We study contact tracing in a new macro-epidemiological model in which infected agents may not show any symptoms of the disease and the availability of tests to detect these asymptomatic spreaders of the virus is limited. Contact tracing is a testing strategy aiming at reconstructing the infection chain of newly symptomatic agents. A coordination failure arises as agents fail to internalize that their individual consumption and labor decisions raise the number of traceable contacts to be tested, threatening the viability of the tracing system. The collapse of the tracing system considerably aggravates the pandemic's toll on the economy and mortality. A timely, limited lockdown solves the coordination failure allowing policymakers to buy time to expand the testing scale and to preserve the tracing system. We provide theoretical underpinnings to the risk of becoming infected in macro-epidemiological models. Our solution method is not affected by curse-of-dimensionality problems.