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A model for economic growth is given by Here K(t) = r₂K(t)}L(t)\ — r₁K(t) İ(t)=r3L(t) •K: the amount of capital. L: the amount of labor force. • r₁: the rate of depreciation of capital. • r2: the rate of reinvestment income. r3: the growth rate of the labor force. Here 11, 12, 13 are all real constant, 0≤r₁ ≤ 1,0 ≤r2 ≤ 1, r3 > 0. (1) If r₁=0, r₂ = 0, find the equation that K satisfies. Explain its economic meaning. (2) If r₂ = 0, find the equation that K satisfies. Explain its economic meaning. K(t) (3) Now assume r₁, 72, 73 >0 strictly. If we define p(t) = as the capital per labor, (a) show that p=r₂p - (₁+rs)p (Hint: Consider #p(t), and use the equation of K(t) and L(t), then use p = to eliminate K and L). (b) Find all fixed points of this 1D dynamical system.

A model for economic growth is given by Here K(t) = r₂K(t)}L(t)\ — r₁K(t) İ(t)=r3L(t) •K: the amount of capital. L: the amount of labor force. • r₁: the rate of depreciation of capital. • r2: the rate of reinvestment income. r3: the growth rate of the labor force. Here 11, 12, 13 are all real constant, 0≤r₁ ≤ 1,0 ≤r2 ≤ 1, r3 > 0. (1) If r₁=0, r₂ = 0, find the equation that K satisfies. Explain its economic meaning. (2) If r₂ = 0, find the equation that K satisfies. Explain its economic meaning. K(t) (3) Now assume r₁, 72, 73 >0 strictly. If we define p(t) = as the capital per labor, (a) show that p=r₂p - (₁+rs)p (Hint: Consider #p(t), and use the equation of K(t) and L(t), then use p = to eliminate K and L). (b) Find all fixed points of this 1D dynamical system.

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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clearly provide solution (Donnot use others' work) A model for economic growth is given by K(t) = rgK(t)*L(t)* –rįK(t) İ(t) = r3L(t) Here K: the amount of capital. • L: the amount of labor force. • 71: the rate of depreciation of capital. r2: the rate of reinvestment income. r3: the growth rate of the labor force. Here r1,r2, 13 are all real constant, 0 <ris 1,0 < r2 < 1, r3 > 0. (1) If rį = 0, r2 = 0, find the equation that K satisfies. Explain its economic meaning. (2) If r2 = 0, find the equation that K satisfies. Explain its economic meaning. (3) Now assume ri,r2, r3 > 0 strictly. If we define p(t) = 8 as the capital per labor, (a) show that p = rapt – (ri + rs)p (Hint: Consider p(t), and use the equation of K(t) and L(t), then use p = 4 to eliminate K and L). (b) Find all fixed points of this 1D dynamical system.
Transcribed Image Text:clearly provide solution (Donnot use others' work) A model for economic growth is given by K(t) = rgK(t)*L(t)* –rįK(t) İ(t) = r3L(t) Here K: the amount of capital. • L: the amount of labor force. • 71: the rate of depreciation of capital. r2: the rate of reinvestment income. r3: the growth rate of the labor force. Here r1,r2, 13 are all real constant, 0 <ris 1,0 < r2 < 1, r3 > 0. (1) If rį = 0, r2 = 0, find the equation that K satisfies. Explain its economic meaning. (2) If r2 = 0, find the equation that K satisfies. Explain its economic meaning. (3) Now assume ri,r2, r3 > 0 strictly. If we define p(t) = 8 as the capital per labor, (a) show that p = rapt – (ri + rs)p (Hint: Consider p(t), and use the equation of K(t) and L(t), then use p = 4 to eliminate K and L). (b) Find all fixed points of this 1D dynamical system.
(c) Show that for any initial condition p(0) = po > 0, the flow converges to a fixed point p* > 0.
Transcribed Image Text:(c) Show that for any initial condition p(0) = po > 0, the flow converges to a fixed point p* > 0.
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