Applications of Micro Theory: Demand Functions
There’s An Actual Purpose to This
All this abstract math stuff has a purpose:
At the start, I mentioned that firms, non-profits, and government agencies all have an incentive to model demand across every major industry
They can ask: “If we raise the price of our product by 2%, how will our sales numbers react?”
Applications
The models used to estimate demand responses are derived from utility maximization
For example, Dutch Bros sets up a utility that they think represents preferences for their coffee and related goods
They can derive and estimate the associated demand function from their sales data
Up to now, we have only solved utility maximization problems with the goal of finding the level of demand (aka we found numbers )
Now we will solve these problems without specifying either price or income
This makes it more general, which has its advantages:
It allows us to analyze how demand responds when prices or income change
The math stays the same, just that we have more unknowns now
How Do We Find Demand Functions?
Let’s begin with the workhorse of this course: Cobb-Douglas
\[\max U(x,y) = x^{\alpha}y^{\beta} \;\; \text{s.t.} \;\; P_{x} \cdot x + P_{y} \cdot y = M\]
Recall how we solve this problem:
Find the MRS and set it equal to the Price Ratio
Solve this equality for one of the goods \(\rightarrow\) Optimality Condition
Plug the Optimality Condition into the Budget Constraint
Use the found demand for one good and plug it into the Optimality Condition or Budget Constraint to find demand for the other good
Cobb-Douglas General Solution - Step 1
\[\max U(x,y) = x^{\alpha}y^{\beta} \;\; \text{s.t.} \;\; P_{x} \cdot x + P_{y} \cdot y = M\]
Find the MRS and the Price Ratio
\[MRS = \dfrac{MU_{x}}{MU_{y}} = \dfrac{\alpha x^{\alpha - 1 y^{\beta}}}{\beta x^{\alpha} y^{\beta - 1}} = \dfrac{\alpha}{\beta} \cdot \dfrac{x^{\alpha - 1 - \alpha}}{y^{\beta - 1 - \beta}} = \dfrac{\alpha}{\beta} \cdot \dfrac{x^{-1}}{y^{-1}} = \dfrac{\alpha}{\beta} \cdot \dfrac{y}{x}\]
\[\text{Price Ratio} = \dfrac{P_{x}}{P_{y}}\]
Set them equal to each other
\[\begin{align*}
\text{MRS} &= \text{Price Ratio} \\
\dfrac{\alpha}{\beta} \cdot \dfrac{y}{x} &= \dfrac{P_{x}}{P_{y}}
\end{align*}\]
Cobb-Douglas General Solution - Step 2
\[\dfrac{\alpha}{\beta} \cdot \dfrac{y}{x} = \dfrac{P_{x}}{P_{y}}\]
Solve for \(y\)
\[\begin{align*}
\dfrac{\alpha}{\beta} \cdot \dfrac{y}{x} &= \dfrac{P_{x}}{P_{y}} \\
\dfrac{y}{x} &= \dfrac{P_{x}}{P_{y}} \cdot \dfrac{\beta}{\alpha} \\
y^{*} &= \dfrac{\beta}{\alpha} \cdot \dfrac{P_{x}}{P_{y}} \cdot x
\end{align*}\]
Cobb-Douglas General Solution - Step 3
\[y^{*} = \dfrac{\beta}{\alpha} \cdot \dfrac{P_{x}}{P_{y}} \cdot x \;\; \& \;\; \text{BC: } P_{x} \cdot x + P_{y} \cdot y = M\]
Solve for the Demand of good \(x\)
\[\begin{align*}
M &= P_{x} \cdot x + P_{y} \cdot y \\
M &= P_{x} \cdot x + P_{y} \cdot \left( \dfrac{\beta}{\alpha} \cdot \dfrac{P_{x}}{P_{y}} \cdot x \right) \\
