Dear Muhammad,

I attach an example from Dornbusch et al Macroeconomics book to illustrate how this can be done.

Combing thses two, you get sacrifice ratio.

the relation between these two can be better understood if you go through

 A. W. H. Phillips: Collected Works in Contemporary Perspective.

Yes, they can be combined together.

The short-run Phillips curve, named after the New Zealand economist A.W.H. Phillips in 1958, shows the trade-off between the inflation rate and unemployment rate, holding constant

1. The expected inflation rate

2. The natural unemployment rate.

The Phillips curve was initially viewed by economists and policymakers as a structural relationship in the economy that depends on the basic behaviour of households and firms and remains unchanged over long periods. Thus, the central bank could permanently reduce the unemployment rate at the expense of higher inflation rate, and vice versa.

Two events changed the view that the Phillips curve represented a stable trade-off for policymakers. First, Milton Friedman and Edmund Phelps refused the assumption that  the expected inflation rate is being constant. They argued that it is more likely that expected inflation adjusts if current inflation rates are different from past inflation rates. They believed that if inflation is higher in 2012 than it has been in the past, households and firms will expect the higher inflation to persist in 2013. Expected inflation would affect both the prices charged by firms and the wages that workers seek. Accordingly, higher expected inflation in 2013 may also increase the actual inflation rate in 2013. Friedman and Phelps believed that as the inflation rate in the United States increased during the 1960s, it was only a matter of time before the stable inverse relationship between unemployment and inflation embodied in the Phillips curve broke down. The second event that weakened the view that the Phillips curve represented a stable trade-off between unemployment and inflation was the increase in both the inflation rate and the unemployment rate during the 1970s (i.e. the stagflation).

The Phillips curve can shift over time in response to supply shocks, such as an increase in the price of oil, and changes in the expected inflation rate. Accounting for these two factors, the Phillips curve remains useful in explaining the short-run trade-off between the unemployment rate and the inflation rate. To capture the effect of changes in the unemployment rate on the inflation rate, it is recommended to look at the gap between the current unemployment rate and the natural rate of unemployment (i.e. the cyclical unemployment). Cyclical unemployment is caused by a business cycle recession raising the unemployment rate above its full employment level. If the current unemployment rate equals the natural rate, the inflation rate typically does not change, holding constant expectations of inflation and the effects of supply shocks. If the current unemployment rate exceeds the natural rate, it would not be easy to find jobs, so wage increases and cost production increases will be limited. Therefore, the inflation rate will decrease. On the other hand, if the current unemployment rate is lower than the natural rate of unemployment, wages are likely to increase pushing up the costs of production. Hence, the inflation rate will increase.

Based upon the above discussion, the following equation for the Phillips Curve could be written.

πt = πet - a (Ut – UN)  – st.                  (1)

where

πt = current inflation.

πet= expected inflation rate.

Ut= current unemployment rate

UN=natural rate of unemployment.

st= variable representing the effects of a supply shock (s will have a negative value for a negative supply shock and a positive value for a positive supply shock.)

a = constant that represents how much the gap between the current rate of unemployment and the natural rate affects the inflation rate.

Equation (1) shows that an increase in expected inflation or a negative supply shock such as an increase in oil prices will shift the Phillips curve up, while a decrease in expected inflation or a positive supply shock such as an increase in the growth rate of productivity will shift the Phillips curve down.

Okun’s law, named after Arthur Okun who served as chairman of the President’s Council of Economic Advisers in the 1960s, summarizes the relationship between cyclical unemployment and the output gap. According to Okun’s Law, as real GDP increases by 1 percentage point relative to potential GDP, cyclical unemployment decreases by 0.5 percentage point (see equation 2 below)

(Ut – UN) = –0.5Ỹt,                                                                                     (2)

Where (Ut – UN) is the difference between current and natural rates of unemployment

Ỹt is the output gap

It is possible to modify the Phillips curve from being a relationship between the inflation rate and the unemployment rate to being a relationship between the inflation rate and the output gap. Equation (2) could be rewritten as shown by equation 3.

Ỹt = –2(Ut – UN).                                                                             (3)

Then, the Okun's law relationship could be substituted for the Phillips curve to obtain:

πt = πet + bỸt – st.                                                                                            (4)

The coefficient b represents the effect of changes in the output gap on the inflation rate. The term bỸt represents the effect of demand shocks on inflation, and st represents the effect of supply shocks on inflation. When real GDP increases relative to potential GDP during expansions, the inflation rate typically increases. On the other hand, when real GDP falls relative to potential GDP during recessions, the inflation rate typically decreases. However, inflation does not always decrease during recessions. Supply shocks that accompany recessions raise firms’ costs and can increase the inflation rate (e.g. stagflation). Increases in productivity are positive supply shocks that can reduce firms’ costs and decrease the inflation rate.

Assuming that households and firms have adaptive expectations, which means that people expect that the value of a variable from last year will occur again this year, the adaptive expectations of the inflation rate is given by equation (5):

                            πet = πt–1.                                           (6)

Thus, the Phillips curve can be rewritten as:

                πt = πt–1 + bỸt – st.                                                (7)

The answer of this question is mainly based upon the following reference:

Hubbard, R.G., O’Brien, A.P. and Rafferty (2014). Macroeconomics, 2ed edition, Pearson

Thanks Every one