On Comparative Dynamics

Lately, there has been an increased interest in stability of growth paths, see e.g., Brock and Scheinkman [1974]. The problem has been stated in terms of properties of stationary paths. In order to appreciate the difficulty of the general stability problem, one must realize that there are two types of "time" involved in the analysis: stability "time” and path “time.” Thus, the appropriate mathematical field is that of differential equations defined on a space of functions rather than a finite dimensional space. Naturally, if one restricts one’s attention to stationary paths, then the usual stability analysis is appropriate. However, we would be then discussing the asymptotic behavior of the asymptotic state of the economy. This note strives to put the problem of path stability in the proper perspective by discussing the much simpler problem of comparative dynamics. Unfortunately this term has been used in the economic growth literature to discuss the basically comparative statics problem of comparing stationary growth paths. By comparative dynamics, we mean the determination of the “direction” of change in the optimal path of decision variables due to a change in the exogenous variables. The traditional method of deriving comparative statics results has been to use second order conditions for optimality. However, if one is willing to assume concavity, these results could be derived in a more direct way by utilizing the fact that a differentiable concave function lies below its tangent plane. We shall use this concept in deriving the main inequalities of this note. By way of motivation, we first derive two inequalities of comparative statics. Then we derive the comparative dynamics results and finally we discuss some economic theoretical examples

On Comparative Dynamics

Lately, there has been an increased interest in stability of growth paths, see e.g., Brock and Scheinkman [1974]. The problem has been stated in terms of properties of stationary paths. In order to appreciate the difficulty of the general stability problem, one must realize that there are two types of "time" involved in the analysis: stability "time” and path “time.” Thus, the appropriate mathematical field is that of differential equations defined on a space of functions rather than a finite dimensional space. Naturally, if one restricts one’s attention to stationary paths, then the usual stability analysis is appropriate. However, we would be then discussing the asymptotic behavior of the asymptotic state of the economy. This note strives to put the problem of path stability in the proper perspective by discussing the much simpler problem of comparative dynamics. Unfortunately this term has been used in the economic growth literature to discuss the basically comparative statics problem of comparing stationary growth paths. By comparative dynamics, we mean the determination of the “direction” of change in the optimal path of decision variables due to a change in the exogenous variables. The traditional method of deriving comparative statics results has been to use second order conditions for optimality. However, if one is willing to assume concavity, these results could be derived in a more direct way by utilizing the fact that a differentiable concave function lies below its tangent plane. We shall use this concept in deriving the main inequalities of this note. By way of motivation, we first derive two inequalities of comparative statics. Then we derive the comparative dynamics results and finally we discuss some economic theoretical examples

PP2A:B56ε is required for eye induction and eye field separation

AbstractEye induction and eye field separation are the earliest events during vertebrate eye development. Both of these processes occur much earlier than the formation of optic vesicles. The insulin-like growth factor (IGF) pathway appears to be essential for eye induction, yet it remains unclear how IGF downstream pathways are involved in eye induction. As a consequence of eye induction, a single eye anlage is specified in the anterior neural plate. Subsequently, this single eye anlage is divided into two symmetric eye fields in response to Sonic Hedgehog (Shh) secreted from the prechordal mesoderm. Here, we report that B56ε regulatory subunit of protein phosphatase 2A (PP2A) is involved in Xenopus eye induction and subsequent eye field separation. We provide evidence that B56ε is required for the IGF/PI3K/Akt pathway and that interfering with the PI3K/Akt pathway inhibits eye induction. In addition, we show that B56ε regulates the Hedgehog (Hh) pathway during eye field separation. Thus, B56ε is involved in multiple signaling pathways and plays critical roles during early development

Modules in the photoreceptor RGS9-1•Gβ5L GTPase-accelerating protein complex control effector coupling, GTPase acceleration, protein folding, and stability

RGS (regulators of G protein signaling proteins regulate G protein signaling by accelerating GTP hydrolysis, but little is known about regulation of GTPase-accelerating protein (GAP) activities or roles of domains and subunits outside the catalytic cores. RGS9-1 is the GAP required for rapid recovery of light responses in vertebrate photoreceptors and the only mammalian RGS protein with a defined physiological function. It belongs to an RGS subfamily whose members have multiple domains, including G gamma -like domains that bind G(beta5) proteins. Members of this subfamily play important roles in neuronal signaling, Within the GAP complex organized around the RGS domain of RGS9-1, we have identified a functional role for the G gamma -like-G(beta 5L) complex in regulation of GAP activity by an effector subunit, cGMP phosphodiesterase gamma and in protein folding and stability of RGS9-1, The C-terminal domain of RGS9-1 also plays a major role in conferring effector stimulation. The sequence of the RGS domain determines whether the sign of the effector effect will be positive or negative. These roles were observed in, vitro using full-length proteins or fragments for RGS9-1, RGS7, G(beta 5S), and G(beta 5s), The dependence of RGS9-1 on Gp, co-expression for folding, stability, and function has been confirmed in vivo using transgenic Xenopus laevis, These results reveal how multiple domains and regulatory polypeptides work together to fine tune G(t alpha) inactivation

Candidate Gene Screen in the Red Flour Beetle Tribolium Reveals Six3 as Ancient Regulator of Anterior Median Head and Central Complex Development

Several highly conserved genes play a role in anterior neural plate patterning of vertebrates and in head and brain patterning of insects. However, head involution in Drosophila has impeded a systematic identification of genes required for insect head formation. Therefore, we use the red flour beetle Tribolium castaneum in order to comprehensively test the function of orthologs of vertebrate neural plate patterning genes for a function in insect head development. RNAi analysis reveals that most of these genes are indeed required for insect head capsule patterning, and we also identified several genes that had not been implicated in this process before. Furthermore, we show that Tc-six3/optix acts upstream of Tc-wingless, Tc-orthodenticle1, and Tc-eyeless to control anterior median development. Finally, we demonstrate that Tc-six3/optix is the first gene known to be required for the embryonic formation of the central complex, a midline-spanning brain part connected to the neuroendocrine pars intercerebralis. These functions are very likely conserved among bilaterians since vertebrate six3 is required for neuroendocrine and median brain development with certain mutations leading to holoprosencephaly