Consider an economy in a continuous time universe: t [0, +). There is a single...

 

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Consider an economy in a continuous time universe: t € [0, +∞). There is a single commodity produced with land. The production technology is Yt = al where yt is output, & is the land input actually used, a > 0 is a technology parameter, and 3 € (0,1) is another fixed parameter. Suppose that production is detrimental to the natural environment. That is, if production is above a certain threshold (y > 0), then land input de- creases, depending on a parameter 0 > 0: i = -0(yt - y) Let the initial land input used be equal to lo > 0. Answer the following: • Does there exist a steady-state * > 0? If yes, find it as a function of parameters. If not, formally explain why the model does not have a nontrivial steady-state? • Is = 0 a steady-state? . Can you draw the time path of lt (i.e., where t is on the horizontal axis)? Consider an economy in a continuous time universe: t € [0, +∞). There is a single commodity produced with land. The production technology is Yt = al where yt is output, & is the land input actually used, a > 0 is a technology parameter, and 3 € (0,1) is another fixed parameter. Suppose that production is detrimental to the natural environment. That is, if production is above a certain threshold (y > 0), then land input de- creases, depending on a parameter 0 > 0: i = -0(yt - y) Let the initial land input used be equal to lo > 0. Answer the following: • Does there exist a steady-state * > 0? If yes, find it as a function of parameters. If not, formally explain why the model does not have a nontrivial steady-state? • Is = 0 a steady-state? . Can you draw the time path of lt (i.e., where t is on the horizontal axis)?