<p>Consider the <a href="https://en.wikipedia.org/wiki/Quasilinear_utility" rel="noreferrer">Wikipedia definition</a> (take from Hal Varian's book) of a quasi-linear utility function</p>
<p><span class="math-container">$$u(w,x_1,...,x_n) = w + h(x_1,...,x_n),$$</span></p>
<p>where <span class="math-container">$h$</span> is strictly concave.</p>
<p>The example in this exercise is according to this definition not a quasi-linear utility function because <span class="math-container">$h(x_1,...,x_n)$</span>, which in this case is <span class="math-container">$x_2^\alpha x_3^{1-\alpha}$</span> is not <strong>strictly</strong> concave but instead concave. This is so because <span class="math-container">$x_2^\alpha x_3^{1-\alpha}$</span> has exponents that sum to 1 instead of being less than one.</p>
<p>The case under consideration is closer to that of perfect substitutes. However, it is a little more complicated because one of the goods is a composite good.</p>
<p>You need to know two things in order to solve the exercise:</p>
<p><strong>A. You need to know how to solve the case of perfect substitutes</strong></p>
<p><strong>B. You need to know how to solve the case of CD preferences</strong></p>
<p>Consider first the case of perfect substitutes</p>
<p><span class="math-container">$$\max_{x_1,z}\ \ u(x_1,z) = x_1 + \bar a z \\ s.t. \ \ x_1 + p_zz = I$$</span></p>
<p>for each coin spend on <span class="math-container">$x_1$</span> the agent gets 1 utility point and for each <span class="math-container">$p_z$</span> coins spent on <span class="math-container">$z$</span> the agent gets <span class="math-container">$\bar a$</span> utility points which amounts to <span class="math-container">$\bar a/p_z$</span> utility point per coin. Hence if <span class="math-container">$1&gt;\bar a/p_z$</span> the agent will only buy <span class="math-container">$x_1$</span> and demand for <span class="math-container">$x_1$</span> is simply <span class="math-container">$I$</span> units because the price of <span class="math-container">$x_1$</span> is standardized to <span class="math-container">$1$</span>. If <span class="math-container">$1&lt;\bar a/p_z$</span> the agent will only buy <span class="math-container">$z$</span> and the demand for <span class="math-container">$I/p_z$</span>. Finally, if <span class="math-container">$1=\bar a/p_z$</span> the agent is indifferent between <span class="math-container">$x_1$</span> and <span class="math-container">$z$</span> and the demand for <span class="math-container">$x_1$</span> is then any amount of units in <span class="math-container">$[0,I]$</span> with the demand for <span class="math-container">$z$</span> being <span class="math-container">$(I-x_1)/p_z$</span>.</p>
<p>Consider next the case of CD-preferences with</p>
<p><span class="math-container">$$h(x_2,x_3) = \left(\frac{x_2}{\alpha}\right)^\alpha \left(\frac{x_3}{1-\alpha}\right)^{1-\alpha}$$</span></p>
<p>Note that the function is homogenous of degree 1 hence has constants returns to scale. In this case, there exists a price for each utility point which can be found by solving the expenditure minimization problem</p>
<p><span class="math-container">$$\min_{x_2,x_3} \ \ p_{2}x_2 + p_{3} x_3 \\ s.t. \ \ \left(\frac{x_2}{\alpha}\right)^\alpha \left(\frac{x_3}{1-\alpha}\right)^{1-\alpha} = z$$</span></p>
<p>when solving this problem you will get the expenditure function</p>
<p><span class="math-container">$$E_h(p_{2},p_{3},z) = (p_{2}^\alpha p_{3}^{1-\alpha}) z,$$</span></p>
<p>which tells you that for each utility point <span class="math-container">$z$</span> the consumer must pay <span class="math-container">$p_{2}^\alpha p_{3}^{1-\alpha}$</span>. Because expenditure is proportional to <span class="math-container">$z$</span> with the constant of proportionality being <span class="math-container">$p_{2}^\alpha p_{3}^{1-\alpha}$</span> you can conceive of <span class="math-container">$p_{2}^\alpha p_{3}^{1-\alpha}$</span> as the price per utility point <span class="math-container">$z$</span>. I therefore adopt the definition</p>
