LAUSR

Capital $$(k_t,t\ge 0)$$(kt,t≥0) in the Solow growth model solves a nonlinear ODE defined by $$b:[0,\infty )\rightarrow \mathbb {R}$$b:[0,∞)→R where $$b^{\prime }(k)\rightarrow \infty $$b′(k)→∞ as $$k\rightarrow 0^{+}$$k→0+. Macroeconomics states Solow’s ODE has a unique positive equilibrium point $$k^{*}$$k∗ which is asymptotically stable. Proving $$k_t\rightarrow k^{*}$$kt→k∗ as $$t\rightarrow \infty $$t→∞ when $$k_0>0$$k0>0 is close to $$k^{*}$$k∗ traditionally rests on a Taylor expansion of $$b(k_t)$$b(kt) around $$k^{*}$$k∗ and approximation of $$|k_t-k^{*}|$$|kt-k∗|. However, this proof ignores the remainder $$R(k_t,k^{*})$$R(kt,k∗) in Taylor’s formula. Failure to establish $$R(k_t,k^{*})\rightarrow 0$$R(kt,k∗)→0 as $$t\rightarrow \infty $$t→∞ means the accepted proof is incorrect. Our paper presents a correct proof of asymptotic stability in Solow’s model, driven by properties of $$(k_t, t\ge 0)$$(kt,t≥0) not linearized stability theory, which is rendered useless by explosion of $$b^{\prime }$$b′. Derivative explosion appears in many investment models of economic dynamics making our method of general interest.