We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to solve a graph problem. We show that the local information cost (LIC) presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST-KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that Ω ( logLICτγlog(P)n) bits are required for solving a graph problem P with a τ-round algorithm that errs with probability at most γ. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST-KT1 model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a multiplicative spanner with stretch 2t − 1 that consists of at most O(n1+ 1 t +ε) edges, where ε = O(1/t2). Our main result is that any O(poly(n))-time algorithm must send at least Ω-( t12 n1+1/2t) bits in the CONGEST model under the KT1 assumption. Previously, only a trivial lower bound of Ω(-n) bits was known for this problem; in fact, this is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem in this setting. A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing a distributed spanner algorithm. In light of the work of King, Kutten, and Thorup (2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with Õ(n) communication complexity. Our result also implies time complexity lower bounds for constructing a spanner in the node-congested clique of Augustine et al. (2019) and in the push-pull gossip model with limited bandwidth.