We constrain the shape of the primordial power spectrum using recent measurements of the cosmic microwave background (CMB) from the Wilkinson Microwave Anisotropy Probe (WMAP) 7-year data and other high-resolution CMB experiments. We also include observations of the matter power spectrum from the luminous red galaxy (LRG) subset DR7 of the Sloan Digital Sky Survey (SDSS). We consider two different models of the primordial power spectrum. The first is the standard nearly scale-invariant spectrum in the form of a generalized power-law parametrized in terms of the spectral amplitude As, the spectral index ns and (possibly) the running parameter nrun. The second spectrum is derived from the Lasenby and Doran (LD) model. The LD model is based on the restriction of the total conformal time available in a closed Universe and the predicted primordial power spectrum depends upon just two parameters. An important feature of the LD spectrum is that it naturally incorporates an exponential fall-off on large scales, which might provide a possible explanation for the lower-than-expected power observed at low multipoles in the CMB. In addition to parameter estimation, we compare both models using Bayesian model selection. We find there is a significant preference for the LD model over a simple power-law spectrum for a CMB-only data set, and over models with an equal number of parameters for all the data sets considered.