<p>It turns out there was a mistake in the problem statement. The question asked that we rank the sectors along the balanced growth path, but only meant that <span class="math-container">$k_t$</span>, <span class="math-container">$c_t$</span>, and <span class="math-container">$A_t$</span> should be in balanced growth. With <span class="math-container">$\bar{a},\bar{s}&gt;0$</span>, there is no balanced growth path, as can be seen in the equations <span class="math-container">$$\frac{\Delta a_t}{a_{t-1}}=g\left(1-\frac{\bar{a}}{a_{t-1}}\right)$$</span>
and
<span class="math-container">$$\frac{\Delta s_t}{s_{t-1}}=g\left(1+\frac{\bar{s}}{s_{t-1}}\right)$$</span></p>
<p>Therefore, my proposed solution is complete and the ordering should be <span class="math-container">$$g_a&lt;g_c=g=g_k&lt;g_s$$</span></p>

CC BY-SA 4.0
2023-12-22T07:05:31.320Z
2023-12-22T07:05:31.320Z
Joseph Basford
<p>I'm going over a past paper for a macro course I'm taking and have gotten stuck on the last part of one of the questions.</p>
<p>Suppose we have a three-sector economy. We are given <span class="math-container">$B_a,B_s&gt;0$</span>, a technology variable <span class="math-container">$A_t$</span> which grows at rate <span class="math-container">$g$</span> and is such that <span class="math-container">$A_0=1$</span>, and initial capital levels <span class="math-container">$k_{m0},k_{s0},k_{a0}$</span>. There is an agricultural good which is produced according to <span class="math-container">$$a_t=B_ak_{at}^{\alpha}(h_{at}A_t)^{1-\alpha}$$</span>
Similarly, a service good is produced according to <span class="math-container">$$s_t=B_sk_{st}^{\alpha}(h_{st}A_t)^{1-\alpha}$$</span> Then there is a manufacturing output which can either be used for investment or consumption <span class="math-container">$$c_t+k_{t+1}-(1-\delta)k_t=k_{mt}^{\alpha}(h_{mt}A_t)^{1-\alpha}$$</span>
Normalise the price of the manufactured good to be 1 and let the price of the agricultural and service good be denoted by <span class="math-container">$P_{at}$</span> and <span class="math-container">$P_{st}$</span>, respectively. Assume labour and capital markets are perfectly competitive with households supplying one unit of labour. We have the labour and capital market clearing conditions <span class="math-container">$$h_{at}+h_{st}+h_{mt}=1$$</span> and <span class="math-container">$$k_{at}+k_{st}+k_{mt}=k_t$$</span> Households have utility of the form <span class="math-container">$$\sum_{t=0}^{\infty}\beta^t\frac{[(a_t-\bar{a})^{\eta}c_t^{\gamma}(s_t+\bar{s})^{\theta}]^{1-\sigma}}{1-\sigma}$$</span> subject to the per-period budget constraint <span class="math-container">$$P_{at}a_t+c_t+k_{t+1}-(1-\delta)k_t+P_{st}s_t=w_t+R_tk_t$$</span></p>
<p>Assume that <span class="math-container">$\eta,\theta,\gamma,\sigma&gt;0$</span>, <span class="math-container">$\eta+\theta+\gamma=1$</span>, and that <span class="math-container">$\bar{a}B_s=\bar{s}B_a$</span>.</p>
<p>The question asks us to find the growth rates of <span class="math-container">$a_t,s_t,c_t,k_{t+1}$</span> and rank the growth rates of the sectors.</p>
<p>So far I have calculated from the first order conditions that <span class="math-container">$$s_t+\bar{s}=\frac{\theta B_s}{\gamma}c_t$$</span> and <span class="math-container">$$a_t-\bar{a}=\frac{\eta B_a}{\gamma}c_t$$</span>
and have found from the budget constraint that <span class="math-container">$$k_{t+1}=k_t^{\alpha}A_t^{1-\alpha}+(1-\delta)k_t-\frac{c_t}{\gamma}$$</span> I think the budget constraint implies that both <span class="math-container">$k_t$</span> and <span class="math-container">$c_t$</span> grow at rate <span class="math-container">$g$</span> since otherwise they don't have balanced growth. However, I can't see how <span class="math-container">$s_t$</span> or <span class="math-container">$a_t$</span> can have balanced growth given the equations I have so I think I must have made a mistake but I can't see where it is and can't tell if it's an algebraic or conceptual one.</p>
<p>Given what I have, I think I get that <span class="math-container">$$\frac{\Delta s_t}{s_{t-1}}=g\left(1+\frac{\bar{s}}{s_{t-1}}\right)$$</span> and <span class="math-container">$$\frac{\Delta a_t}{a_{t-1}}=g\left(1-\frac{\bar{a}}{a_{t-1}}\right)$$</span></p>
<p>This would give the ordering <span class="math-container">$g_{at}&lt;g_c=g_k&lt;g_{st}$</span>, but this is wrong since this isn't a balanced growth path as <span class="math-container">$g_{at}$</span> and <span class="math-container">$g_{st}$</span> depend on time.</p>
<p>Could someone direct me as to where I am going wrong or how I can go about solving this problem?</p>

<macroeconomics><balanced-growth>
Balanced Growth Path in a 3-Sector Economy
Joseph Basford
# Balanced Growth Path in a 3-Sector Economy

(Asked by: username_0 on 2023-12-15)

I'm going over a past paper for a macro course I'm taking and have gotten stuck on the last part of one of the questions.

