<p>What is the depreciation rate (<span class="math-container">$\delta$</span>)?</p>

<p>Assumptions:</p>

<ul>
<li><span class="math-container">$Y_t = K_t^{\alpha} (A_tL_t)^{1-\alpha}$</span>,</li>
<li><span class="math-container">$\alpha = 0.5$</span> (<span class="math-container">$\alpha$</span> = capital income share),</li>
<li><span class="math-container">$A_0 =1$</span> and <span class="math-container">$g = 0$</span> (<span class="math-container">$g$</span> = growth rate of technology),</li>
<li><span class="math-container">$s = 0.2$</span> (<span class="math-container">$s$</span> = savings rate),</li>
<li><span class="math-container">$n = 0.05$</span> (<span class="math-container">$n$</span> = growth rate of the labor force/population).</li>
</ul>

<p>Capital in the next period is equal to the capital from the current period, minus the part that depreciates, plus the extra investment (savings rate times production):
<span class="math-container">$$K_{t+1} = K_t(1 - \delta) + s K_t^{\alpha} (A_t L_t)^{1-\alpha}$$</span></p>

<p>Multiply and divide LHS by <span class="math-container">$A_{t+1}L_{t+1}$</span> and multiply and divide RHS by <span class="math-container">$A_{t}L_{t}$</span>.</p>

<p><span class="math-container">$$\frac{K_{t+1}}{A_{t+1}L_{t+1}} A_{t+1}L_{t+1} = 
A_t L_t \left[(1 - \delta) \frac{K_t}{A_t L_t} + s \frac{K_t^{\alpha} (A_t L_t)^{1-\alpha}}{A_t L_t} \right] $$</span></p>

<p>Then note that <span class="math-container">$A_t = A_0 (1+g)^t$</span> and <span class="math-container">$L_t = L_0 (1 + n)^t$</span> and capital per effective worker is <span class="math-container">$\tilde{k}_t = \frac{K_t}{A_t L_t}$</span>. Using this, rewrite the above equation to obtain:
<span class="math-container">$$\tilde{k}_{t+1} = \frac{(1 - \delta) \tilde{k}_t + s \tilde{k}^\alpha_t}{(1+g)(1+n)}$$</span></p>

<p>On the balanced growth path <span class="math-container">$\tilde{k}_t = \tilde{k}_{t+1} = \tilde{k}^\ast$</span>. Using this, rewrite the above to get:
<span class="math-container">$$\tilde{k}^\ast = \left( \frac{s}{g+n+gn+\delta} \right)^{ \frac{1}{1-\alpha} } $$</span></p>

<p>Since you implicitly assume that <span class="math-container">$A_t = 1, \forall t$</span> this balanced growth path capital per effective worker is equal to capital per worker:</p>

<p><span class="math-container">$$\tilde{k}^\ast = \frac{K}{AL} = \frac{K}{L} = \left( \frac{s}{g+n+gn+\delta} \right)^{ \frac{1}{1-\alpha} } = \left( \frac{0.2}{0.05+\delta} \right)^{2} $$</span></p>

<p>As you can see this <span class="math-container">$\frac{K}{L}$</span> is stable.</p>

<p>For consumption per worker, we can consume what we don't invest:
<span class="math-container">$$C = Y(1 - s)$$</span>
<span class="math-container">$$\frac{C}{L} = \frac{Y}{L} (1 - s)$$</span></p>

<p>Plugging the <span class="math-container">$\tilde{k}^\ast$</span> from above into the production function gives us:
<span class="math-container">$$\tilde{y}^\ast = \frac{Y}{L} = \left( \frac{s}{g+n+gn+\delta} \right)^{\frac{\alpha}{1-\alpha}} $$</span></p>

<p>So,
<span class="math-container">$$\frac{C}{L} = \left( \frac{s}{g+n+gn+\delta} \right)^{\frac{\alpha}{1-\alpha}} (1 -s) = \frac{0.16}{0.05 + \delta}$$</span></p>

CC BY-SA 4.0
2019-04-28T22:23:22.287Z
2019-04-28T22:29:19.810Z
2019-04-28T22:29:19.810Z
Mark
<p>The question is this:</p>

