6.3. The multiplier-accelerator. (Samuelson, 1939) Consider the following model of income determination. (1) Consumption depends on the previous period's income: \(C_t=a+bY_{t-1}\). (2) the desired capital stock (or inventory stock) is proportional to the previous period's output: \(K_t^\star=cY_{t-1}\). (3) Investment equals the difference between the desired capital stock and the stock inherited from the previous period: \(I_t = K_t^\star-K_{t-1} = K_t^\star-cY_{t-2}\). (4) Government purchases are constant: \(G_t=\bar{G}\). (5) \(Y_t=C_t+I_t+G_t\).
a) Express \(Y\) in terms of \(Y_{t-1}\)\(Y_{t-2}\), and the parameters of the model.
\(Y_t = a + bY_{t-1}+K_t-cY_{t-2}+\bar{G}\)
\(Y_t = a + bY_{t-1}+cY_{t-1}-cY_{t-2}+\bar{G}\)
\(Y_t = a + (b+c)Y_{t-1}-cY_{t-2}+\bar{G}\)
b) Suppose b=0.9 and c=0.5. Suppose there is a one time disturbance to government purchases; specifically, suppose that G is equal to \(\bar{G} +1\) in period t and is equal to \(\bar{G}\) in all other periods. How does this shock affect output over time?
We have:
\(Y_t=a+1.4Y_{t-1}-0.5Y_{t-2}+\bar{G}\)
Let's now assume that government expenditures increase by one unit at period t. We then have:
\(Y_t=a+1.4Y_{t-1}-0.5Y_{t-2}+\bar{G}+1\)
This means that the change of output with a one unit increase in G will be of one at the year of the increase:
\(\Delta Y_t = 1\)
Which means that the change in output in time t+1 resulting from the increase in G is equal to:
\(\Delta Y_{t+1}=a+1.4\Delta Y_t -0.5Y_{t-1}+\bar{G} = 1.4\)
In period t+2:
\(\Delta Y_{t+2}=a+1.4\Delta Y_{t+1} -0.5 \Delta Y_{t}+\bar{G} = 1.46\)
In period t+3:
\(\Delta Y_{t+3}=a+1.4\Delta Y_{t+2} -0.5 \Delta Y_{t+1}+\bar{G} = 1.344\)
If we continue, we find that \(\Delta Y_{t+4} = 1.1516\), \(\Delta Y_{t+5} = 0.94024\)\(\Delta Y_{t+6} = 0,740536\). We used an excel spreadsheet using the coefficients given for b and c and assuming that a=1 to illustrate the impact of a one unit increase in G on output: