Given the following aggregate production function: Y = K0.25 (AL) 0.75, where technology A grows at a fixed rate: =g>0 (a) Obtain the marginal product of capital algebraically, also discussing the second derivative. (b) Transform the production function into efficiency-worker terms, showing howy depends on=k. (c) Working with the Solow model, where capital evolution follows K=1-8K, investment is a fixed proportion of income I = SY, production is given by the Cobb Douglas function above, and the workforce grows at a positive rate, algebraically derive the fundamental Solow equation in efficiency-worker terms. Show all workings.
Given the following aggregate production function: Y = K0.25 (AL) 0.75, where technology A grows at a fixed rate: =g>0 (a) Obtain the marginal product of capital algebraically, also discussing the second derivative. (b) Transform the production function into efficiency-worker terms, showing howy depends on=k. (c) Working with the Solow model, where capital evolution follows K=1-8K, investment is a fixed proportion of income I = SY, production is given by the Cobb Douglas function above, and the workforce grows at a positive rate, algebraically derive the fundamental Solow equation in efficiency-worker terms. Show all workings.
Chapter1: Making Economics Decisions
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Transcribed Image Text:Given the following aggregate production function: Y = K0.25 (AL) 0.75, where technology A grows at a
fixed rate: =g>0
(a) Obtain the marginal product of capital algebraically, also discussing the second derivative.
depends
(b) Transform the production function into efficiency-worker terms, showing how
к
on=k.
(c) Working with the Solow model, where capital evolution follows K=1-8K, investment is a
fixed proportion of income I = sy, production is given by the Cobb Douglas function above,
and the workforce grows at a positive rate, algebraically derive the fundamental Solow
equation in efficiency-worker terms. Show all workings.
(d) Present and discuss the graphical counterpart to this equation, explaining the dynamics of
capital per efficiency worker that result from the logic of the model.
(e) Working with the fundamental equation you derived in part (c), find the equilibrium solution
for k algebraically. Relate this to your diagram in part (d).
(f) Find the steady state solution algebraically for output per efficiency worker, ỹ, where
(i = 1,2). Show all workings.
(g) Outline and explain the behaviour of per worker variables along the model's balanced growth
path.
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