Consider the economy of Avataria, which can be described by the Solow model. Avataria has the depreciation rates of 5% and the production function

1. (8 points) Consider the economy of Avataria, which can be described by the Solow model. Avataria has the depreciation rates of 5% and the production function

Y(K,L) = (K)1/4(L)3/4.

(a) Show that the production function in Avataria exhibits the constant return to scale property.

(b) Assume that Avataria has the investment rate of 10%. Calculate the steady-state capital-labor ratio, output per worker, and consumption per worker. Also, what is the growth rate of the total output in Avataria in the steady state?

(c) Now assume that Avataria has the new investment rate, which increases its steady state output per worker by 50%. Calculate this new investment rate.

(d) Finally, assume that Avataria has the old investment rate of 10% but now its producers rely on capital relatively more so that the production function is

Y(K,L) = (K)1/3(L)2/3.

How does this change affect output per capita in steady state compared to part (b)? Calculate its exact value.

2. (2 points) This problem is based on the growth equation (see page 10 of the textbook).

(a) How long will it take for Avataria with an annual growth rate of 3.5% to increase its income by 6 times?

(b) What was the growth rate in Twilightia, if its income decreased from $8,000,000 in 2010 to $2,500,000 in 2020?