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Multiplier: Concept, Types, and Derivation of each type of Multiplier

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Rohan Byanjankar
Rohan Byanjankar

The document contains the concept of multiplier, its various types, and derivation of each type of multiplier such as investment multiplier, government expenditure multiplier, import and export multiplier, tax (autonomous and induced) and transfer payment multiplier in two, three and four sector economy...

Economy & Finance
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Multiplier | Macroeconomics Rohan Byanjankar | 1 Concept of Multiplier Multiplier is the numeric value that helps to determine the change in national income due to change in autonomous investment in an economy. Multiplier simply means Keynesian Multiplier, that is, Investment Multiplier. In other words, investment multiplier reveals the action of income to increased investment. Government Expenditure multiplier is the numeric value that helps to determine the change in national income due to change in autonomous Government Expenditure in an economy. Government Expenditure multiplier reveals the action of income to increased Government Expenditure. Export multiplier is the numeric value that helps to determine the change in national income due to change in autonomous Export in an economy. Export multiplier reveals the action of income to increased Export. Import multiplier is the numeric value that helps to determine the change in national income due to change in autonomous Import in an economy. Import multiplier reveals the action of income to increased import. Tax multiplier is the numeric value that helps to determine the change in national income due to change in autonomous Taxation in an economy. Import multiplier reveals the action of income to increased taxation. Transfer Payment multiplier is the numeric value that helps to determine the change in national income due to change in autonomous transfer payment in an economy. Import multiplier reveals the action of income to increased import. So, from these definition, Multiplier (k) = Change in Income Change in Investment = ∆Y ∆I Government Expenditure Multiplier (k 𝐺) = Change in Income Change in Government Expenditure = ∆Y ∆G
Multiplier | Macroeconomics Rohan Byanjankar | 2 Export Multiplier (k 𝑋) = Change in Income Change in Exports = ∆Y ∆X Import Multiplier (k 𝑀) = Change in Income Change in Imports = ∆Y ∆M Tax Multiplier (k 𝑇) = Change in Income Change in Tax = ∆Y ∆T Transfer Payment Multiplier (k 𝑅) = Change in Income Change in Transfer Payment = ∆Y ∆R Multiplier in Two Sector Economy The Keynesian multiplier is also called Two Sector Multiplier, that is, there is only with sectors in an economy: Household sector and Business Sector. Derivation of Keynesian multiplier 𝑌 = 𝐶 + 𝐼 In an Keynesian Economy, 𝐶 = 𝐶𝑎 + 𝐶𝑖 = 𝐶𝑎 + 𝑏𝑌 [𝑏 = 𝑀𝑃𝐶] 𝐼 = 𝐼𝑎 So, 𝑌 = 𝐶𝑎 + 𝑏𝑌 + 𝐼𝑎 𝑌(1 − 𝑏) = 𝐶𝑎 + 𝐼𝑎 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎) National Income equals to 1 1−𝑏 proportion of summation of Autonomous Consumption and Autonomous Investment. Thus, as MPC increases, National Income also increases and vice-versa.