M &= P_{x} \cdot x + \dfrac{\beta}{\alpha} \cdot P_{x} \cdot x \\
M &= P_{x} \cdot x \left( 1 + \dfrac{\beta}{\alpha} \right) \\
\vdots \\
x^{*} &= \dfrac{M}{P_{x}} \cdot \dfrac{\alpha}{\alpha + \beta}
\end{align*}\]
Cobb-Douglas General Solution - Step 4
\[x^{*} = \dfrac{M}{P_{x}} \cdot \dfrac{\alpha}{\alpha + \beta}\]
Use found demand of one good to find demand for the other good
\[\begin{align*}
y^{*} &= \dfrac{\beta}{\alpha} \cdot \dfrac{P_{x}}{P_{y}} \cdot \color{red}{x^{*}} \\
y^{*} &= \dfrac{\beta}{\alpha} \cdot \dfrac{P_{x}}{P_{y}} \cdot \color{red}{\dfrac{M}{P_{x}} \cdot \dfrac{\alpha}{\alpha + \beta}} \\
y^{*} &= \dfrac{\beta}{\alpha + \beta} \cdot \dfrac{M}{P_{y}}
\end{align*}\]
Cobb-Douglas General Solution - Trick
When \(x^{*}\) and \(y^{*}\) are functions, we call them Demand Functions
\[x^{*} = \dfrac{\alpha}{\alpha + \beta} \cdot \dfrac{M}{P_{x}}\;\;\;\;\;\; \& \;\;\;\;\;\; y^{*} = \dfrac{\beta}{\alpha + \beta} \cdot \dfrac{M}{P_{y}} \]
Knowing a C-D utility looks like \(U(x,y) = x^{\alpha} y^{\beta}\) what do you notice about these functions?
Let me show you a trick for Cobb-Douglas utility functions
If you ever use this in an exam, be sure to explain why this shortcut works (bullet point 1)
Demand Functions
In general, the demand for \(x\) and \(y\) are functions of three variables:
\(P_{x}\) , \(P_{y}\) , and \(M\)
This allows us to write demand functions as:
\(x^{*} = f(P_{x}, P_{y}, M)\) \(y^{*} = f(P_{x}, P_{y}, M)\) We are interested in how demand responds to changes to all three arguments
Techniques to Find Demand Functions
We will learn how to find Demand Functions
Graphically
Expansion Paths and Engel Curves
Mathematically
Derivatives and Elasticities
Visualizing the Change
Let’s start with a our standard optimized utility graph
What would happen if we Increase
Visualizing the Change
With an increase in income, our budget constraint will:
We find the new Maximizing Point
More Goods
The previous graphs tell us
When income \((M)\) goes up, you can now reach a higher IC and be better off (relatively)
This graph also makes an important assumption
Both goods ,\(x\) and \(y\) , are desirable
We call these Normal Goods
Normal Goods are desirable, which simply means that if we have more income we will consume more of them
Other Types of Goods?
Does it always have to be the case that consumpution of both \(x\) and \(y\) has to increase in response to an increase in income?
No! There are goods we call Inferior Goods
These goods are consumed less when income increases
A classic example of this is Maruchan ramen noodles
Inferior Goods - Graph
Let \(y\) be an inferior good, the optimized utility graph would look something like
Maximized Utility
Increase in Income & Inferior Good \(y\)
Inferior Goods - Graph Intuition
With an inferior good (let’s keep saying \(y\) ) there is a predictable shift when income increases
The new bundle must be in this region of the budget
Can Both Goods Be Inferior?
As your income increases, can you buy less of both goods?
No! Why Not?