<p><span class="math-container">$$p_z := p_{2}^\alpha p_{3}^{1-\alpha},$$</span></p>
<p>in which case the expenditure function becomes</p>
<p><span class="math-container">$$E_h(p_{2},p_{3},z) = p_z z.$$</span></p>
<hr />
<p>Let us now finally consider the problem in the assignment</p>
<p><span class="math-container">$$\max_{x_1,x_2,x_3} \ \ x_1 + x_2^\alpha x_3^{1-\alpha} \\ s.t. \ \ x_1 + p_2x_2 + p_3x_3 = I.$$</span></p>
<p>You can multiply and divide the second summand of the utility function with <span class="math-container">$\alpha^\alpha$</span> and <span class="math-container">$(1-\alpha)^{1-\alpha}$</span> to get</p>
<p><span class="math-container">$$ x_1 + \bar a \underbrace{\left[\left(\frac{x_2}{\alpha}\right)^\alpha \left(\frac{x_3}{1-\alpha}\right)^{1-\alpha}\right]}_{:=z} = x_1 + \bar a z$$</span></p>
<p>where <span class="math-container">$\bar a=\alpha^\alpha (1-\alpha)^{1-\alpha}$</span>.</p>
<p>You can now see that the agent faces the choice of spending money on <span class="math-container">$x_1$</span> getting one utility point per coin or spending money on <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> getting <span class="math-container">$\bar a$</span> utility points per <span class="math-container">$p_z = p_2^\alpha p_3^{1-\alpha}$</span> coins. Using the logic from the case of perfect substitutes it follows that if</p>
<p><span class="math-container">$$1&gt;\bar a/p_z = \alpha^\alpha(1-\alpha)^{1-\alpha}/(p_2^\alpha p_3^{1-\alpha})$$</span></p>
<p>the consumer only spends money on <span class="math-container">$x_1$</span> and demand is <span class="math-container">$x_1(p_1,p_2,p_3,I) = I$</span> while <span class="math-container">$x_2(p_1,p_2,p_3,I) = x_3(p_1,p_2,p_3,I) =0$</span>.</p>
<p>In the case where</p>
<p><span class="math-container">$$1&lt;\bar a/p_z = \alpha^\alpha(1-\alpha)^{1-\alpha}/(p_2^\alpha p_3^{1-\alpha})$$</span></p>
<p>the consumer spend all money on <span class="math-container">$z$</span> which is <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> and as always with CD preferences <span class="math-container">$\alpha$</span> share is spent on <span class="math-container">$x_2$</span> and <span class="math-container">$(1-\alpha)$</span> share on <span class="math-container">$x_3$</span> so the demand is</p>
<p><span class="math-container">$x_1(p_1,p_2,p_3,I) = 0$</span> while <span class="math-container">$x_2(p_1,p_2,p_3,I) = \frac{\alpha I}{p_2}$</span> and <span class="math-container">$x_3(p_1,p_2,p_3,I) =\frac{(1-\alpha)I}{p_3}$</span>.</p>
<p>Finally, in the case where</p>
<p><span class="math-container">$$1=\bar a/p_z = \alpha^\alpha(1-\alpha)^{1-\alpha}/(p_2^\alpha p_3^{1-\alpha})$$</span></p>
<p>it does not matter how the consumer allocates spending between <span class="math-container">$x_1$</span> and <span class="math-container">$z$</span>, but any amount spent on <span class="math-container">$z$</span> is spent on <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> and must be distributed between <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> optimally. Hence, demand for <span class="math-container">$x_1(p_1,p_2,p_3,I) \in [0,I]$</span> and <span class="math-container">$x_2(p_1,p_2,p_3,I) = \frac{\alpha(I-x_1)}{p_2}$</span> while <span class="math-container">$x_3(p_1,p_2,p_3,I) = \frac{(1-\alpha)(I-x_1)}{p_3}$</span>.</p>
<hr />
<p>You will have trouble using FOC's because they do not capture the border solution where the consumer chooses to spend all money on <span class="math-container">$x_1$</span> or on the combined good <span class="math-container">$z$</span> that is <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span>.</p>