Multiplier | Macroeconomics Rohan Byanjankar | 3 Proof of Keynesian Multiplier 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎) 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 = 𝑌 + ∆𝑌 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 = 𝐼𝑎 + ∆𝐼 So, 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + ∆𝐼) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎) + 1 1 − 𝑏 × ∆𝐼 𝑌 + ∆𝑌 = 𝑌 + 1 1 − 𝑏 × ∆𝐼 ∆𝑌 = 1 1 − 𝑏 × ∆𝐼 ∆𝑌 ∆𝐼 = 1 1 − 𝑏 = 1 1 − 𝑀𝑃𝐶 = 1 𝑀𝑃𝑆 Thus, the magnitude of multiplier is directly related to Marginal Propensity to Consume, and inversely related to Marginal Propensity to Save. Multiplier in Three Sector Economy In three sector economy, Government Sector is added to components of Two Sector Economy. 𝑌 = 𝐶 + 𝐼 + 𝐺 In an Keynesian Economy, 𝐶 = 𝐶𝑎 + 𝐶𝑖 = 𝐶𝑎 + 𝑏𝑌 [𝑏 = 𝑀𝑃𝐶] 𝐼 = 𝐼𝑎
Multiplier | Macroeconomics Rohan Byanjankar | 4 𝐺 = 𝐺̅ 𝑜𝑟 𝐺𝑐 So, 𝑌 = 𝐶𝑎 + 𝑏𝑌 + 𝐼𝑎 + 𝐺𝑐 𝑌(1 − 𝑏) = 𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐) National Income equals to 1 1−𝑏 proportion of summation of Autonomous Consumption, Autonomous Investment, and constant Government Expenditure. Proof of Multiplier in Three Sector Economy 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐) 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 = 𝑌 + ∆𝑌 𝑇𝑜𝑡𝑎𝑙 𝐺𝑜𝑣𝑒𝑟𝑛𝑚𝑒𝑛𝑡 𝐸𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑢𝑟𝑒 = 𝐺𝑐 + ∆𝐺 So, 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + ∆𝐺) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + ∆𝐺) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐) + 1 1 − 𝑏 × ∆𝐺 𝑌 + ∆𝑌 = 𝑌 + 1 1 − 𝑏 × ∆𝐺 ∆𝑌 = 1 1 − 𝑏 × ∆𝐺 ∆𝑌 ∆𝐺 = 1 1 − 𝑏 Multiplier in Four Sector Economy
Multiplier | Macroeconomics Rohan Byanjankar | 5 In four sector economy, Foreign Sector is added to components of Three Sector Economy. 𝑌 = 𝐶 + 𝐼 + 𝐺 + 𝑋 − 𝑀 In an Keynesian Economy, 𝐶 = 𝐶𝑎 + 𝐶𝑖 = 𝐶𝑎 + 𝑏𝑌 [𝑏 = 𝑀𝑃𝐶] 𝐼 = 𝐼𝑎 𝐺 = 𝐺̅ 𝑜𝑟 𝐺𝑐 𝑋 = 𝑋𝑐 𝑀 = 𝑀𝑜 + 𝑚𝑌 [𝑚 = 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖𝑚𝑝𝑜𝑟𝑡] So, 𝑌 = 𝐶𝑎 + 𝑏𝑌 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜 − 𝑚𝑌 𝑌(1 − 𝑏 + 𝑚) = 𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) National Income equals to 1 1−𝑏+𝑚 proportion of summation of Autonomous Consumption, Autonomous Investment, constant Government Expenditure, Exports and autonomous Imports. Proof of Export Multiplier in Four Sector Economy 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 = 𝑌 + ∆𝑌 𝑇𝑜𝑡𝑎𝑙 𝐸𝑥𝑝𝑜𝑟𝑡𝑠 = 𝑋𝑐 + ∆𝑋 So, 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 + ∆𝑋 − 𝑀𝑜)
Multiplier | Macroeconomics Rohan Byanjankar | 6 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜 + ∆𝑋) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) + 1 1 − 𝑏 + 𝑚 × ∆𝑋 𝑌 + ∆𝑌 = 𝑌 + 1 1 − 𝑏 + 𝑚 × ∆𝑋 ∆𝑌 = 1 1 − 𝑏 + 𝑚 × ∆𝑋 ∆𝑌 ∆𝑋 = 1 1 − 𝑏 + 𝑚 Proof of Import Multiplier in Four Sector Economy 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 = 𝑌 + ∆𝑌 𝑇𝑜𝑡𝑎𝑙 𝐼𝑚𝑝𝑜𝑟𝑡𝑠 = 𝑀𝑐 + ∆𝑀 So, 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 + ∆𝑋 − 𝑀𝑜) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜 − ∆𝑀) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) − 1 1 − 𝑏 + 𝑚 × ∆𝑀 𝑌 + ∆𝑌 = 𝑌 − 1 1 − 𝑏 + 𝑚 × ∆𝑀 ∆𝑌 = − 1 1 − 𝑏 + 𝑚 × ∆𝑀 ∆𝑌 ∆𝑀 = − 1 1 − 𝑏 + 𝑚
Multiplier | Macroeconomics Rohan Byanjankar | 7 First Fiscal Model In first fiscal model, Government introduces autonomous taxation. Thus, Tax (T) = To Disposable Income (Yd) = Y − To Now, Two Sector Economy 𝑌 = 𝐶 + 𝐼 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝐼𝑎 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜) − 1 1 − 𝑏 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 Three Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺
Multiplier | Macroeconomics Rohan Byanjankar | 8 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝐼𝑎 + 𝐺 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜) − 1 1 − 𝑏 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 Four Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 + 𝑋 − 𝑀 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏(𝑇𝑜 + ∆𝑇))
Multiplier | Macroeconomics Rohan Byanjankar | 9 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜) − 1 1 − 𝑏 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 + 𝑚 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 + 𝑚 Second Fiscal Model In second fiscal model, Government introduces autonomous taxation as well as transfer payment. Thus, Tax (T) = To Transfer Payment (R) = Ro Disposable Income (Yd) = Y − To + Ro Now, Two Sector Economy 𝑌 = 𝐶 + 𝐼 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 + 𝑅𝑜) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝐼𝑎 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) Suppose income is increased by ∆𝑌 and transfer payment is increased by ∆𝑅 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 + 𝑏(𝑅𝑜 + ∆𝑅))
Multiplier | Macroeconomics Rohan Byanjankar | 10 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝑏∆𝑅) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) + 1 1 − 𝑏 × 𝑏∆𝑅 𝑌 + ∆𝑌 = 𝑌 + 𝑏 1 − 𝑏 × ∆𝑅 ∆𝑌 ∆𝑅 = 𝑏 1 − 𝑏 Three Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 + 𝑅𝑜) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝐼𝑎 + 𝐺 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) Suppose income is increased by ∆𝑌 and transfer payment is increased by ∆𝑅 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 + 𝑏(𝑅𝑜 + ∆𝑅)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝑏∆𝑅) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) + 1 1 − 𝑏 × 𝑏∆𝑅 𝑌 + ∆𝑌 = 𝑌 + 𝑏 1 − 𝑏 × ∆𝑅 ∆𝑌 ∆𝑅 = 𝑏 1 − 𝑏 Four Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 + 𝑋 − 𝑀
Multiplier | Macroeconomics Rohan Byanjankar | 11 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 + 𝑅𝑜) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) Suppose income is increased by ∆𝑌 and transfer payment is increased by ∆𝑅 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 + 𝑏(𝑅𝑜 + ∆𝑅)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝑏∆𝑅) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) + 1 1 − 𝑏 × 𝑏∆𝑅 𝑌 + ∆𝑌 = 𝑌 + 𝑏 1 − 𝑏 + 𝑚 × ∆𝑅 ∆𝑌 ∆𝑅 = 𝑏 1 − 𝑏 + 𝑚 Third Fiscal Model In third fiscal model, government imposes tax based on level of income. So, Tax (T) = To + tY Disposable Income (Yd) = Y − To − tY Now, Two Sector Economy 𝑌 = 𝐶 + 𝐼 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 − 𝑡𝑌) + 𝐼𝑎
Multiplier | Macroeconomics Rohan Byanjankar | 12 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 − 𝑏𝑡𝑌 + 𝐼𝑎 𝑌 = 1 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜) − 1 1 − 𝑏 + 𝑏𝑡 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 + 𝑏𝑡 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 + 𝑏𝑡 Three Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 − 𝑡𝑌) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 − 𝑏𝑡𝑌 + 𝐼𝑎 + 𝐺 𝑌 = 𝑏 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 𝑏 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 𝑏 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 𝑏 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜) − 𝑏 1 − 𝑏 + 𝑏𝑡 × 𝑏∆𝑇
Multiplier | Macroeconomics Rohan Byanjankar | 13 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 + 𝑏𝑡 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 + 𝑏𝑡 Four Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 + 𝑋 − 𝑀 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 − 𝑡𝑌) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 − 𝑏𝑡𝑌 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜) − 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × ∆𝑇 ∆𝑌 ∆𝑇 = − 1 1 − 𝑏 + 𝑏𝑡 + 𝑚