The Budget Constraint does not allow it
Recall that the BC looks like: \(P_{x} \cdot x + P_{y} \cdot y = M\)
If your income increases, and you decrease the amount of \(x\) you consume (\(x\) is an inferior good), then there is a lot of income leftover that is required to be spent
The only other option is to increase your consumption of \(y\)
In fact, if increasing your income has no effect on how much \(x\) you consume, you would still have to increase your consumption of \(y\)
Engel Curves
This curve describes the relationship between \(M^{*}\) and \(y^{*}\) , with \(M\) on the vertical axis and \(y^{*}\) on the horizontal axis
Let’s build this by parts, starting with a bunch of income levels and the associated \(y^{*}\) consumption levels
Engel Curves
If we connect all our optimal levels of \(y^{*}\) we find our Engel Curve
Engel Curves
The important thing to consider about Engel Curves is the sign of the slope \(\rightarrow\) It tells us if a good is normal or inferior
A Positive Slope means that when Income goes up, \(y^{*}\) increases and we say \(y\) is a normal good
A Negative Slope means that when Income goes up, \(y^{*}\) decreases and we say \(y\) is an inferior good
Can a good be an inferior good over all income levels?
Engel Curves - Inferior Goods Always?
Can in be the case that no matter what my current income is, an increase in income will decrease my consumption of a good?
It is impossible!
Even a good that is usually inferior, or better said, inferior over common income levels must be a normal good over low enough levels of income
I’ll show you an example of an Engel Curve with a negative slope and we’ll see why
Engel Curves - Inferior Goods Always?
Think about our Instant Ramen example. When in college, the more money you have, the more you buy Instant Ramen (so it is a normal good).
But once you graduate and get that promised pay increase from graduating, you buy Instant Ramen less and less the higher your Income becomes (inferior good)
Engel Curves - Mathematically
We can formalize Engel Curves as:
Engel Curves are graphs of the Demand Function
\(x^{*} = f(P_{x},P_{y},M)\)
Where we hold prices fixed and flip the axis
Engel Curves - Mathematically
Take our Cobb-Douglas example
\[U(x,y) = x^{\alpha}y^{\beta} \;\;\; \text{where} \;\;\; \alpha = \beta = P_{x} = 1\]
Also recall that
\[x^{*} = \dfrac{\alpha}{\alpha + \beta} \cdot \dfrac{M}{P_{x}}\]
Solving for \(M\) yields:
\[\begin{align*}
x^{*} &= \dfrac{\alpha}{\alpha + \beta} \cdot \dfrac{M}{P_{x}} \\
x^{*} &= \dfrac{1}{2} \cdot M = \dfrac{M}{2} \\
x^{*} &= \dfrac{M}{2} \rightarrow M = 2x^{*}
\end{align*}\]
What’s the slope of this demand tell us?
\[\begin{align*}
\dfrac{\partial x}{\partial M} = \dfrac{1}{2} > 0
\end{align*}\]
Engel Curves - Using Derivatives
We can show whether a good is normal or not over all income levels by taking the derivative with respect to \(M\)
\[\text{if} \;\; \dfrac{\partial x^{*}}{\partial M} > 0 \Rightarrow \text{Normal Good}\]
\[\text{if} \;\; \dfrac{\partial x^{*}}{\partial M} < 0 \Rightarrow \text{Inferior Good}\]
Back to Income Changes
We just saw that we can find whether a good is a normal or inferior by taking the partial derivative of the demand function w.r.t. income
By knowing what category the good falls into, we can quickly estimate how demand will react to changes in income
But this only tells us up or down, not magnitudes
To find that, we will look at elasticities
Elasticities
They tell us how responsive demand is to income changes (later we will see price changes)
They follow the formula:
\[E_{x^{*},M} = \dfrac{\partial x^{*}}{\partial M} \cdot \dfrac{M}{x^{*}}\]
There are 3 steps to finding an elasticity
Take the partial derivative of the good w.r.t. \(M\)
