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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?

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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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