# [Economics] Demand for minimum of $4$ different goods

## Solution 1:

You can solve it sequentially by noting the nesting structure of the utility function $$U$$.

So first note that the utility function combines utility functions you are probably already familiar with $$U=\min\{u_1,u_2\}$$ of complements and $$u_1=\sqrt{x+y}$$ and $$u_2 = z+w$$ both of which are perfect substitutes. Where $$u_1$$ and $$u_2$$ are nested within $$U$$.

The solution procedure is based on the following argument: However much money $$m_{u_1}$$ the agent decides to spend on $$u_1 =\sqrt{x+y}$$ she must be minimizing the cost of getting $$u_1$$ utility points. Similarly, however much money $$m_{u_2}$$ the agent spends on $$u_2= z + w$$ it must be the case that the agent minimize cost. So first find the expenditure functions for these inner problem.

Secondly the agent must the maximize the outer utility $$\min\{u_1,u_2\}$$ subject to the expenditures of the inner problems adding up to total income.

Solve the second inner problem:

The second inner problem is $$max_{z,w} \{z+w \lvert p_zz+p_ww\leq m_{u_2}\},$$

where $$m_{u_2}$$ is money spend on $$u_2$$.

How do you get the most utility for the buck when utility is $$u_2 = z + w$$? You simply buy the cheapest good so the price of one point of $$u_2$$ utility is $$p_{u_2} := \min\{p_z,p_w\}$$. The demand for $$z$$ and $$w$$ depends on the amount of money you decide to spend on $$u_2$$ assumed for now to be $$m_{u_2}$$. Given this amount demand for $$z$$ is

$$z = \frac{m_{u_2}}{p_z} 1[p_z

and

$$w = \frac{m_{u_2}}{p_w} 1[p_w

I leave it to you to handle the threshold case where $$p_z=p_w$$. How much utility $$u_2$$ do you get? Well this must be $$u_2 = m_{u_2}/\min\{p_z,p_w\}$$ - this is the value function for this inner utility maximization problem - hence it must be the case that

$$m_{u_2} = \min\{p_z,p_w\} u_2,$$

which is the expenditure function for this inner problem.

Solve the first inner problem:

The first inner problem is $$max_{x,y} \{\sqrt{x+y} \lvert p_xx+p_yy\leq m_{u_1}\},$$

where $$m_{u_1}$$ is money spend on $$u_1$$.

How do you get the most utility for the buck when utility is $$u_1 = \sqrt{x+y}$$? It is perfect substitutes so you buy the cheapest good so demand is

$$x = \frac{m_{u_1}}{p_x} 1[p_x

and

$$y = \frac{m_{u_1}}{p_y} 1[p_y

where $$m_{u_1}$$ is amount of money spend on $$u_1$$ assumed known. How much utility $$u_1$$ do you get? That must be $$u_1 = \sqrt{m_{u_1}/\min\{p_x,p_y\}}$$ such that

$$m_{u_1} = \min\{p_x,p_y\} u_1^2$$

Solve outer problem:

The outer problem is

$$\max_{u_1,u_2}: \ \ \ \min\{u_1,u_2\} \\[8pt] s.t. \ \ \ \ m_{u_1} + m_{u_2} = m$$

for which we rewrite the budget constraint to

$$\min\{p_x,p_y\} u_1^2 + \min\{p_z,p_w\} u_2 = m$$

and then use that $$u_1=u_2$$ because the problem is perfect complements. You then have two equations in two unknowns $$u_1$$ and $$u_2$$. Solve for $$u_1$$ and $$u_2$$ and plug solutions back into expenditure functions for inner problems to get $$m_{u_1}$$ and $$m_{u_2}$$ which you then plug into the equations for the demand.

## Solution 2:

To provide another answer with less equations:

Consider first that the inner utility function $$u_1 = \sqrt{x+y}$$ and $$u_2 = z + w$$ are perfect substitutes implying that consumer only buy the cheapest of $$x$$ and $$y$$ and similarly only the cheapest of $$z$$ and $$w$$. Because prices are given we know that $$y$$ is cheaper than $$x$$ and $$w$$ is cheaper than $$z$$. Using this we can simply set $$x=0$$ and $$z=0$$ and get the utility maximization problem

Maximize $$U(y,w) = \min\{\sqrt y,w\}$$

which implies that $$\sqrt y = w$$ for any solution.

Therefore there are two equations in two unknowns

$$p_y y + p_w w = m \Rightarrow 2y + w = 1$$

and $$\sqrt y = w$$

Hence it follows that

$$2y + \sqrt y = 1$$

which is satisfied for $$y=1/4$$ and since $$\sqrt y = w$$ it follows that $$w = \sqrt(1/4) = 1/2$$. Adding that $$x=0$$ and $$z=0$$ we have all demands.