Distribution, capital intensity and public debt-to-GDP ratio: an input output—stock flow consistent model

Abstract

The paper analyzes the relationship between the interest rate and the public debt-to-GDP ratio through the lens of the Classical-Keynesian approach. We focus on the value dimension as a transmission channel of monetary policy, modeling how a change in the interest rate set by the central bank affects the economy’s capital intensity and, in turn, debt ratios. We do so by developing a Stock-Flow Consistent Supermultiplier model (SFC-SM) based on a simplified Input–Output structure of production, showing that the effect of an increase in the interest rate on public debt-to-GDP ratio will depend on the impact exerted by the shock on the capital intensity through changes in relative prices. Lastly, we calibrate the model, showing the possible emergence of reverse capital deepening; past a threshold, any base rate hike produces an increase in the public debt-to-GDP ratio by decreasing the capital intensity of the economy.

Appendices

Appendix 1. Balance sheet and transaction matrix

Balance sheet and transaction matrix

Assets	Workers	Capitalists	Sector A	Sector B	Sector C	Bank	Government	CB	$\sum$
Check deposits	${+M1}_{w}$	${+M1}_{cap}$	${+M1}_{a}$	${+M1}_{c}$	${+M1}_{c}$	$-M1$			0
Time deposits	$+{M1}_{w}$	$+{M1}_{cap}$				$-M2$			0
HPM						$+{H}_{b}$		$-H$	0
Advances						$-A$		$-A$	0
Loans			$-{L}_{a}$	$-{L}_{b}$	$-{L}_{c}$	$+L$			0
Fixed Capital			$+{K}_{{f}_{a}}$	$+{K}_{{f}_{b}}$	$+{K}_{{f}_{c}}$				$+{K}_{f}$
Public bonds		$+{B}_{h,cap}$					$-B$	$+{B}_{cb}$	0
Net wealth	${-V}_{h,w}$	${-V}_{h,cap}$	$-{V}_{a}$	$-{V}_{b}$	$-{V}_{c}$	0	$+GD$	0	$-{K}_{f}$
$\sum$	0	0	0	0	0	0	0	0	0

	Workers	Capitalists	Sector A	Sector B	Sector C	Government	Bank		Central Bank		$\sum$
							Current	Capital	Current	Capital
Consumption	$-{C}_{w}$	$-{C}_{cap}$			$+C$						0
Investments in A			$+I$	$-{I}_{b}$	$-{I}_{c}$						0
Investments in B			$-{I}_{a}$	$+I$							0
Public expenditure					$+G$	$-G$					0
Wages	$+W$		$-{W}_{a}$	$-{W}_{b}$	$-{W}_{c}$						0
Taxes	$-{T}_{w}$	$-{T}_{cap}$				$+T$					0
Profits		$+{Div}_{F}$	$-{Div}_{a}$	$-{Div}_{b}$	$-{Div}_{c}$						0
Profits B		$+{Div}_{B}$					$-{Div}_{B}$				0
Profits BC						${+F}_{cb}$			${-F}_{cb}$		0
Int. Deposit	$+{r}_{m,t-1}{M2}_{w,t-1}$	$+{r}_{m,t-1}{M2}_{c,t-1}$					$-{r}_{m,t-1}{M2}_{t-1}$				0
Int. Loans			$-{r}_{l-1}{L}_{a,t-1}$	$-{r}_{l-1}{L}_{b,t-1}$	$-{r}_{l-1}{L}_{c,t-1}$		$+{r}_{l-1}{L}_{t-1}$				0
Int. Bond		$+{i}_{t-1}{B}_{h,t-1}$				$-{i}_{t-1}{B}_{t-1}$			$+{i}_{r}{B}_{bc,t-1}$		0
Int. Reserves							$+{r}_{r-1}{H}_{t-1}$		$-{r}_{r-1}{H}_{t-1}$		0
Int. Advances							$-{r}_{a,t-1}{A}_{t-1}$		$+{r}_{a,t-1}{A}_{t-1}$		0
∆Deposit time	$-{\Delta M2}_{w}$	$-{\Delta M2}_{cap}$						$+\Delta M2$			0
∆ Deposit check	$-{\Delta M1}_{w}$	$-{\Delta M1}_{w}$						$+\Delta M1$			0
∆ Loans			$+{\Delta L}_{a}$	$+{\Delta L}_{b}$	$+{\Delta L}_{c}$			$-\Delta L$			0
∆ Bonds		$-{\Delta B}_{h}$				$+\Delta B$			${-\Delta B}_{bc}$		0
∆ Reserves								$-\Delta H$	$+\Delta H$		0
$\sum$	0	0	0	0	0	0	0	0	0	0	0

