Micro-Raman spectroscopy is used to monitor the cooling-induced compressive deformation of carbon nanotubes embedded in an epoxy matrix. Young's modulus of single- and multiwall nanotubes may then be derived from a concentric cylinder model for thermal stresses, using the D^*-band shift for each tube type. The resulting values of the elastic moduli are in very good agreement with predicted theoretical values, and with the published experimental data set of Treacy et al., Nature (London) 381, 678 (1996).