We study the delay differential equation \(x'(t) = a [x(t) - x(t - 1)] - g(x(t - \tau )) \) where \(a>0\) , \(\tau >0\) , and \(g:\mathbb {R}\ni u\mapsto u |u|^\kappa \in \mathbb {R}\) with \(\kappa >0\) . This equation is a modification of the Brunovský–Erdélyi–Walther price model by incorporating a reaction delay \(\tau >0\) . For any \(a>0\) and \(\kappa >0\) , by the Kaplan–Yorke method and the homogeneity of the nonlinear function g, we find a countable and dense set of delays \(\tau \) in \((0,\infty )\) for which there exists a periodic solution. A consequence is that global asymptotic stability of the zero solution cannot be expected if \(a\in (0,1)\) , in contrast to the case \(\tau =0\) . For \(a\in (0,1)\) and \(\tau \in (0,1]\) , an \(R_1>0\) is constructed such that 0 attracts the ball of radius \(R_1\) with center at 0. Local asymptotic stability of the zero solution follows as well. It is also shown that, for any \(a\in (0,1)\) , the region of attraction of 0 tends to the whole phase space \(C([-1,0],\mathbb {R})\) as \(\tau \rightarrow 0^+.\)
