When we consider the Savings-Investment model what happens when the government runs a budget deficit and how does this impact the real rate of interest and savings (private)?

To explore this question we have to revisit the GDP model of Y = C + I + G + NI but ignoring net exports so we have Y – C – G = I. We will also explore this graph that assumes savings from consumers has increased.

To determine the impact on savings we have to get disposable income and subtract from consumption. Thus, we define disposable income (YD) as income (Y) plus transfers (TR) minus taxes (TA)
YD = Y + TR - TA
Therefore we can write:
(YD - C) = (TA - TR - G) = I
or
S = S (private) + S (govt) = I

Notice that term TA – TR – G , is equal to government revenue minus government spending, which is the government budgetary surplus (if it is positive) or budgetary deficit (if it is negative). When the government-is running a budgetary surplus, government savings is positive. Thus the equation states that for aggregate demand equilibrium savings equals investment. Savings is composed of private savings S (prviate)= YD-C and government savings S (govt) = TA – TR -G.

When the government runs a budget deficit, government savings is negative. Therefore, the S curve, which is composed of private savings and government savings, shifts to the left. The result is that a government budget deficit causes higher real interest rate and lower total savings.

However, this result assumes that private savings is not affected by the change in government savings. If private savings increases, the curve may not shift at all, depending on the magnitude of the change in private savings relative to the change in the budget deficit.