Question

# Consider an economy described by the production function: Y = F(K, L) = K0.3L0.7. Assume that...

Consider an economy described by the production function:

Y = F(K, L) = K0.3L0.7.

Assume that the depreciation rate is 5 percent per year. Make a table showing steady-state capital per worker, output per worker, and consumption per worker for saving rates of 0 percent, 10 percent, 20 percent, 30 percent, and so on. Round your answers to two decimal places. (You might find it easiest to use a computer spreadsheet then transfer your answers to this table.)

 Steady State Values for Various Saving Rates s k* y* c* Depreciation Rate: 0.0 (0.05) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

What saving rate maximizes output per worker? What saving rate maximizes consumption per worker?

A saving rate of  percent maximizes output per worker. A saving rate of  percent maximizes consumption per worker.

Y = K0.3L0.7

Output per worker (y) = Y / L = (K/L)0.3 = k0.3 where k = K/L

When s: Savings rate, in steady state,

s / = k / y

s / = k / k0.3

s / = k0.7

Capital per worker (k*) = [s / ](1/0.7) = [s / ]1.43

Output per worker (y*) = k0.3 = [s / ](0.3/0.7) = [s / ]0.43

Consmption per worker (c*) = yY - sy* = y* x (1 - s) = [s / ]0.43 x (1 - s)

When = 0.05,

k* = (s / 0.05)1.43

y* = (s / 0.05)0.43

c* = (s / 0.05)0.43 x (1 - s)

Therefore:

 s k* y* c* 0 0 0 0 0.1 2.69 1.35 1.21 0.2 7.26 1.82 1.45 0.3 12.96 2.16 1.51 0.4 19.56 2.45 1.47 0.5 26.92 2.69 1.35 0.6 34.93 2.91 1.16 0.7 43.55 3.11 0.93 0.8 52.71 3.29 0.66 0.9 62.38 3.47 0.35 1 72.52 3.63 0.00

A savings rate of 100% maximizes output per worker.

A savings rate of 30% maximizes consumption per worker.

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