  A cornerstone of the theory of lambda-calculus is that intersection types
characterise termination properties. They are a flexible tool that can be
adapted to various notions of termination, and that also induces adequate
denotational models.
  Since the seminal work of de Carvalho in 2007, it is known that multi types
(i.e. non-idempotent intersection types) refine intersection types with
quantitative information and a strong connection to linear logic. Typically,
type derivations provide bounds for evaluation lengths, and minimal type
derivations provide exact bounds.
  De Carvalho studied call-by-name evaluation, and Kesner used his system to
show the termination equivalence of call-by-need and call-by-name. De
Carvalho's system, however, cannot provide exact bounds on call-by-need
evaluation lengths.
  In this paper we develop a new multi type system for call-by-need. Our system
produces exact bounds and induces a denotational model of call-by-need,
providing the first tight quantitative semantics of call-by-need.
