Determinants of Maximum Sustainable Government Debt 85 If public expenditure shares are sufficiently small (i.e., g C u < 1 ˛ and g C u < 1 ˛), for: (i) b D 0 and b D 0, one non-trivial steady state exists, (ii) 0 < b < bmax and 0 < b < .b /max , two non-trivial steady states exist. Proposition 2 Stability of steady state If public expenditure shares are sufficiently small (i.e., g C u < 1 ˛ and gCu < 1 ˛), and if Home and Foreign debt ratios are below some upper limits (i.e., b < bmax and b < .b /max ) but non-negative, the higher steady state equilibrium (i.e., where k is larger) is saddle-path stable and the lower steady state is saddle-path unstable. To determine the stability properties of the economic system, the dynamic equations (17), (20), and (22) have to be analyzed near the steady state equilibria. This can be done by means of the eigenvalues of the Jacobian matrix for the respective steady states. The derivation of the elements of the Jacobian matrix can be found, for example, in Farmer and Schelnast (2013, p. 326f).3 Extensive numerical simulations reveal that for the lower steady state, two eigenvalues are outside the unit circle, and for the higher steady state, only one steady state is outside the unit circle. Hence, the lower steady state is saddle-path unstable, while the higher steady state is saddle-path stable. Without productive government capital, this property can be proved analytically for the case of equal private capital elasticities in Home and Foreign (see Farmer and Schelnast 2013, p. 315) and zero debt in Home and Foreign (i.e. b D b D 0) and also for differing private capital elasticities in Home and Foreign (see Farmer and Schelnast 2013, p. 339f). 3 Maximum Sustainable Debt A debt is called sustainable if a debt to output ratio can be maintained by retaining the pre-determined tax structure.4 Hence, if the debt ratio is lower than the maximum sustainable debt ratio, no governmental interventions are necessary to prevent an economic collapse. The aim of this section is to define how the maximum sustainable debt ratio is influenced by different economic parameters, such as Foreign’s debt ratio, and governmental productive and unproductive expenditures, Home’s and Foreign’s savings rates, and a change in Home’s and Foreign’s production elasticities. As Proposition 1 states, there are two steady state equilibria if debt ratios are sufficiently small. A higher debt ratio shifts the curve F(k) downwards.5 Hence, 3 For details on the elements of the Jacobian matrix, please consult the authors. 4 This assumption is also applied in Rankin and Roffia (2003). 5 Proved in 3.2.2.1 in Schelnast (2013).