A two-dimensional model of polymer chain folding invented by Zwanzig and Lauritzen is here studied using a grand ensemble and transfer matrix method. Due to the character of the model, there are no extensive parameters in the grand ensemble and the dispersion in system size is large, raising doubts about the validity and usefulness of the ensemble. We find it possible to define a thermodynamic limit such that it leads to near equivalence between the canonical and grand ensembles in the limit of large systems. The transfer matrix in this case is a nonlocal operator on a space of L2 functions, and the eigenvalue equation is a homogeneous Fredholm integral equation of the second kind which can be completely solved in terms of Bessel functions. The grand partition function can then be expressed as a sum of powers of the known eigenvalues. It is an easy matter to reproduce the second-order phase transition in the canonical ensemble found in the original work on the model. The investigation is extended to yield the probability densities describing the length of a segment and the correlations among segments. The concept of a local width of the folded chain is found to break down at higher temperatures, while critical correlations are characterized by infinite range, as expected. Apart from physical and methodological implications, the new solution provides striking illustrations of some basic ideas concerning phase transitions.