Multiply the partial derivative by the ratio of the input variable to the response variable
What is changing to what we are estimating to chagne Substitute the original demand equation and simplify
Elasticities - Example
Let the Demand for \(x^{*}\) be:
\[x^{*} = \dfrac{M}{P_{x}}\]
1 - Take the partial derivative w.r.t. Income
\[\begin{align*}
\dfrac{\partial x}{\partial M} = \dfrac{1}{P_{x}}
\end{align*}\]
Elasticities - Example
The partial derivative is:
\[\dfrac{\partial x}{\partial M} = \dfrac{1}{P_{x}}\]
2 - Multiply the partial derivative by the ratio of input variable to the response variable \(\left(\dfrac{M}{x^{*}} \right)\)
\[\begin{align*}
E_{x^{*},M} = \dfrac{\partial x^{*}}{\partial M} \cdot \dfrac{M}{x^{*}} \rightarrow \dfrac{1}{P_{x}} \cdot \dfrac{M}{x^{*}}
\end{align*}\]
Elasticities - Example
\[
E_{x^{*},M} = \dfrac{\partial x^{*}}{\partial M} \cdot \dfrac{M}{x^{*}} \rightarrow \dfrac{1}{P_{x}} \cdot \dfrac{M}{x^{*}}
\]
3 - Substitute the original demand equation in and simplify
\[\begin{align*}
E_{x^{*},M} &= \dfrac{1}{P_{x}} \cdot \dfrac{M}{\color{red}{x}} = \dfrac{1}{P_{x}} \cdot \dfrac{M}{\color{red}{\dfrac{M}{P_{x}}}} = \dfrac{1}{P_{x}} \cdot P_{x} = \dfrac{P_{x}}{P_{x}} = 1
\end{align*}\]
So we would say that good \(x\) has an Income Elasticity of 1
Elasticities: What do They Mean?
They tell us how to translate a proportional change in \(x\) into a proportional change in \(y\)
So for the elasticity
\[E_{x,y} = \eta \]
\(y\) goes up by \(10\%\) , \(x\) increases by \(\eta \cdot 10\%\)
Equivalently, if \(y\) goes down by \(20\%\) , \(x\) decreases by \(\eta \cdot 20\%\)
Elasticities can be
Positive, negative, zero, or infinite
When considering Income Elasticities we know that
Elasticities: Responsiveness
We can describe how responsive goods are in terms of elasiticity
If the absolute value greater than 1 \((> 1)\) we say that there is an elastic
\(E_{x,M} = 2 \rightarrow M \uparrow 10\%, \; x \uparrow 20\%\) If the absolute value less than 1 \((<1)\) we say that there is an inelastic
\(E_{x,M} = 0.5 \rightarrow M \uparrow 10\%, \; x \uparrow 5\%\) If the absolute value equal to 1 \((=1)\) we say that the response is unit elastic
If the elastiscity is exactly equal to 0 we call this perfectly inelastic
If the elasticity is infinite \((\infty)\) we call this perfectly elastic
Price Changes are Important
Changing the price of a good and seeing what changes that causes is one of the most important types of changes we can consider
This is what people in the real world most want to know about
An important thing to have in mind throughout this section is:
If you hear/read “Demand Curve” or “Elasticity” without specifying if it’s about price or income, the default is Own-Price Elasticity
Price Changes
Let’s recall an important effect of price changes on the Budget Constraint
What happens to the Budget Constraint when then price of a normal good \(x\) falls?
What happens to the slope?
\[\dfrac{-P_{x}}{P_{y}} \; \rightarrow \; \downarrow P_{x} \; \rightarrow \; ???\]
Individual Demand
Similar to the Engel Curve, we can connect these optimal bundles to get an important curve
This is an Individual Demand Curve
Individual Demand Curve
If we clean it up a bit, we get our typical downward sloping Demand Curve
Notice I changed the axis slightly: I put Price on the vertical axis and Quantity of \(x\) demanded on the horizontal axis
Income Changes (Again)
If you recall from EC 201 we learned about demand shifters when income increases.
If we repeat this exercise with an increased income we can observe the Demand Shift
Individual Demand (Mathematically)
As we have done before, we also show the Demand Curve Properties through derivatives!
