Question

Suppose \(D\) is real government debt, \(s\) the primary budget surplus \(T-G\) (that is, excluding interest payments on debt), \(i\) the real interest rate, \(Y\) real output and \(g\) the rate of output growth. The debt burden \(D / Y\) rises with debt but falls with output and the ability to repay debt. Let \(\Delta\) denote the increase in a variable. (a) If \(\Delta(D / Y)=(\Delta D / D)-(\Delta Y / Y)\), show that the debt \(/\) GDP ratio shrinks only if \(s / D>i-\) g. (b) Suppose all debt is cash, paying no interest. Show that the above relationship becomes \(s / D>(g+\pi)\).

Short answer

(a) The debt-to-GDP ratio shrinks if \( \frac{s}{D} > i - g \). (b) When debt pays no interest, \( \frac{s}{D} > g + \pi \).

Step-by-step solution

Understanding the Debt to GDP Ratio Change

The expression given is \( \Delta\left(\frac{D}{Y}\right) = \frac{\Delta D}{D} - \frac{\Delta Y}{Y} \) which describes the change in the debt-to-GDP ratio as a function of the change in debt and output.

Linking Change in Debt to Primary Surplus and Interest

We know that the change in debt \( \Delta D \) can be expressed as the interest on existing debt minus the primary surplus, thus \( \Delta D = iD - s \). This is crucial because primary surplus \( s \) reduces debt, while interest \( iD \) increases it.

Expressing Output Growth

The change in output \( \Delta Y \) depends on the growth rate of the economy, \( g \), such that \( \Delta Y = gY \).

Substituting into the Ratio Change Equation

Substitute \( \Delta D = iD - s \) and \( \Delta Y = gY \) into the expression for \( \Delta\left(\frac{D}{Y}\right) \) to get: \[ \Delta\left(\frac{D}{Y}\right) = \frac{iD - s}{D} - g. \] Simplifying this gives \( \frac{iD - s}{D} - g = i - \frac{s}{D} - g \).

Condition for Debt-to-GDP Ratio to Decrease

For the debt-to-GDP ratio to shrink, we require \( \Delta\left(\frac{D}{Y}\right) < 0 \). This implies \( i - \frac{s}{D} - g < 0 \) or equivalently \( \frac{s}{D} > i - g \).

Considering Debt as Cash (Debt Pays No Interest)

If all debt is cash, meaning it pays no interest, \( i = 0 \). The expression becomes \( \frac{s}{D} - g < 0 \). Simplifying gives \( \frac{s}{D} > g \).

Incorporating Inflation into the Interest-Free Scenario

Inflation \( \pi \) affects the real value of output, effectively increasing the growth rate to \( g + \pi \). Thus, for the debt-to-GDP ratio to decrease, we need \( \frac{s}{D} > g + \pi \).

Key concepts

Primary Budget Surplus

The primary budget surplus represents the difference between government revenues and expenditures, excluding interest payments on existing debt. In simpler terms, it's how much money a government has left over after covering its operational costs but before paying interest on its debts.
If a government regularly achieves a primary budget surplus, it indicates that it has additional funds which can be used to reduce existing debt.
This surplus is crucial in the context of the debt-to-GDP ratio because it creates a cushion to offset interest payments, thereby lowering the overall debt burden. For a lower debt-to-GDP ratio, a primary surplus should be high enough to not only cover interest payments but also contribute to reducing the principal debt amount.
In the equation for the debt-to-GDP ratio shrinkage, the primary surplus must exceed the difference between the real interest rate and the output growth rate. This means achieving financial discipline through reduced spending, increased taxes, or a combination of both strategies to maintain a healthy surplus.

Real Interest Rate

The real interest rate is the nominal interest rate adjusted for inflation. It represents the true cost of borrowing or the genuine yield on savings. The real interest rate significantly impacts the economy as it influences both consumer and business spending.
If the real interest rate is high, borrowing becomes costly, discouraging expenditure and investment, while saving becomes more attractive. Conversely, a low real interest rate implies cheaper borrowing costs, stimulating spending and investment but possibly reducing incentives to save.
In terms of debt management, governments must be wary of the real interest rate because it affects the cost of servicing national debt. When the real interest rate exceeds the output growth rate, it can imply an increasing debt obligation relative to the economic output. More of the government's budget needs to be allocated to interest payments instead of other critical areas like healthcare or infrastructure.
Maintaining a balance such that the growth rate exceeds the real interest rate helps in sustaining a manageable debt-to-GDP ratio.

Output Growth Rate

The output growth rate, often referred to as GDP growth rate, reflects how fast an economy is expanding. A strong output growth rate indicates a healthy and robust economic environment, potentially increasing employment opportunities and wealth.
Output growth is pivotal for improving the debt-to-GDP ratio because as the economy grows, it generates more revenue without necessarily increasing debt.
With consistent growth, the same debt level represents a smaller share of a bigger economy, thereby lowering the debt burden. This also implies greater governmental ability to service and reduce debt without additional borrowing.
A higher growth rate can absorb new debt more effectively, supporting financial health and enabling governments to invest in future productivity enhancements. It's essential to foster conditions that promote healthy growth, such as innovation, infrastructure, and education, which can push long-term output growth rates upward.