<p>Let production function $$F(K_t,L_t)=K_t^aL_t^{1-a}$$</p>

<p>$$max[c^*=f(k^*)-(δ+n)k^*]$$ with respect to $k^*$</p>

<p>Then $MPK=a(k^*)^{a-1}$</p>

<p>So $k_G= (\frac{a}{(δ+n)})^{1/1-a}$ which is golden rule level of capital stock per capita. </p>

<p>Now calculate consumption at golden rule level </p>

<p>$$c_G=f(k_G)-(δ+n)k_G$$
$$c_G=(k_G)^a-(δ+n)k_G$$</p>

<p>$$c_G=(\frac{a}{(δ+n)})^{a/1-a}-(δ+n)(\frac{a}{(δ+n)})^{1/1-a}$$</p>

<p>$$c_G=(\frac{a}{(δ+n)})^{a/1-a}[1-(δ+n)(\frac{a}{(δ+n)})]$$</p>

<p>$$c_G=(\frac{a}{(δ+n)})^{a/1-a}[1-a]$$
$$c_G=k_G^a[1-a]$$</p>

<p>Ow consider to calculate wage </p>

<p>$w=f(k)-kf’(k)=(1-a)f(k)$</p>

<p>Then, </p>

<p>$$c_G=k_G^a[1-a]=w_G$$</p>

<p>$$c_G=w_G$$</p>

<p>Now consider aggregate version,</p>

<p>$C_G/L=W_G/L$</p>

<p>so,</p>

<p>$C_G=W_G$</p>

<p>I am confused at the aggregate transformation.</p>

<p>I add this question with its answer in order to contribute to the website. Thus, I will be glad if you add your opinions about my solution as well.</p>

CC BY-SA 4.0
2018-07-02T20:12:17.570Z
2018-07-02T20:12:17.570Z
Enjoyecon
<blockquote>
  <p>In a perfectly competitive Solow economy with physical capital accumulation, population growth and a Cobb-Douglas production function, show that the “golden rule” steady-state would be reached if at every period aggregate consumption coincides with the aggregate labour income.</p>
</blockquote>

<macroeconomics><solow><steady-state>
Solow model golden rule with my exact answer
Enjoyecon
# Solow model golden rule with my exact answer

(Asked by: username_0 on 2018-07-02)

> In a perfectly competitive Solow economy with physical capital accumulation, population growth and a Cobb-Douglas production function, show that the “golden rule” steady-state would be reached if at every period aggregate consumption coincides with the aggregate labour income.

**Tags:** macroeconomics, solow, steady-state

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## 1 answers

### Answer by username_0 on 2018-07-02. Score: 1 | Accepted answer

Let production function $$F(K\_t,L\_t)=K\_t^aL\_t^{1-a}$$

$$max[c^\*=f(k^\*)-(δ+n)k^\*]$$ with respect to $k^\*$

Then $MPK=a(k^\*)^{a-1}$

So $k\_G= (\frac{a}{(δ+n)})^{1/1-a}$ which is golden rule level of capital stock per capita.

Now calculate consumption at golden rule level

$$c\_G=f(k\_G)-(δ+n)k\_G$$
$$c\_G=(k\_G)^a-(δ+n)k\_G$$

$$c\_G=(\frac{a}{(δ+n)})^{a/1-a}-(δ+n)(\frac{a}{(δ+n)})^{1/1-a}$$

$$c\_G=(\frac{a}{(δ+n)})^{a/1-a}[1-(δ+n)(\frac{a}{(δ+n)})]$$

$$c\_G=(\frac{a}{(δ+n)})^{a/1-a}[1-a]$$
$$c\_G=k\_G^a[1-a]$$

Ow consider to calculate wage

$w=f(k)-kf’(k)=(1-a)f(k)$

Then,

$$c\_G=k\_G^a[1-a]=w\_G$$

$$c\_G=w\_G$$

Now consider aggregate version,

$C\_G/L=W\_G/L$

so,

$C\_G=W\_G$

I am confused at the aggregate transformation.

I add this question with its answer in order to contribute to the website. Thus, I will be glad if you add your opinions about my solution as well.


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