Let’s show that the Demand Curve is downward sloping
First, recall that the optimal consumption for a normal good \(y\) from a Cobb-Douglas Utility Function is:
\[
y^{*} = \dfrac{\beta}{\alpha + \beta} \cdot \dfrac{M}{P_{y}} = \dfrac{\beta M}{\alpha + \beta} \cdot P_{y}^{-1}
\]
Second, we take the derivative w.r.t. Price
\[
\dfrac{\partial y}{\partial P_{y}} = \dfrac{\beta M}{\alpha + \beta} \cdot (-1)P_{y}^{-2} = \dfrac{-\beta M}{\alpha + \beta} \cdot P_{y}^{-2} = \dfrac{-\beta M}{\alpha + \beta} \cdot \dfrac{1}{P_{y}^{2}} < 0
\]
C-D behaves as expected, the demand curve is downward sloping
Price Elasticity
This concept is identical to income elasticity
I will refer to the Elasticity of Demand as the elasticity of \(x\) with respect to \(P_{x}\)
The formula is
\[
E_{x^{*},P_{x}} = \dfrac{\partial x^{*}}{\partial P_{x}} \cdot \dfrac{P_{x}}{x^{*}}
\]
Elasticity of Demand - Example
Let
\[
E_{x^{*},P_{x}} = \dfrac{\partial x^{*}}{\partial P_{x}} \cdot \dfrac{P_{x}}{x^{*}} \;\; \text{where} \;\; x^{*} = \dfrac{M}{2P_{x}}
\]
Find \(\;\; \dfrac{\partial x^{*}}{\partial P_{x}}\)
\[
\dfrac{\partial x^{*}}{\partial P_{x}} = \dfrac{-M}{2P_{x}^{2}}
\]
Plug in our known values to the elasticity formula
\[
E_{x^{*},P_{x}} = \dfrac{-M}{2P_{x}^{2}} \cdot \dfrac{P_{x}}{\color{red}{x}} = \dfrac{-M}{2P_{x}^{2}} \cdot \dfrac{P_{x}}{\color{red}{\dfrac{M}{2P_{x}}}} = \dfrac{-P_{x}}{P_{x}} = -1
\]
Elasticity of Demand - Example 2
\[
y^{*} = \dfrac{20}{P_{y}} - 10
\]
Find \(\;\; \dfrac{\partial y^{*}}{\partial P_{y}}\)
\[
\dfrac{\partial y^{*}}{\partial P_{y}} = \dfrac{-20}{P_{y}^{2}}
\]
Plug in our known values to the elasticity formula
\[
E_{y^{*}, P_{y}} = \dfrac{-20}{20 - 10 P_{y}}
\]
Elasticity Results
\[
E_{y^{*}, P_{y}} = \dfrac{-20}{20 - 10 P_{y}}
\]
We will allow elasticities to look like functions, with some restrictions:
Income Change Example
Let’s put it all together. Given
\[
U(x,y) = 12 \cdot ln(x) + y
\]
Find \(x^{*}\) and \(y^{*}\)
Draw a graph on the \((x,y)\) plane with 3 ICs and the respective utility maximizing budget constraint
Draw Engel Curves for both \(x^{*}\) and \(y^{*}\)
Find the Function, I’ll guide us through the graphing
Use the derivatives with respect to income for both \(x^{*}\) and \(y^{*}\) and classify the type of good (Normal or Inferior)
Find the Elasticities of Demand with respect to income for both \(x^{*}\) and \(y^{*}\) and classify them (Elastic, Inelastic, etc)
Find \(x^{*}\) and \(y^{*}\)
\[
U(x,y) = 12 \cdot ln(x) + y \;\;\;\; \text{subject to} \;\;\;\; P_{x} \cdot x + P_{y} \cdot y = M
\]
Find MRS and set equal to Price Ratio
Plug in \(\; x^{*} \;\) into BC to find \(\; y^{*}\)
3 ICs and Their Respective Utility Max BCs
Engel Curve for \(x^{*}\)
Engel Curve for \(y^{*}\)
Classify Goods as Normal or Inferior
Elasticities of Demand w.r.t Income
Changes in Quantity Demanded
There are two major driving factors in changes in quantity when price changes:
Change that results from a change in the relative price of the two goods \(\rightarrow\) Substitution Effect
Change that results from a change in the purchasing power of the consumer’s income \(\rightarrow\) Income Effect
However, we cannot
We observed the combined
So we have:
\[
\text{Total Effect} = \text{Substitution Effect} + \text{Income Effect}
\]
Substitution and Income Effect
Substitution effect
It is the change in bundle resulting from a change in the relative price of two goods
The relative price of both goods is simply the slope of the Budget Constraint
Forcing No Change in Relative Income
Change in Relative Income
We can try to balance things out
The goal is to find where the consumer does not
Substitution Effect
We want to find where the consumer feels indifferent after price changes?
