### Abstract:

We consider a time-dependent spatial economic model for capital in which the region's production function is a parameter. This forward model predicts the distribution of capital of a region based on that region's production function. We will solve the inverse problem based on this model, i.e. given data describing the capital of a region we wish to determine the production function through discretization. Inverse problems are generally ill-posed, which in this case means that if the data describing the capital are changed slightly, the solution of the inverse problem could change dramatically. The solution we seek is therefore a probability distribution of parameters. However, this probability distribution is complex, and at best we can describe some of its features. We describe the solution to this inverse problem using two different techniques, Markov chain Monte Carlo (Metropolis Algorithm ) and least squares optimization, and compare summary statistics coming from each method.