CC BY-SA 4.0
2022-02-19T16:50:53.163Z
2022-02-19T16:50:53.163Z
Jesper Hybel
<p><a href="https://i.sstatic.net/t1SfL.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/t1SfL.png" alt="enter image description here" /></a></p>
<p>Actually I don't know how to solve such utility maximization problem, only know using FOC and budget constraint to solve for demand. I will appreciate it if someone tell me the procedure facing such problem.</p>

CC BY-SA 4.0
2022-02-19T13:44:35.093Z
It is [quasilinear utility](https://en.wikipedia.org/wiki/Quasilinear_utility#:~:text=In%20economics%20and%20consumer%20theory,the%20Marshallian%2FWalrasian%20demand%20for). What have you tried?
Golden_Ratio
CC BY-SA 4.0
2022-02-19T15:41:37.113Z
"using FOC and budget constraint to solve for demand" would work here for each good, so go for it!
Giskard
<microeconomics><optimization><demand>
Two-Stage Utility Maximization Problem
Barry
# Two-Stage Utility Maximization Problem

(Asked by: username_0 on 2022-02-19)

[![enter image description here](https://i.sstatic.net/t1SfL.png)](https://i.sstatic.net/t1SfL.png)

Actually I don't know how to solve such utility maximization problem, only know using FOC and budget constraint to solve for demand. I will appreciate it if someone tell me the procedure facing such problem.

**Tags:** microeconomics, optimization, demand

> **Comments:**
> **username_1**: It is [quasilinear utility](https://en.wikipedia.org/wiki/Quasilinear_utility#:~:text=In%20economics%20and%20consumer%20theory,the%20Marshallian%2FWalrasian%20demand%20for). What have you tried?  
> **username_2**: "using FOC and budget constraint to solve for demand" would work here for each good, so go for it!  

---

## 1 answers

### Answer by username_3 on 2022-02-19. Score: 8 | Accepted answer

Consider the [Wikipedia definition](https://en.wikipedia.org/wiki/Quasilinear_utility) (take from Hal Varian's book) of a quasi-linear utility function

$$u(w,x\_1,...,x\_n) = w + h(x\_1,...,x\_n),$$

where $h$ is strictly concave.

The example in this exercise is according to this definition not a quasi-linear utility function because $h(x\_1,...,x\_n)$, which in this case is $x\_2^\alpha x\_3^{1-\alpha}$ is not **strictly** concave but instead concave. This is so because $x\_2^\alpha x\_3^{1-\alpha}$ has exponents that sum to 1 instead of being less than one.

The case under consideration is closer to that of perfect substitutes. However, it is a little more complicated because one of the goods is a composite good.

You need to know two things in order to solve the exercise:

**A. You need to know how to solve the case of perfect substitutes**

**B. You need to know how to solve the case of CD preferences**

Consider first the case of perfect substitutes

$$\max\_{x\_1,z}\ \ u(x\_1,z) = x\_1 + \bar a z \\ s.t. \ \ x\_1 + p\_zz = I$$

for each coin spend on $x\_1$ the agent gets 1 utility point and for each $p\_z$ coins spent on $z$ the agent gets $\bar a$ utility points which amounts to $\bar a/p\_z$ utility point per coin. Hence if $1>\bar a/p\_z$ the agent will only buy $x\_1$ and demand for $x\_1$ is simply $I$ units because the price of $x\_1$ is standardized to $1$. If $1<\bar a/p\_z$ the agent will only buy $z$ and the demand for $I/p\_z$. Finally, if $1=\bar a/p\_z$ the agent is indifferent between $x\_1$ and $z$ and the demand for $x\_1$ is then any amount of units in $[0,I]$ with the demand for $z$ being $(I-x\_1)/p\_z$.