We need to find a bundle on the same Indifference Curve
Finding the Substitution Effect
Find the New Budget Constraint
Find the Hypothetical Budget Constraint
Find the Optimal Bundle under Original Utility and Hypothetical BC
Find the Difference between Hypothetical and Original Bundles
Finding the Substitution Effect - Step 1
Let’s say \(P_{y}\) decreased. Find the New Budget Constraint
Finding the Substitution Effect - Step 2
Find the Hypothetical Budget Constraint
We figuratively grab the new BC and shift it over
Finding the Substitution Effect - Step 3
Find the Optimal Bundle under Original Utility and Hypothetical
Finding the Substitution Effect - Step 4
Find the Difference between Hypothetical and Original Bundles
The Substitution Effect is the movement from \(\; A \;\) to \(\; \hat{A} \;\)
Substitution Effect - Example
Find the Substitution Effect of the following graph
Substitution Effect - Example Solution
Find the Substitution Effect of the following graph
Income Effect
It is the change in bundle resulting from a change in the purchasing power of the consumer’s income
The relative income (purchasing power) can change even when income does not
We want to find the change in consumption from an increase in purchasing power if the relative prices were always at the new ratio
Finding the Income Effect
Find the New Budget Constraint & the Optimal Bundle
Find the Hypothetical Budget Constraint
Find the Optimal Bundle under Original Utility and Hypothetical Budget Constraint
Find the Difference between Hypothetical and New Bundle
Luckily it is a lot of the same steps so it is not a lot of extra work
Finding the Income Effect - Step 1
Find the New Budget Constraint & the Optimal Bundle
Finding the Income Effect - Step 2
Find the Hypothethical Budget Constraint
Finding the Income Effect - Step 3
Find the Optimal Bundle under Original Utility and Hypothetical Budget Constraint
Finding the Income Effect - Step 4
Find the Difference Between Hypothetical and New Bundle
Substitution and Income Effect Summary
The Substitution Effect is the effect of changing the slope of the budget line without changing the indifference curve you started on
The Income Effect is the difference between the Substitution Effect and the new utility maximizing point
Total Effect
Remember: In the real world we can only observe the Total Effect
We try to decompose what drives people decisions and that’s how we arrive at the Sub & Inc Effects
Individual Demand \(\Rightarrow\) Market Demand
Up to Now
All the work we have done until now has been to understand and characterize where demand for a good comes from at an individual level
But now, we want to talk about how markets behave and where prices come from
To do so, we need a measure of Aggregate or Market Demand
Market Demand
We will begin by assuming the following demand function for ONE consumer in a market
\[
x^{*} = f(P_{x},M) = M - P_{x}
\]
If I said that this market was made up of 2 consumers with identical utility functions and incomes how can I get market demand?
