beta \simeq 2 .
Our model equates with defining two distinct SFEs : a global mass-varying SFE averaged over the whole cloud ( clump ) , and a local mass-independent SFE measured over the central cluster-forming region .
While the global SFE relates the mass function of clouds to that of embedded-clusters , the local SFE rules cluster evolution after residual star-forming gas expulsion .
That the cluster mass function slope does not change through early cluster evolution implies a mass-independent local SFE and , thus , the same mass function index for cluster-forming regions and embedded-clusters , that is , \beta = \beta _ { \star } .
Our model can therefore reproduce the observed cluster mass function index \beta _ { \star } \simeq 2 .

For the same model parameters , the radius distribution also steepens from clouds ( clumps ) to embedded-clusters , which contributes to explaining observed cluster radius distributions .