<p>Suppose there is a production function of <span class="math-container">$Q=F(K, L)=\sqrt{K}\sqrt{L}$</span>. Suppose that MPC=0.8 and the labor force is growing at a rate of 0.05 per year. Also labor force participation rate is 100%.</p>

<p>a. What is the steady state equilibrium capital labor ratio?
b. Is this equilibrium stable?
c. What is the corresponding level of consumption per worker?</p>

<p>Any help would be appreciated. </p>

<macroeconomics><labor-economics><solow><capital>
Looking for help with Solow Model Question
Stanley Hudson
# Looking for help with Solow Model Question

(Asked by: username_0 on 2019-04-28)

The question is this:

Suppose there is a production function of $Q=F(K, L)=\sqrt{K}\sqrt{L}$. Suppose that MPC=0.8 and the labor force is growing at a rate of 0.05 per year. Also labor force participation rate is 100%.

a. What is the steady state equilibrium capital labor ratio?
b. Is this equilibrium stable?
c. What is the corresponding level of consumption per worker?

Any help would be appreciated.

**Tags:** macroeconomics, labor-economics, solow, capital


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## 1 answers

### Answer by username_1 on 2019-04-28. Score: 1

What is the depreciation rate ($\delta$)?

Assumptions:

* $Y\_t = K\_t^{\alpha} (A\_tL\_t)^{1-\alpha}$,
* $\alpha = 0.5$ ($\alpha$ = capital income share),
* $A\_0 =1$ and $g = 0$ ($g$ = growth rate of technology),
* $s = 0.2$ ($s$ = savings rate),
* $n = 0.05$ ($n$ = growth rate of the labor force/population).

Capital in the next period is equal to the capital from the current period, minus the part that depreciates, plus the extra investment (savings rate times production):
$$K\_{t+1} = K\_t(1 - \delta) + s K\_t^{\alpha} (A\_t L\_t)^{1-\alpha}$$

Multiply and divide LHS by $A\_{t+1}L\_{t+1}$ and multiply and divide RHS by $A\_{t}L\_{t}$.

$$\frac{K\_{t+1}}{A\_{t+1}L\_{t+1}} A\_{t+1}L\_{t+1} =
A\_t L\_t \left[(1 - \delta) \frac{K\_t}{A\_t L\_t} + s \frac{K\_t^{\alpha} (A\_t L\_t)^{1-\alpha}}{A\_t L\_t} \right] $$

Then note that $A\_t = A\_0 (1+g)^t$ and $L\_t = L\_0 (1 + n)^t$ and capital per effective worker is $\tilde{k}\_t = \frac{K\_t}{A\_t L\_t}$. Using this, rewrite the above equation to obtain:
$$\tilde{k}\_{t+1} = \frac{(1 - \delta) \tilde{k}\_t + s \tilde{k}^\alpha\_t}{(1+g)(1+n)}$$

On the balanced growth path $\tilde{k}\_t = \tilde{k}\_{t+1} = \tilde{k}^\ast$. Using this, rewrite the above to get:
$$\tilde{k}^\ast = \left( \frac{s}{g+n+gn+\delta} \right)^{ \frac{1}{1-\alpha} } $$

Since you implicitly assume that $A\_t = 1, \forall t$ this balanced growth path capital per effective worker is equal to capital per worker:

$$\tilde{k}^\ast = \frac{K}{AL} = \frac{K}{L} = \left( \frac{s}{g+n+gn+\delta} \right)^{ \frac{1}{1-\alpha} } = \left( \frac{0.2}{0.05+\delta} \right)^{2} $$

As you can see this $\frac{K}{L}$ is stable.

For consumption per worker, we can consume what we don't invest:
$$C = Y(1 - s)$$
$$\frac{C}{L} = \frac{Y}{L} (1 - s)$$

Plugging the $\tilde{k}^\ast$ from above into the production function gives us:
$$\tilde{y}^\ast = \frac{Y}{L} = \left( \frac{s}{g+n+gn+\delta} \right)^{\frac{\alpha}{1-\alpha}} $$

So,
$$\frac{C}{L} = \left( \frac{s}{g+n+gn+\delta} \right)^{\frac{\alpha}{1-\alpha}} (1 -s) = \frac{0.16}{0.05 + \delta}$$


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