Consider next the case of CD-preferences with

$$h(x\_2,x\_3) = \left(\frac{x\_2}{\alpha}\right)^\alpha \left(\frac{x\_3}{1-\alpha}\right)^{1-\alpha}$$

Note that the function is homogenous of degree 1 hence has constants returns to scale. In this case, there exists a price for each utility point which can be found by solving the expenditure minimization problem

$$\min\_{x\_2,x\_3} \ \ p\_{2}x\_2 + p\_{3} x\_3 \\ s.t. \ \ \left(\frac{x\_2}{\alpha}\right)^\alpha \left(\frac{x\_3}{1-\alpha}\right)^{1-\alpha} = z$$

when solving this problem you will get the expenditure function

$$E\_h(p\_{2},p\_{3},z) = (p\_{2}^\alpha p\_{3}^{1-\alpha}) z,$$

which tells you that for each utility point $z$ the consumer must pay $p\_{2}^\alpha p\_{3}^{1-\alpha}$. Because expenditure is proportional to $z$ with the constant of proportionality being $p\_{2}^\alpha p\_{3}^{1-\alpha}$ you can conceive of $p\_{2}^\alpha p\_{3}^{1-\alpha}$ as the price per utility point $z$. I therefore adopt the definition

$$p\_z := p\_{2}^\alpha p\_{3}^{1-\alpha},$$

in which case the expenditure function becomes

$$E\_h(p\_{2},p\_{3},z) = p\_z z.$$

---

Let us now finally consider the problem in the assignment

$$\max\_{x\_1,x\_2,x\_3} \ \ x\_1 + x\_2^\alpha x\_3^{1-\alpha} \\ s.t. \ \ x\_1 + p\_2x\_2 + p\_3x\_3 = I.$$

You can multiply and divide the second summand of the utility function with $\alpha^\alpha$ and $(1-\alpha)^{1-\alpha}$ to get

$$ x\_1 + \bar a \underbrace{\left[\left(\frac{x\_2}{\alpha}\right)^\alpha \left(\frac{x\_3}{1-\alpha}\right)^{1-\alpha}\right]}\_{:=z} = x\_1 + \bar a z$$

where $\bar a=\alpha^\alpha (1-\alpha)^{1-\alpha}$.

You can now see that the agent faces the choice of spending money on $x\_1$ getting one utility point per coin or spending money on $x\_2$ and $x\_3$ getting $\bar a$ utility points per $p\_z = p\_2^\alpha p\_3^{1-\alpha}$ coins. Using the logic from the case of perfect substitutes it follows that if

$$1>\bar a/p\_z = \alpha^\alpha(1-\alpha)^{1-\alpha}/(p\_2^\alpha p\_3^{1-\alpha})$$

the consumer only spends money on $x\_1$ and demand is $x\_1(p\_1,p\_2,p\_3,I) = I$ while $x\_2(p\_1,p\_2,p\_3,I) = x\_3(p\_1,p\_2,p\_3,I) =0$.

In the case where

$$1<\bar a/p\_z = \alpha^\alpha(1-\alpha)^{1-\alpha}/(p\_2^\alpha p\_3^{1-\alpha})$$

the consumer spend all money on $z$ which is $x\_2$ and $x\_3$ and as always with CD preferences $\alpha$ share is spent on $x\_2$ and $(1-\alpha)$ share on $x\_3$ so the demand is

$x\_1(p\_1,p\_2,p\_3,I) = 0$ while $x\_2(p\_1,p\_2,p\_3,I) = \frac{\alpha I}{p\_2}$ and $x\_3(p\_1,p\_2,p\_3,I) =\frac{(1-\alpha)I}{p\_3}$.

Finally, in the case where

$$1=\bar a/p\_z = \alpha^\alpha(1-\alpha)^{1-\alpha}/(p\_2^\alpha p\_3^{1-\alpha})$$

it does not matter how the consumer allocates spending between $x\_1$ and $z$, but any amount spent on $z$ is spent on $x\_2$ and $x\_3$ and must be distributed between $x\_2$ and $x\_3$ optimally. Hence, demand for $x\_1(p\_1,p\_2,p\_3,I) \in [0,I]$ and $x\_2(p\_1,p\_2,p\_3,I) = \frac{\alpha(I-x\_1)}{p\_2}$ while $x\_3(p\_1,p\_2,p\_3,I) = \frac{(1-\alpha)(I-x\_1)}{p\_3}$.

---

You will have trouble using FOC's because they do not capture the border solution where the consumer chooses to spend all money on $x\_1$ or on the combined good $z$ that is $x\_2$ and $x\_3$.


---

