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Stagflation, a portmanteau of the words stagnation and inflation, is a term in general use within modern macroeconomics used to describe a period of out-of-control price inflation combined with slow-to-no output growth, rising unemployment, and eventually recession.
Stagflation is a problem because the two principal tools for directing the economy, fiscal policy and monetary policy, offer only trade offs between growth and inflation. A central bank can either slow growth to reduce inflationary pressures, or it can allow general increases in price to occur in order to stimulate growth. Stagflation creates a dilemma in that efforts to correct stagnation only worsen inflation, and vice versa. The dilemma in monetary policy is instructive. The central bank can make one of two choices, each with negative outcomes. First, the bank can choose to stimulate the economy and create jobs by increasing the money supply (by purchasing government debt), but this risks boosting the pace of inflation. The other choice is to pursue a tight monetary policy (reducing government debt purchases in order to raise interest rates) to reduce inflation, at the risk of higher unemployment and slower output growth.
The problem for fiscal policy is far less clear. Both revenues and expenditures tend to rise with inflation, all else equal, while they fall as growth slows. Unless there is a differential impact on either revenues or spending due to stagflation, the impact of stagflation on the budget balance is not altogether clear. As a policy matter, there is one school of thought that the best policy mix is one in which government stimulates growth through increased spending or reduced taxes while the central bank fights inflation through higher interest rates. In reality coordinating fiscal and monetary policy is not an easy task.
This situation is definitely a problem as people face inflation but the economy is in recession and people might be unemployed.