Market Demand \((Q_{D})\) is the sum of all individual demands
\[
Q_{D} = x^{*} + x^{*} = 2x^{*} \; \rightarrow \; 2(M - P_{x}) = 2M - 2P_{x}
\]
Market Demand - Graphically
Individual Demand
\[\begin{align*}
x^{*} &= M - P_{x} \\
P_{x} &= M - x^{*}
\end{align*}\]
Market Demand
\[\begin{align*}
Q_{D} &= 2M - 2P_{x} \\
2P_{x} &= 2M - Q_{D} \\
P_{x} &= M - \dfrac{Q_{D}}{2}
\end{align*}\]
Let’s Make It More Complicated
We will call our 2 agents Jose and Maria
Now we will assume the same demand \(x^{*} = M - P_{x}\) but they have different incomes \((M)\) :
Jose has \(M = 6\)
Maria has \(M = 10\)
It is still the case that market demand is the sum of both individuals
\[
Q_{D} = x^{*}_{J} + x^{*}_{M} = 10 - P_{x} + 6 - P_{x} = 16 - 2P_{x}
\]
But there is an issue here!
What is it?
Once \(P_{x} \geq 6\) , Jose demands 0 and only Maria is left in the market
Complicated Market Demand
Individual Demands
\[\begin{align*}
x^{*}_{J} &= 6 - P_{x} \\
P_{x} &= 6 - x^{*}_{J}
\end{align*}\]
\[\begin{align*}
x^{*}_{M} &= 10 - P_{x} \\
P_{x} &= 10 - x^{*}_{M}
\end{align*}\]
Market Demand
\[\begin{align*}
Q_{D} &= 16 - 2P_{x} \\
2P_{x} &= 16 - Q_{D} \\
P_{x} &= 8 - \dfrac{Q_{D}}{2}
\end{align*}\]
Market Demand
Whenever you add up linear demand curves across individuals with different \(P_{x}\) intercepts in the demand curves, you will get a small kink in the market demand function
Note: The only linear demand curves we have are Perfect Substitutes and Quasi-linear so you should be able to identify them
Market Demand - Example 2
Let there be 2 agents with the following utilities
\[
U_{a} = ln(x) + y \;\;\;\; \& \;\;\;\; U_{b} = 2ln(x) + y \;\;\; ; \;\;\; P_{y} = 1
\]
Find the market demand curve (Hint: Find \(x_{a}^{*}\) and \(x_{b}^{*}\) )
\[\begin{align*}
x_{a}^{*} &\; : \; MRS = \dfrac{P_{x}}{P_{y}} \; \rightarrow \; \dfrac{MU_{x}}{MU_{y}} = \dfrac{P_{x}}{1} \; \rightarrow \; \dfrac{1}{x^{*}_{a}} = P_{x} \; \rightarrow \; x^{*}_{a} = \dfrac{1}{P_{x}} \\
x_{b}^{*} &\; : \; MRS = \dfrac{P_{x}}{P_{y}} \; \rightarrow \; \dfrac{MU_{x}}{MU_{y}} = \dfrac{P_{x}}{1} \; \rightarrow \; \dfrac{2}{x^{*}_{b}} = P_{x} \; \rightarrow \; x^{*}_{b} = \dfrac{2}{P_{x}}
\end{align*}\]
\[\begin{align*}
Q_{D} &\; : \; x^{*}_{a} + x^{*}_{b} = \dfrac{1}{P_{x}} + \dfrac{2}{P_{x}} = \dfrac{3}{P_{x}} \; \rightarrow \; Q_{D} = \dfrac{3}{P_{x}} \; \rightarrow \; P_{x} = \dfrac{3}{Q_{D}}
\end{align*}\]
Market Demand - Example 2 Graph
Individual Demand Functions
\[\begin{align*}
P_{x} = \dfrac{1}{x^{*}_{a}}
\end{align*}\]
\[\begin{align*}
P_{x} = \dfrac{2}{x^{*}_{b}}
\end{align*}\]
Market Demand Function
\[\begin{align*}
P_{x} = \dfrac{3}{Q_{D}}
\end{align*}\]
Consumer Theory
All we have seen up to now is the basics of Consumer Theory
A quick example of an application can be:
Let’s say you are interested in the demand for energy drinks
You build utility functions for a couple different groups in the market that you believe represent their preferences for energy drinks and other consumption
You solve the maximization problem and understand how the optimal solutions change when inputs change
Then you add up the solutions across groups in the market to measure the relationsihp between price and demand
