Question

Consider an economy described by the production function:

*Y = F(K, L) = K ^{0.3}L^{0.7}.*

**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.

Answer #1

Y = K^{0.3}L^{0.7}

Output per worker (y) = Y / L = (K/L)^{0.3} =
k^{0.3} where k = K/L

When s: Savings rate, in steady state,

s / = k / y

s /
= k / k^{0.3}

s /
= k^{0.7}

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

Output per worker (y*) = k^{0.3} = [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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