The authors study the invariant curves for the difference equation \[ x_{n+1}=cx_{n}+f(x_{n}-x_{n-1}),\quad n\in \mathbb N, \tag{1} \] which is modelled from macroeconomics. The dynamics of Eq. (1) can be described equivalently by the planar mapping \( F:\mathbb R^{2}\rightarrow \mathbb R^{2} \)defined by \( F(x,y)=(y,cy+f(y-x))\). Its invariant curves \( \Gamma :y=\varphi(x) \) satisfy the equation \( cy+f(y-x)=\varphi(y) \), which leads to the iterative functional equation \[ \varphi(\varphi(x))=c \varphi(x)+f(\varphi(x)-x).\tag{2} \] First, the authors investigate the linear case, viz, \( f=kx+\rho\). For \( \rho\neq 0\), all solutions of (2) with \( f=kx+\rho \) are given under non-hyperbolic characteristic roots by piecewise construction, see [J. Matkowski and W. Zhang, Acta Math. Sin., New Ser. 13, No. 3, 421–432 (1997; Zbl 0881.39011)]. For nonlinear \(f\), the Banach contraction principle is employed to obtain results concerning existence of invariant curves for (2). Continuous dependence of solutions of (2) on \( f \) is discussed and an example is given to illustrate the results obtained for nonlinear case.