Appendix 2. Data sources

• Adjusted wage share, percentage of GDP at current factor costs, United States, https://economy-finance.ec.europa.eu/economic-research-and-databases/economic-databases/ameco-database_en

• Federal Debt: Total Public Debt as Percent of Gross Domestic Product (GFDEGDQ188S), Percent of GDP, Seasonally Adjusted, https://fred.stlouisfed.org/series/GFDEGDQ188S

• Government Consumption Expenditure and gross investment in billions of dollars, seasonally adjusted, quarterly data, BEA, NIPA Table 1.1.5., https://bit.ly/34DlOsj

• Gross Domestic Product in billions of dollars, seasonally adjusted, quarterly data, BEA, NIPA Table 1.1.5., https://bit.ly/34DlOsj

• Gross private domestic investment (nonresidential) in billions of dollars, seasonally adjusted, quarterly data, BEA, NIPA Table 1.1.5., https://bit.ly/34DlOsj

• Implicit price deflator for government consumption expenditures and gross investment, seasonally adjusted, quarterly data, BEA, NIPA Table 1.1.9., https://bit.ly/2z6230N

• Implicit price deflator for gross domestic product, seasonally adjusted, quarterly data, BEA, NIPA Table 1.1.9., https://bit.ly/2z6230N

• Implicit price deflator for Gross private domestic investment (non residential), seasonally adjusted, quarterly data, BEA, NIPA Table 1.1.9., https://bit.ly/2z6230N

• Nonfinancial Corporate Business; Debt Securities and Loans; Liability, Level (BCNSDODNS), Billions of Dollars, Seasonally Adjusted, https://fred.stlouisfed.org/series/BCNSDODNS

• Real Gross Domestic Product, Billions of Chained 2012 Dollars, Seasonally Adjusted Annual Rate (GDPC1), https://fred.stlouisfed.org/series/GDPC1

All weblinks last accessed on November 20, 2022.

Appendix 3. Parameter values and steady-state results

See Table 1

Table 1 Parameter values

Appendix 4. Reverse capital deepening and the public debt-to-GDP ratio

As discussed in Sect. 4, the model is able to generate the phenomenon of reverse capital deepening. Consistently with the argument exposed in the paper, an increase in the interest rate would then lead to a decrease in the public debt-to-GDP ratio. Under these circumstances, the model exhibits the motion shown in Fig. 6.

Fig. 6

Impact of an increase in interest rate on macroeconomic variables

Appendix 5. Computation of intersectoral multipliers

The intersectoral multipliers can be computed adopting the Leontief Inverse matrix (L):

$${L=\left(I-A\right)}^{-1}={\left[\begin{array}{ccc}1& -\vartheta & 0\\ -\beta & 1& 0\\ -\gamma & 0& 1\end{array}\right]}^{-1}=\left[\begin{array}{ccc}\frac{1}{1-\vartheta \beta }& \frac{\alpha }{1-\vartheta \beta }& 0\\ \frac{\beta }{1-\vartheta \beta }& \frac{1}{1-\vartheta \beta }& 0\\ \frac{\gamma }{1-\vartheta \beta }& \frac{\alpha \gamma }{1-\vartheta \beta }& 1\end{array}\right]$$

To produce one unit of consumer good it is required to produce a gross amount of A and B equal to:

$${m}_{a}=\frac{1+\beta +\gamma }{1-\alpha \beta }$$

$${m}_{b}=\frac{1+\alpha (1+\gamma )}{1-\alpha \beta }$$

If the ratio $\frac{{m}_{a}}{{m}_{b}}=\frac{1+\beta +\gamma }{1+\alpha (1+\gamma )}$ is higher than one, basic commodity A has the highest intersectoral multiplier, the opposite applies if the ratio is lower than one.