Mathematical representation - Inflation Unemployment Trade-off:
Suppose that firms correctly perceive the state of demand in the economy and the rate of price inflation. Then the actual real wage rate in the economy would always be equal to the real wage rate on the basis of which firms decide how much labour to demand. Then, if the economy is to remain at a fixed rate of unemployment, the real wage rate must be constant so that the rate of growth of the money wage rate must equal the rate of price inflation. That is, if w(t) denotes the money wage rate andp(4the price level in the economy in period t:
w(t)/p(t) = w(t+1)/p(t+1) = w(a positive constant), so that
[w(t+1)-w(t)]/w(t) = [p(t+1)-p(t)]/p(t)....(1)
Remember that [w(t+1)-w(t)]/w(t) gives the growth rate in nominal or money wage in period t.
However, suppose that the fixed rate of unemployment is lower than the natural rate. Then, for the amount of labour corresponding to this rate of unemployment to be supplied, suppose that a higher real wage rate must be expected by workers. Workers therefore will supply the amount of labour corresponding to this rate of unemployment, if and only if
w(t+l)/p^{e}(t+l) = w' . . .(2)
Where p^{e}(t+1) denotes the price level expected by workers in period t+1.
By rearranging terms in (2) we find that p^{e}(t+1) = (1/w')[(t+1) . . .(3)
That is, [p^{e}(t+1) -p(t)]/p(t) = (1/w')[w(t+1)/p(t)] - 1
That is, [p^{e}(t+1)- p(t)]/p(t) = (w/w')[p(t+1)/p(t)] - 1
That is, (w'/w)[p^{e}(t+1)- p(t)]/p(t) = [(w'/w) - 1] = [p(t+1) -p(t)]/p(t) ....(4)
Since w'/w> 1 it follows that the rate of growth of nominal demand in the economy must be such that the actual rates of nod wage and price inflation, given by [p(t+1)-p(t)]/p(t), must always be greater than the expected rate of price inflation [p^{e}(t+1)- p(t)]/p(t).
If despite the actual rate of price inflation king greater than the expected rate of price inflation, the workers expected that the hate of price inflation remain the same overtime, then the actual rate of price inflation on required to maintain the given level of unemployment would be constant over time. There would be a stable relation between the rate of unemployment and the rate of inflation as given by the Phillips curve.