In this paper, we introduce and analyze an extension to the matching problem on a weighted bipartite graph (i.e. the assignment problem): Assignment with Type Constraints. Here, the two parts of the graph are each partitioned into subsets, called types and blocks respectively; we seek a matching with the largest sum of weights under the constraint that there is a pre-specified cap on the number of vertices matched in every type-block pair. Our primary motivation stems from the large-scale public housing program run by the state of Singapore, accounting for over 70% of its residential real estate. To promote ethnic diversity within its housing projects, Singapore imposes ethnicity quotas: the population is divided into ethnicity-based groups and each new housing development into blocks of flats such that each group must not own more than a certain percentage of flats in a block. However, other domains use similar hard capacity constraints to maintain diversity: these include matching prospective students to schools or medical residents to hospitals. Limiting agents' choices for ensuring diversity in this manner naturally entails some welfare loss. One of our goals is to study the tradeoff between diversity and (utilitarian) social welfare in such settings. We first show that, while the classic assignment program is polynomial-time computable, adding diversity constraints makes the problem computationally intractable; however, we identify a 1 2-approximation algorithm, as well as reasonable assumptions on the structure of utilities (or weights) which permit poly-time algorithms. Next, we provide two upper bounds on the price of diversity-a measure of the loss in welfare incurred by imposing diversity constraints-as functions of natural problem parameters. We conclude the paper with simulations based on publicly available data from two diversity-constrained allocation problems-Singapore Public Housing and Chicago School Choice-which shed light on how the constrained maximization as well as lottery-based variants perform in practice.