Methodological expansion for “Debt-Driven GDP Growth”

Based on Ednaldo Silva’s LinkedIn post “Debt-Driven GDP Growth”, we examine the relationship between government debt and GDP growth. The analysis starts from a simple GDP identity, derives a structural equation for government spending, and transforms it into an empirical regression framework that captures the dynamic interaction between debt, autonomous factors, and GDP.

1. The GDP identity

(1)\quad Y = A + (G-T)

GDP consists of autonomous factors A and the government budget balance (G-T).
(https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL)

2. Structural equation for government spending

(2)\quad G = T + b\,\Delta D - rD_{(-1)}

• Tax revenue T.
• A share b of the new debt ΔD (for investments).
• Debt payment with interest payments r on old debts D(-1).

(https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL)

We substitute equation (2) into (1):

\begin{aligned}
(3)\quad Y &= A + (G-T) \\
  &= A + \left[b\,\Delta D - rD_{(-1)}\right]
\end{aligned}

Equation (3) is the reduced form.
(https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL)

To obtain a regression equation for GDP growth, we calculate the first difference (ΔY = Y_t – Y_t-1) of equation (3):

For time t:

Y_t = A_t + b ΔD_t – r D_t-1

For time t-1:

Y_t-1 = A_t-1 + b ΔD_t-1 – r D_t-2

Explanation:

• Y_t is GDP at time t.
• A_t are the autonomous factors at time t.
• ΔD_t = D_t – r D_t-1 is the change in debt at time t.
• D_t-1 are the debts at time t-1.

Subtraction (first derivative) yields:

ΔY_t = Y_t – Y_t-1 = [A_t + b ΔD_t – r D_t-1] – [A_t-1 + b ΔD_t-1 – r D_t-2]
= [A_t – A_t-1] + b [ΔD_t – ΔD_t-1] – r [D_t-1 – D_t-2]

It follows that:

• ΔA = A_t – A_t-1,
• b Δ²D_t = b [ΔD_t – ΔD_t-1],
• ΔD_t = D_t – r D_t-1,
• ΔD_t-1 = ΔD_t-1 – r D_t-2
• -r ΔD_t-1 = -r [D_t-1 – D_t-2]

⇒ ΔD_t-1 is the change in debt in the previous period.

Consolidation:

(4)\quad Y_t = A_t + b\,\Delta D_t - rD_{t-1}

(https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL)

3. Simplified regression form

The problem with equation (4) is that it cannot be estimated directly, because:

1. The second derivative Δ²D_t is difficult to interpret.
2. The parameters b (proportion of debt for investment) and r (interest rate) are structurally linked.

Solution: Rewrite in regression form:

We define new coefficients β₁, β₂, β₃, which we can estimate directly.

We obtain an empirical regression model:

(5)\quad \Delta Y_t
=
\beta_1 \Delta A_t
+
\beta_2 \Delta D_t
+
\beta_3 \Delta D_{t-1}
+
\varepsilon_t

(https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL)

Connection between (4) and (5):

We replace Δ²D_t with ΔD_t and ΔD_t-1 because Δ²D_t = ΔD_t – ΔD_t-1.

We substitute this into equation (4):

ΔY_t = ΔA_t + b (ΔD_t – ΔD_t-1) – r ΔD_t-1
= ΔA_t + b ΔD_t – b ΔD_t-1 – r ΔD_t-1
= ΔA_t + b ΔD_t + (-b -r) ΔD_t-1

If we add an error term εₜ (which includes measurement errors, specification errors, etc.), we get:

ΔY_t = ΔA_t + b ΔD_t + (-b -r) ΔD_t-1 + εₜ

Comparing this with equation (5), we obtain:

ΔY_t = β₁ ΔA_t + β₂ ΔD_t + β₃ ΔD_t-1 + εₜ, with the following connections:

With β₁ we denote the coefficient of ΔA_t and thus the effect of changes in autonomous factors on GDP growth. In theory, β₁ < 1 should apply, because if the autonomous factors A increase by one unit, GDP Y increases mechanically by exactly one unit, since A directly and completely enters into Y. This means that in the empirical estimation, we test whether β₁ deviates statistically from 1.

The parameter β₂ = b is the effect of current debt changes ΔD_t on GDP growth ΔY from the structural model. If β₂ = b > 0, this means more new debt and thus more spending, which leads to higher GDP.

The coefficient β₃ = -b-r is the effect of the delayed debt change on GDP growth. This combines both structural parameters b and r. β₃ should be negative if r > 0 and b is small, because a higher debt burden causes more interest payments and leads to lower GDP.

The residuum εₜ contains all factors that the model does not explain.

4. Koyck Transformation

“To simplify the regression further, we move to a level GDP equation:”

(6)\quad Y_t
=
\alpha Y_{t-1}
+
\beta_1 \Delta A_t
+
\beta_2 \Delta D_t
+
\beta_3 \Delta D_{t-1}
+
\varepsilon_t

(https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL)

With theoretical expectations:

• α < 1: Stable or slow effect. GDP inherits part of its previous level, but not completely.
• α close to 1: Highly persistent, GDP changes only slowly. 
• β₁ > 0: More tax revenue makes more spending possible.
• β₂ > 0: New debt enables higher government spending (fiscal capacity). This stimulates GDP growth in the short term.
• β₃ < 0: A higher debt burden leads to a smaller budget due to interest payments. This interest burden reduces the available fiscal scope.

(https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL)

The Toyck transformation allows for:

1. Modeling the dynamics, because Y_t depends on Y_t-1 (autoregressive component).
2. Distinguishing between short-term and long-term effects.
3. Capturing persistence in the GDP process.
4. Establishing connections between theoretical models by Kalecki (1954) and Domar (1944, 1947).

5. Conclusion

We have shown how government debt affects GDP. The model structure with a lagged dependent variable generates persistence and captures the dynamic relationship between debt and GDP.

References

Ednaldo Silva’s LinkedIn post “Debt-Driven GDP Growth” https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL

Ednaldo Silva’s blog post “Measurements and Uncertainties in Transfer Pricing” https://edgarstat.com/blog/measurements-and-uncertainties-in-transfer-pricing/