Multiplier | Macroeconomics Rohan Byanjankar | 14

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Multiplier: Concept, Types, and Derivation of each type of Multiplier

  • 1. Multiplier | Macroeconomics Rohan Byanjankar | 1 Concept of Multiplier Multiplier is the numeric value that helps to determine the change in national income due to change in autonomous investment in an economy. Multiplier simply means Keynesian Multiplier, that is, Investment Multiplier. In other words, investment multiplier reveals the action of income to increased investment. Government Expenditure multiplier is the numeric value that helps to determine the change in national income due to change in autonomous Government Expenditure in an economy. Government Expenditure multiplier reveals the action of income to increased Government Expenditure. Export multiplier is the numeric value that helps to determine the change in national income due to change in autonomous Export in an economy. Export multiplier reveals the action of income to increased Export. Import multiplier is the numeric value that helps to determine the change in national income due to change in autonomous Import in an economy. Import multiplier reveals the action of income to increased import. Tax multiplier is the numeric value that helps to determine the change in national income due to change in autonomous Taxation in an economy. Import multiplier reveals the action of income to increased taxation. Transfer Payment multiplier is the numeric value that helps to determine the change in national income due to change in autonomous transfer payment in an economy. Import multiplier reveals the action of income to increased import. So, from these definition, Multiplier (k) = Change in Income Change in Investment = ∆Y ∆I Government Expenditure Multiplier (k 𝐺) = Change in Income Change in Government Expenditure = ∆Y ∆G
  • 2. Multiplier | Macroeconomics Rohan Byanjankar | 2 Export Multiplier (k 𝑋) = Change in Income Change in Exports = ∆Y ∆X Import Multiplier (k 𝑀) = Change in Income Change in Imports = ∆Y ∆M Tax Multiplier (k 𝑇) = Change in Income Change in Tax = ∆Y ∆T Transfer Payment Multiplier (k 𝑅) = Change in Income Change in Transfer Payment = ∆Y ∆R Multiplier in Two Sector Economy The Keynesian multiplier is also called Two Sector Multiplier, that is, there is only with sectors in an economy: Household sector and Business Sector. Derivation of Keynesian multiplier 𝑌 = 𝐶 + 𝐼 In an Keynesian Economy, 𝐶 = 𝐶𝑎 + 𝐶𝑖 = 𝐶𝑎 + 𝑏𝑌 [𝑏 = 𝑀𝑃𝐶] 𝐼 = 𝐼𝑎 So, 𝑌 = 𝐶𝑎 + 𝑏𝑌 + 𝐼𝑎 𝑌(1 − 𝑏) = 𝐶𝑎 + 𝐼𝑎 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎) National Income equals to 1 1−𝑏 proportion of summation of Autonomous Consumption and Autonomous Investment. Thus, as MPC increases, National Income also increases and vice-versa.
  • 3. Multiplier | Macroeconomics Rohan Byanjankar | 3 Proof of Keynesian Multiplier 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎) 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 = 𝑌 + ∆𝑌 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 = 𝐼𝑎 + ∆𝐼 So, 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + ∆𝐼) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎) + 1 1 − 𝑏 × ∆𝐼 𝑌 + ∆𝑌 = 𝑌 + 1 1 − 𝑏 × ∆𝐼 ∆𝑌 = 1 1 − 𝑏 × ∆𝐼 ∆𝑌 ∆𝐼 = 1 1 − 𝑏 = 1 1 − 𝑀𝑃𝐶 = 1 𝑀𝑃𝑆 Thus, the magnitude of multiplier is directly related to Marginal Propensity to Consume, and inversely related to Marginal Propensity to Save. Multiplier in Three Sector Economy In three sector economy, Government Sector is added to components of Two Sector Economy. 𝑌 = 𝐶 + 𝐼 + 𝐺 In an Keynesian Economy, 𝐶 = 𝐶𝑎 + 𝐶𝑖 = 𝐶𝑎 + 𝑏𝑌 [𝑏 = 𝑀𝑃𝐶] 𝐼 = 𝐼𝑎
