The gross domestic product is the sum of private consumption, investments, government expenditures and net exports, which is exports minus imports. The marginal propensity to consume, MPC, is the ratio of the change in consumption to change in income. The marginal propensity to save, MPS, is the ratio of the change in savings to change in income. The relationship between a change in the aggregate expenditure -- for example, through increased government expenditures -- and the corresponding change in the GDP is defined by the GDP multiplier, which is equal to 1 / (1 - MPC).
Compute MPC and MPS. Note that MPC plus MPS is equal to 1, meaning what is not consumed is saved. For example, if consumption changes by $10 billion for a $40 billion change in income, MPC equals 0.25 ($10 billion / $40 billion) and MPS equals 0.75 (1 - 0.25).
Calculate the GDP multiplier, which is 1 / (1 - MPC). Substituting 1 - MPS for MPC, the multiplier is equal to 1 / (1 - 1 + MPS), which is 1 / MPS. Continuing with the example, the multiplier is equal to 1 / 0.75, which is about 1.33.
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Solve for GDP when there is a change in the aggregate expenditure. Concluding the example, if government expenditures and private investments increase by $50 billion, the GDP increases by $66.5 billion ($50 billion x 1.33).