Worldwide, countries have been restricting work and social activities to counter an emerging public health crisis due to the coronavirus pandemic. These measures have caused dramatic increases in unemployment in the short run, with an expected deepening of the recession in the long run. Some commentators argue that the "draconian measures" will do more harm than good due to the economic contraction, despite a large literature that finds mortality rates decline during recessions. We estimate the relationship between unemployment, a widely accepted proxy for economic climate, and mortality in Australia, a country with universal health care. Using administrative time-series data on mortality that varies by state, age, sex, and cause of death collected for the years 1979-2017, we find no relationship between unemployment and mortality on average. However, we observe beneficial health effects in economic downturns for young men aged 25 to 34 associated with a reduction in vehicle transport accidents. Our estimates imply 425 fewer deaths if Reserve Bank of Australia expectations of a doubling of unemployment rates are realized by the end of 2020. For the early 1980s, we also find a procyclical pattern in the mortality rates of infants. However, this pattern disappears starting from the mid-1980s, coincident with the full implementation of universal health care in Australia in 1984. Our results suggest that universal health care may insulate individuals from the health effects of macroeconomic fluctuations. We conclude that the economic recession is an unlikely mediator for pandemic-related deaths in Australia.