  • 4. Multiplier | Macroeconomics Rohan Byanjankar | 4 𝐺 = 𝐺̅ 𝑜𝑟 𝐺𝑐 So, 𝑌 = 𝐶𝑎 + 𝑏𝑌 + 𝐼𝑎 + 𝐺𝑐 𝑌(1 − 𝑏) = 𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐) National Income equals to 1 1−𝑏 proportion of summation of Autonomous Consumption, Autonomous Investment, and constant Government Expenditure. Proof of Multiplier in Three Sector Economy 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐) 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 = 𝑌 + ∆𝑌 𝑇𝑜𝑡𝑎𝑙 𝐺𝑜𝑣𝑒𝑟𝑛𝑚𝑒𝑛𝑡 𝐸𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑢𝑟𝑒 = 𝐺𝑐 + ∆𝐺 So, 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + ∆𝐺) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + ∆𝐺) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐) + 1 1 − 𝑏 × ∆𝐺 𝑌 + ∆𝑌 = 𝑌 + 1 1 − 𝑏 × ∆𝐺 ∆𝑌 = 1 1 − 𝑏 × ∆𝐺 ∆𝑌 ∆𝐺 = 1 1 − 𝑏 Multiplier in Four Sector Economy
  • 5. Multiplier | Macroeconomics Rohan Byanjankar | 5 In four sector economy, Foreign Sector is added to components of Three Sector Economy. 𝑌 = 𝐶 + 𝐼 + 𝐺 + 𝑋 − 𝑀 In an Keynesian Economy, 𝐶 = 𝐶𝑎 + 𝐶𝑖 = 𝐶𝑎 + 𝑏𝑌 [𝑏 = 𝑀𝑃𝐶] 𝐼 = 𝐼𝑎 𝐺 = 𝐺̅ 𝑜𝑟 𝐺𝑐 𝑋 = 𝑋𝑐 𝑀 = 𝑀𝑜 + 𝑚𝑌 [𝑚 = 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖𝑚𝑝𝑜𝑟𝑡] So, 𝑌 = 𝐶𝑎 + 𝑏𝑌 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜 − 𝑚𝑌 𝑌(1 − 𝑏 + 𝑚) = 𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) National Income equals to 1 1−𝑏+𝑚 proportion of summation of Autonomous Consumption, Autonomous Investment, constant Government Expenditure, Exports and autonomous Imports. Proof of Export Multiplier in Four Sector Economy 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 = 𝑌 + ∆𝑌 𝑇𝑜𝑡𝑎𝑙 𝐸𝑥𝑝𝑜𝑟𝑡𝑠 = 𝑋𝑐 + ∆𝑋 So, 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 + ∆𝑋 − 𝑀𝑜)
  • 6. Multiplier | Macroeconomics Rohan Byanjankar | 6 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜 + ∆𝑋) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) + 1 1 − 𝑏 + 𝑚 × ∆𝑋 𝑌 + ∆𝑌 = 𝑌 + 1 1 − 𝑏 + 𝑚 × ∆𝑋 ∆𝑌 = 1 1 − 𝑏 + 𝑚 × ∆𝑋 ∆𝑌 ∆𝑋 = 1 1 − 𝑏 + 𝑚 Proof of Import Multiplier in Four Sector Economy 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 = 𝑌 + ∆𝑌 𝑇𝑜𝑡𝑎𝑙 𝐼𝑚𝑝𝑜𝑟𝑡𝑠 = 𝑀𝑐 + ∆𝑀 So, 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 + ∆𝑋 − 𝑀𝑜) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜 − ∆𝑀) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺𝑐 + 𝑋𝑐 − 𝑀𝑜) − 1 1 − 𝑏 + 𝑚 × ∆𝑀 𝑌 + ∆𝑌 = 𝑌 − 1 1 − 𝑏 + 𝑚 × ∆𝑀 ∆𝑌 = − 1 1 − 𝑏 + 𝑚 × ∆𝑀 ∆𝑌 ∆𝑀 = − 1 1 − 𝑏 + 𝑚
  • 7. Multiplier | Macroeconomics Rohan Byanjankar | 7 First Fiscal Model In first fiscal model, Government introduces autonomous taxation. Thus, Tax (T) = To Disposable Income (Yd) = Y − To Now, Two Sector Economy 𝑌 = 𝐶 + 𝐼 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝐼𝑎 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜) − 1 1 − 𝑏 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 Three Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺
  • 8. Multiplier | Macroeconomics Rohan Byanjankar | 8 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝐼𝑎 + 𝐺 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜) − 1 1 − 𝑏 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 Four Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 + 𝑋 − 𝑀 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏(𝑇𝑜 + ∆𝑇))
