If a country is running a government budget surplus, why is ( \(\mathrm{T}-\mathrm{G}\) ) on the left side of the saving-investment identity?

A government budget surplus occurs when the tax revenues (T) collected by the government are greater than its government spending (G). In other words, the government's income exceeds its expenditures. Mathematically, we can represent a government budget surplus as: \[ \mathrm{T} > \mathrm{G} \]

The saving-investment identity is an economic principle that states that total saving in an economy must equal total investment. The identity helps us understand the relationship between savings, investment, and economic growth. The saving-investment identity can be expressed as: \[ S = I \] where S represents total saving in the economy and I represents total investment in the economy.

Total saving (S) in an economy can be divided into private saving and public saving. Private saving is the amount of income households save after consuming goods and services and paying taxes. Public saving is the amount of tax revenue the government collects minus its spending on goods and services (also known as the government budget balance). Mathematically, we can write this as: \[ S = S_{\text{private}} + S_{\text{public}} \]

Now we will substitute the expressions for private and public saving into the saving-investment identity. Private saving can be written as the difference between disposable income (Y - T) and consumption (C), while public saving is the government budget balance, which is the difference between tax revenues (T) and government spending (G). Thus, the saving-investment identity becomes: \[ (Y - \mathrm{T} - C) + (\mathrm{T} - \mathrm{G}) = I \]

In a country with a government budget surplus, tax revenues exceed government spending, i.e., \(\mathrm{T} > \mathrm{G}\). This means that the government is saving more than it is spending, which contributes to the total saving in the economy. In the saving-investment identity, the term \((\mathrm{T} - \mathrm{G})\) represents the public saving component. We see that \((\mathrm{T} - \mathrm{G})\) is on the left side of the identity because it adds to total saving (S), while the right side of the identity represents total investment (I). In a country with a government budget surplus, the presence of \((\mathrm{T} - \mathrm{G})\) on the left side of the saving-investment identity reflects the fact that the surplus is increasing the overall saving in the economy.