Two-Stage Utility Maximization Problem

Question

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.

Answer 1

Consider the Wikipedia definition (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

$$E_h(p_{2},p_{3},z) = p_z z.$$

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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 $x_3(p_1,p_2,p_3,I) = \frac{(1-\alpha)(I-x_1)}{p_3}$.

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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 $x_3$.