  We study the effect of variations in dust size distribution and composition
on the correlation between the spectral shape of extinction (parameterized by
$R_{\textrm{V}}$) and far-infrared dust emissivity (parameterized by the
power-law index $\beta$). Starting from the size distribution models proposed
by Weingartner & Draine (2001), using the dust absorption and emission
properties derived by Laor & Draine (1993) for carbonaceous and silicate
grains, and by Li & Draine (2001) for PAH grains, we calculate the extinction
and compare it with the reddening vector derived by Schlafly et al. (2016). An
optimizer and an MCMC are used to explore the space of available parameters for
the size distributions. We find that larger grains are correlated with high
$R_{\textrm{V}}$. However, this trend is not enough to explain the
emission-extinction correlation observed by Schlafly et al. (2016). For the
$R_{\textrm{V}}-\beta$ correlation to arise, we need to impose explicit priors
for the carbonaceous and silicate volume priors as functions of
$R_{\textrm{V}}$. The results show that a composition with higher ratio of
carbonaceous to silicate grains leads to higher $R_{\textrm{V}}$ and lower
$\beta$. A relation between $E(\textrm{B}-\textrm{V})/\tau_{353}$ and
$R_{\textrm{V}}$ is apparent, with possible consequences for the recalibration
of emission-based dust maps as a function of $R_{\textrm{V}}$.