  • 9. Multiplier | Macroeconomics Rohan Byanjankar | 9 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜) − 1 1 − 𝑏 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 + 𝑚 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 + 𝑚 Second Fiscal Model In second fiscal model, Government introduces autonomous taxation as well as transfer payment. Thus, Tax (T) = To Transfer Payment (R) = Ro Disposable Income (Yd) = Y − To + Ro Now, Two Sector Economy 𝑌 = 𝐶 + 𝐼 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 + 𝑅𝑜) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝐼𝑎 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) Suppose income is increased by ∆𝑌 and transfer payment is increased by ∆𝑅 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 + 𝑏(𝑅𝑜 + ∆𝑅))
  • 10. Multiplier | Macroeconomics Rohan Byanjankar | 10 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝑏∆𝑅) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) + 1 1 − 𝑏 × 𝑏∆𝑅 𝑌 + ∆𝑌 = 𝑌 + 𝑏 1 − 𝑏 × ∆𝑅 ∆𝑌 ∆𝑅 = 𝑏 1 − 𝑏 Three Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 + 𝑅𝑜) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝐼𝑎 + 𝐺 𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) Suppose income is increased by ∆𝑌 and transfer payment is increased by ∆𝑅 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 + 𝑏(𝑅𝑜 + ∆𝑅)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝑏∆𝑅) 𝑌 + ∆𝑌 = 1 1 − 𝑏 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) + 1 1 − 𝑏 × 𝑏∆𝑅 𝑌 + ∆𝑌 = 𝑌 + 𝑏 1 − 𝑏 × ∆𝑅 ∆𝑌 ∆𝑅 = 𝑏 1 − 𝑏 Four Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 + 𝑋 − 𝑀
  • 11. Multiplier | Macroeconomics Rohan Byanjankar | 11 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 + 𝑅𝑜) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) Suppose income is increased by ∆𝑌 and transfer payment is increased by ∆𝑅 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 + 𝑏(𝑅𝑜 + ∆𝑅)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜 + 𝑏∆𝑅) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 + 𝑏𝑅𝑜) + 1 1 − 𝑏 × 𝑏∆𝑅 𝑌 + ∆𝑌 = 𝑌 + 𝑏 1 − 𝑏 + 𝑚 × ∆𝑅 ∆𝑌 ∆𝑅 = 𝑏 1 − 𝑏 + 𝑚 Third Fiscal Model In third fiscal model, government imposes tax based on level of income. So, Tax (T) = To + tY Disposable Income (Yd) = Y − To − tY Now, Two Sector Economy 𝑌 = 𝐶 + 𝐼 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 − 𝑡𝑌) + 𝐼𝑎
  • 12. Multiplier | Macroeconomics Rohan Byanjankar | 12 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 − 𝑏𝑡𝑌 + 𝐼𝑎 𝑌 = 1 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 − 𝑏𝑇𝑜) − 1 1 − 𝑏 + 𝑏𝑡 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 + 𝑏𝑡 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 + 𝑏𝑡 Three Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 − 𝑡𝑌) + 𝐼𝑎 + 𝐺 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 − 𝑏𝑡𝑌 + 𝐼𝑎 + 𝐺 𝑌 = 𝑏 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 𝑏 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 𝑏 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 𝑏 1 − 𝑏 + 𝑏𝑡 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 − 𝑏𝑇𝑜) − 𝑏 1 − 𝑏 + 𝑏𝑡 × 𝑏∆𝑇
  • 13. Multiplier | Macroeconomics Rohan Byanjankar | 13 𝑌 + ∆𝑌 = 𝑌 − 𝑏 1 − 𝑏 + 𝑏𝑡 × ∆𝑇 ∆𝑌 ∆𝑇 = − 𝑏 1 − 𝑏 + 𝑏𝑡 Four Sector Economy 𝑌 = 𝐶 + 𝐼 + 𝐺 + 𝑋 − 𝑀 𝑌 = 𝐶𝑎 + 𝑏(𝑌𝑑) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏(𝑌 − 𝑇𝑜 − 𝑡𝑌) + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 𝐶𝑎 + 𝑏𝑌 − 𝑏𝑇𝑜 − 𝑏𝑡𝑌 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑚𝑌 𝑌 = 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜) Suppose income is increased by ∆𝑌 and tax is increased by ∆𝑇 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏(𝑇𝑜 + ∆𝑇)) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜 − 𝑏∆𝑇) 𝑌 + ∆𝑌 = 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × (𝐶𝑎 + 𝐼𝑎 + 𝐺 + 𝑋 − 𝑀𝑜 − 𝑏𝑇𝑜) − 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × 𝑏∆𝑇 𝑌 + ∆𝑌 = 𝑌 − 1 1 − 𝑏 + 𝑏𝑡 + 𝑚 × ∆𝑇 ∆𝑌 ∆𝑇 = − 1 1 − 𝑏 + 𝑏𝑡 + 𝑚
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