Suppose a country's per worker production function is yt = f(kt) = kt^1/2, where yt is output per worker and kt is capital per worker. Households save a constant fractions of their income (0 < s < 1); output can be either consumed or invested. Capital depreciates at a rate δ = 0.1. Let capital per worker evolve such that kt+1 = it + (1 - 8)kt, where it is investment per worker. Assume there is no population or technological growth.

(a) Write down the Solow model.

(b) Solve for the steady-state values of capital per worker, output per worker, investment per worker, and consumption per worker when s = 0.2

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(c) Solve for the steady-state values of capital per worker, output per worker, investment per worker, and consumption per worker when s = 0.4

(a) The Solow model is a well-known neoclassical growth model used in economics to study long-run economic growth. It involves differential equations and requires numerical analysis to solve for steady-state values. (b) The steady-state values of capital per worker, output per worker, investment per worker, and consumption per worker when s = 0.2 is 1.6. (c) The steady-state values of capital per worker, output per worker, investment per worker, and consumption per worker when s = 0.4 is 2.4.

(a) The Solow model states that capital stock per worker evolves in response to saving and investment behaviors, with capital depreciation and technological change taken into account. The model explains why growth rates of output per worker can differ over time and across countries.

(b) When s = 0.2, steady-state values of capital per worker (k*), output per worker (y*), investment per worker (i*), and consumption per worker (c*) can be calculated as:
k* = (sf(k*) + (1 - δ)k*)/y* = 0.2(k*^1/2)/(k*^1/2) + 0.9k*/(k*^1/2) = 0.2(k*^1/2) + 0.9(k*^1/2)
0.1k*^1/2 = 0.2
k* = 4
y* = f(k*) = k*^1/2 = 2
i* = δk* = 0.4
c* = (1 - s)y* = 1.6

(c) When s = 0.4, steady-state values of capital per worker (k*), output per worker (y*), investment per worker (i*), and consumption per worker (c*) can be calculated as:
k* = (sf(k*) + (1 - δ)k*)/y* = 0.4(k*^1/2)/(k*^1/2) + 0.9k*/(k*^1/2) = 0.4 + 0.9(k*^1/2)
0.1k*^1/2 = 0.4
k* = 16
y* = f(k*) = k*^1/2 = 4
i* = δk* = 1.6
c* = (1 - s)y* = 2.4

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