Suppose we have a three-sector economy. We are given $B\_a,B\_s>0$, a technology variable $A\_t$ which grows at rate $g$ and is such that $A\_0=1$, and initial capital levels $k\_{m0},k\_{s0},k\_{a0}$. There is an agricultural good which is produced according to $$a\_t=B\_ak\_{at}^{\alpha}(h\_{at}A\_t)^{1-\alpha}$$
Similarly, a service good is produced according to $$s\_t=B\_sk\_{st}^{\alpha}(h\_{st}A\_t)^{1-\alpha}$$ Then there is a manufacturing output which can either be used for investment or consumption $$c\_t+k\_{t+1}-(1-\delta)k\_t=k\_{mt}^{\alpha}(h\_{mt}A\_t)^{1-\alpha}$$
Normalise the price of the manufactured good to be 1 and let the price of the agricultural and service good be denoted by $P\_{at}$ and $P\_{st}$, respectively. Assume labour and capital markets are perfectly competitive with households supplying one unit of labour. We have the labour and capital market clearing conditions $$h\_{at}+h\_{st}+h\_{mt}=1$$ and $$k\_{at}+k\_{st}+k\_{mt}=k\_t$$ Households have utility of the form $$\sum\_{t=0}^{\infty}\beta^t\frac{[(a\_t-\bar{a})^{\eta}c\_t^{\gamma}(s\_t+\bar{s})^{\theta}]^{1-\sigma}}{1-\sigma}$$ subject to the per-period budget constraint $$P\_{at}a\_t+c\_t+k\_{t+1}-(1-\delta)k\_t+P\_{st}s\_t=w\_t+R\_tk\_t$$

Assume that $\eta,\theta,\gamma,\sigma>0$, $\eta+\theta+\gamma=1$, and that $\bar{a}B\_s=\bar{s}B\_a$.

The question asks us to find the growth rates of $a\_t,s\_t,c\_t,k\_{t+1}$ and rank the growth rates of the sectors.

So far I have calculated from the first order conditions that $$s\_t+\bar{s}=\frac{\theta B\_s}{\gamma}c\_t$$ and $$a\_t-\bar{a}=\frac{\eta B\_a}{\gamma}c\_t$$
and have found from the budget constraint that $$k\_{t+1}=k\_t^{\alpha}A\_t^{1-\alpha}+(1-\delta)k\_t-\frac{c\_t}{\gamma}$$ I think the budget constraint implies that both $k\_t$ and $c\_t$ grow at rate $g$ since otherwise they don't have balanced growth. However, I can't see how $s\_t$ or $a\_t$ can have balanced growth given the equations I have so I think I must have made a mistake but I can't see where it is and can't tell if it's an algebraic or conceptual one.

Given what I have, I think I get that $$\frac{\Delta s\_t}{s\_{t-1}}=g\left(1+\frac{\bar{s}}{s\_{t-1}}\right)$$ and $$\frac{\Delta a\_t}{a\_{t-1}}=g\left(1-\frac{\bar{a}}{a\_{t-1}}\right)$$

This would give the ordering $g\_{at}<g\_c=g\_k<g\_{st}$, but this is wrong since this isn't a balanced growth path as $g\_{at}$ and $g\_{st}$ depend on time.

Could someone direct me as to where I am going wrong or how I can go about solving this problem?

**Tags:** macroeconomics, balanced-growth


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## 1 answers

### Answer by username_0 on 2023-12-22. Score: 1 | Accepted answer

It turns out there was a mistake in the problem statement. The question asked that we rank the sectors along the balanced growth path, but only meant that $k\_t$, $c\_t$, and $A\_t$ should be in balanced growth. With $\bar{a},\bar{s}>0$, there is no balanced growth path, as can be seen in the equations $$\frac{\Delta a\_t}{a\_{t-1}}=g\left(1-\frac{\bar{a}}{a\_{t-1}}\right)$$
and
$$\frac{\Delta s\_t}{s\_{t-1}}=g\left(1+\frac{\bar{s}}{s\_{t-1}}\right)$$

Therefore, my proposed solution is complete and the ordering should be $$g\_a<g\_c=g=g\_k<g\_s$$


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