High School

Assume that a country's per-worker production is given by [tex]y = 2 \cdot k^{1/2}[/tex], where [tex]y[/tex] is output per worker and [tex]k[/tex] is capital per worker. Assume also that 20 percent of capital depreciates per year ([tex]\delta = 0.20[/tex]) and there is no population growth or technological change. Please show your results step by step.

a. If the saving rate ([tex]s[/tex]) is 0.4, what are capital per worker, production per worker, and consumption per worker in the steady state?

b. Solve for steady-state capital per worker, production per worker, and consumption per worker with [tex]s = 0.6[/tex].

c. Solve for steady-state capital per worker, production per worker, and consumption per worker with [tex]s = 0.8[/tex].

d. Is it possible to save too much? Why?

Answer :

Let's analyze the given economics problem step-by-step by using the Solow Growth Model, which helps us understand how different savings rates affect an economy's capital per worker, output per worker, and consumption per worker in the steady state.

First, we'll use the steady-state condition in the Solow Model:

  1. Part (a): Solving for Steady-State with s = 0.4

    In the steady state, savings (s) and depreciation (b4) balance out. Therefore:

    [tex]s \cdot y = 3b4 \cdot k[/tex]

    Given:

    • Production function per worker: [tex]y = 2 \sqrt{k}[/tex]
    • Savings rate [tex](s) = 0.4[/tex]
    • Depreciation rate [tex](3b4) = 0.2[/tex]

    Substituting [tex]y[/tex] from the production function:

    [tex]0.4 \cdot 2 \sqrt{k} = 0.2 \cdot k[/tex]

    Simplifying gives:

    [tex]0.8 \sqrt{k} = 0.2k[/tex]

    Dividing both sides by [tex]\sqrt{k}[/tex],

    [tex]0.8 = 0.2 \sqrt{k}[/tex]

    Solving for [tex]\sqrt{k}[/tex]:

    [tex]\sqrt{k} = 4[/tex]

    Square both sides:

    [tex]k = 16[/tex]

    Now calculate [tex]y[/tex] using [tex]k = 16[/tex]:

    [tex]y = 2 \times \sqrt{16} = 8[/tex]

    Consumption per worker, [tex]c[/tex], is everything not saved:

    [tex]c = (1 - s) \cdot y = (1 - 0.4) \cdot 8 = 4.8[/tex]

  2. Part (b): Solving for Steady-State with s = 0.6

    [tex]0.6 \cdot 2 \sqrt{k} = 0.2 \cdot k[/tex]

    [tex]1.2 \sqrt{k} = 0.2k[/tex]

    [tex]1.2 = 0.2 \sqrt{k}[/tex]

    [tex]\sqrt{k} = 6[/tex]

    [tex]k = 36[/tex]

    [tex]y = 2 \times 6 = 12[/tex]

    [tex]c = (1 - 0.6) \cdot 12 = 4.8[/tex]

  3. Part (c): Solving for Steady-State with s = 0.8

    [tex]0.8 \cdot 2 \sqrt{k} = 0.2 \cdot k[/tex]

    [tex]1.6 \sqrt{k} = 0.2k[/tex]

    [tex]1.6 = 0.2 \sqrt{k}[/tex]

    [tex]\sqrt{k} = 8[/tex]

    [tex]k = 64[/tex]

    [tex]y = 2 \times 8 = 16[/tex]

    [tex]c = (1 - 0.8) \cdot 16 = 3.2[/tex]

  4. Part (d): Is it possible to save too much? Why?

    Yes, it is possible to save too much. In the Solow model, if the savings rate is very high, it leads to lower current consumption, as seen in part (c) where increasing the savings rate to 0.8 resulted in lower consumption per worker compared to lower savings rates. This highlights the trade-off between current and future consumption, and beyond a certain point, high savings can decrease the overall well-being of workers by reducing their immediate consumption.

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