# utility maximization with nested Cobb–Douglas–CES preferences

I'm trying to understand the following paper: Hsieh & Ossa: A global view of productivity growth in China (2016). The pdf can be found here: https://faculty.chicagobooth.edu/chang-tai.hsieh/research/hsieh_ossa_jie.pdf

I struggle to understand the utility maximization problem right at the beginning of the paper. Can someone show me how to maximize the following utility with respect to $$x_{ijs}$$:

$$U_j=\prod_s^S\left(\sum_i^N \int_0^{M_is^e}x_{ijs}(\nu_{is})^{\frac{\sigma_s-1}{\sigma_s}}d\nu_{is} \right)^{\frac{\sigma_s}{\sigma_s-1}\mu_{js}}$$

The end result is:

$$x_{ijs}=\frac{p_{ijs}^{-\sigma_s}}{P_{js}^{1-\sigma_s}}\mu_{js}E_j$$

where $$P_{js}=\left(\sum_i^N M_{is}^ep_{ijs}^{1-\sigma_s}\right)^\frac{1}{1-\sigma_s}$$

$$N$$: number of countries

$$S$$: number of industries

$$M_{is}^e$$: number of entrants in industry s of country i

$$x_{ijs}$$: quantity of an industry s variety from country i consumed in country j

$$\mu_{js}$$: fraction of country j income spent on industry s varieties

$$\sigma_s>1$$: elasticity of substitution between industry s varieties

$$p_{ijs}$$: price of an industry s variety from country i in country j

$$P_{js}$$: ideal price index in industry s of country j

$$E_j$$: total expenditure in country j

• I dont understand why is there an integral in the utility function. I see that the amount of output purchased by j's consumers from the s-th sector in i ($x_{ijs}(ν_{is})$) depends on the number of entrants in s in i. But I dont understand why do they integrate all the values if$x_{ijs}$ from $x_{ijs}(0)$ to $x_{ijs}(M^e_{is})$, since they only consume $x_{ijs}(M^e_{is})$. Could you explain? Aug 22 '19 at 0:05

## 1 Answer

Take the following utility function: $$\prod_{s \in S} \left(\sum_{i \in N} x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}} \right)^{\frac{\sigma_s}{\sigma_s-1}\mu_{s}}$$

Which is a CES nested in a Cobb-Douglas.

Consider the sub-utility function: $$Q_{s} = \left(\sum_{i \in N} x_{i,s}^{\frac{\sigma_s - 1}{\sigma_s}}\right)^{\frac{\sigma_s}{\sigma_s - 1}}.$$ Then we can re-write the utility function succinctly as: $$U = \prod_{s \in S} Q_s^{\mu_s}$$ To solve the utility maximisation problem. Notice that $$U$$ is separable in the subgroups. As such, we can solve the problem using two stage budgeting.

1. In the first step, we can determine the optimal allocation within each subgroup $$s$$ by maximize the sub-utility functions $$Q_s$$ given the total expenditure $$E_s$$ given on each subgroup $$s$$.
2. In the second step, we determine the optimal allocation of income across each subgroup $$s$$.

For Step 1, we solve the following problem: $$\max\left(\sum_{i \in N} x_{i,s}^{\frac{\sigma_s - 1}{\sigma_s}}\right)^{\frac{\sigma_s}{\sigma_s - 1}} \text{subject to } \sum_{i \in N} p_{i,s} x_{i,s} = E_s.$$ This is a CES utility function. The first order condition gives: $$Q_s \frac{x_{i,s}^{-\frac{1}{\sigma_s}}}{\sum_{w \in N} x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}} = \lambda p_{i,s}.$$ Then: $$x_{i,s} = \lambda^{-\sigma_s} p_{i,s}^{-\sigma_s} \left(\frac{\sum_w x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}}{Q_s}\right)^{-1\sigma_s}$$ Multiplying by $$p_{i,s}$$ and adding over all $$i \in N$$ gives: \begin{align*} E_s = \sum_{i \in N} p_{i,s} x_{i,s} &= \lambda^{-\sigma_s} \left(\frac{\sum_w x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}}{Q_s}\right)^{-\sigma_s} \sum_{i \in N} p_{i,s}^{1 - \sigma_s}\\ &= \lambda^{-\sigma_s} \left(\frac{\sum_w x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}}{Q_s}\right)^{-\sigma_s} P_s^{1 - \sigma_s} \end{align*} Where $$E_{s} = \sum_{i\in N} p_{i,s} x_{i,s}$$ and $$P_{s}^{1- \sigma_s} = \sum_{i \in N} p_{i,s}^{1- \sigma_s}$$.

Then: $$x_{i,s} = E_{s} \frac{p_{i,s}^{-\sigma_s}}{P_{s}^{1-\sigma_s}} \tag{1}$$ Let's compute the value of $$Q_s$$: \begin{align*} &x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}} = E_{s}^{\frac{\sigma_s-1}{\sigma_s}} \left(\frac{p_{i,s}^{-\sigma_s}}{P_{s}^{1 -\sigma_s}}\right)^{\frac{\sigma_s-1}{ \sigma_s}},\\ \to &\sum_{i \in N} x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}}= E_{j,s}^{\frac{\sigma_s-1}{\sigma_s}} \left(\frac{1}{P_{s}^{1- \sigma_s}}\right)^{\frac{\sigma_s-1}{\sigma_s}} \sum_{i \in N} p_{i,s}^{(1-\sigma_s)},\\ \to& \sum_{i \in N} x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}} = E_{s}^{\frac{\sigma_s-1}{\sigma_s}} P_{s}^{\frac{1 - \sigma_s}{\sigma_s}}\\ \to & \left(\sum_i x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}}\right)^{\frac{\sigma_s}{\sigma_s-1}} = E_{s} P_{s}^{-1}\\ \to& P_{s} Q_s = E_{s}. \end{align*} This shows that the price index $$P_s$$ and the quantity index $$Q_s$$ are exact price and quanty indices for this subgroup (as is expected as we were maximising a CES utility function).

Let's now go to step 2. Notice that the budget constraint can be rewritten as: $$E = \sum_{s} E_s = \sum_s P_s Q_s.$$ We want to determine $$E_s$$ to maximize: $$U = \prod_{j \in S} Q_{s}^{\mu_{j,s}} \text{ s.t. } \sum_s P_{s} Q_{s} = E.$$ This is a simple Cobb-Douglas utility function. As such, we know that expenditure shares are proportional to income: $$P_s Q_s = E_s = \frac{\mu_s}{\sum_w \mu_w} E$$ Finally, substitute this into the equation $$(1)$$: $$x_{i,s} = \frac{\mu_{s}}{\sum_w \mu_w} E \frac{p_{i,s}^{-\sigma_s}}{P_{s}^{1- \sigma_s}}$$