# Balanced Growth Path in a 3-Sector Economy

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 $$g_{at}, but this is wrong since this isn't a balanced growth path as $$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?

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