Demand [Micro Economics]

Micro Economics 

- Chapter 6. Demand

This time we will cover the topic of Demand functions and how the ordinary demands for goods change with in prices and income changes. The lecture will began with an overview of the plan to conduct comparative statics analysis on ordinary demand functions by varying prices and income levels.
The concepts of price-offer-curves, income-offer-curves, and Engel curves will be introduce to analyze the effects of price and income changes on quantity demanded. The analysis focused on studying the behavior of ordinary demand functions under different Utility preferences, including (1)Cobb-Douglas, (2)perfect complements, (3)perfect substitutes, and (4)quasi-linear preferences.
For each preference type, the ordinary demand functions, price offer curves, income offer curves, and Engel curves were derived and analyzed. Based on the analysis, goods were categorized as either Ordinary or Giffen based on the relationship between their own price & quantity demanded.
Similarly, goods were classified as either Normal or Inferior based on the relationship between income & quantity demanded. The cross-price effects were also discussed, defining goods as either gross substitutes or gross complements based on how the price of one good affects the quantity demanded of the other.
The discussion covered various examples and graphical representations to illustrate the concepts. Analytical expressions and equations were provided for the ordinary demand functions, inverse demand functions, price offer curves, income offer curves, and Engel curves under different utility preferences.

*conduct (n.)행동 (v.)수행하다

Overview

1. Comparative Statics Analysis 

The lecture began with an overview of the plan to conduct comparative statics analysis on ordinary demand functions. The main question to be addressed was how the ordinary demands for goods change as prices and income change. The exogenous variables in the model were identified as prices and income, while the endogenous variables were the budget constraint and the resulting budget set. The preferences were represented by given utility functions. The plan was to fix some exogenous parameters and vary one parameter at a time to study the changes in ordinary demands.

2. Own Price Changes and Price Offer Curves 

The analysis started with studying the effects of changes in the own price of a good on its quantity demanded, holding other prices and income constant. The concept of a price offer curve was introduced, which traces the utility-maximizing bundles as the price of a good changes while other exogenous variables remain constant. The ordinary demand curve was derived from the price offer curve by plotting the quantity demanded against the price. The analysis was conducted for different utility preferences, including Cobb-Douglas, perfect complements, and perfect substitutes. The behavior of price offer curves and ordinary demand curves was examined for each preference type.

3. Income Changes, Income Offer Curves, and Engel Curves 

The discussion then shifted to analyzing the effects of income changes on quantity demanded, holding prices constant. The concept of an income offer curve was introduced, which traces the utility-maximizing bundles as income changes while prices remain fixed. Engel curves were defined as plots of quantity demanded against income. The analysis was conducted for different utility preferences, and the behavior of income offer curves and Engel curves was examined. Goods were classified as either normal or inferior based on the relationship between income and quantity demanded.

4. Categorization of Goods: Ordinary vs. Giffen and Normal vs. Inferior 

Based on the analysis of price and income changes, goods were categorized into different types. A good was considered ordinary if its quantity demanded increased as its own price decreased, and Giffen if its quantity demanded increased as its own price increased. Goods were classified as normal if their quantity demanded increased with income, and inferior if their quantity demanded decreased with income. A summary table was provided to illustrate the categorization of goods based on the relationships between price, quantity demanded, and income.

5. Cross-Price Effects and Gross Substitutes vs. Gross Complements 

The discussion concluded with an analysis of cross-price effects, which examine how the price of one good affects the quantity demanded of another good. If an increase in the price of one good leads to an increase in the quantity demanded of another good, the two goods are considered gross substitutes. Conversely, if an increase in the price of one good leads to a decrease in the quantity demanded of another good, the two goods are considered gross complements. Examples and analytical expressions were provided to illustrate the cross-price effects for different utility preferences.

Properties of Demand Functions

The consumer’s demand functions give the optimal amounts of each of the goods as a function of the prices and income faced by the consumer. We write the demand functions as

x₁ =x₁(p₁,p₂,m) 

x₂ =x₂(p₁,p₂,m)

In this chapter we will examine how the demand for a good changes as prices and income change. Studying how a choice responds to changes in the economic environment is known as comparative statics, which we first described in Chapter 1. “Comparative” means that we want to compare two situations: before and after the change in the economic environment.

Which means, Comparative statics analysis of ordinary demand functions 

= the study of how ordinary demands x₁*(p₁,p₂,y) and x₂*(p₁,p₂,y) change as prices p₁, p₂ and income y change.

** p₁,p₂ mean price of commodity 1,2. m means maximum budget, y means preference.

6.4 Ordinary Goods and Giffen Goods 

Let us now consider price changes. Suppose that we decrease the price of good 1 and hold the price of good 2 and money income fixed. Then what can happen to the quantity demanded of good 1? Intuition tells us that the quantity demanded of good 1 should increase when its price decreases. Indeed this is the ordinary case, as depicted below.

6.5 The Price Offer Curve and the Demand Curve 

Own-Price Changes (→ Price Offer Curve)

How does x₁*(p₁,p₂,y) change as p1 changes, holding p₂ and y constant?

: Suppose only p₁ increases, from p₁' to p₁'' and then to p₁'''.

: x₁ goes left, when price of x₁ gets higher.

Then, we would arrange p₁=p₁' case & p₁=p₁'' case & p₁=p₁''' case. Then picture will looks like this ↓

The upon 2 graphs are The Price offer curve(left) & Demand curve(right).
left Panel contains a price offer curve, which depicts the optimal choices as the price of good 1 changes. right Panel contains the associated demand curve, which depicts a plot of the optimal choice of good 1 as a function of its price.

Suppose that we let the price of good 1 change while we hold p₂ and income fixed. Geometrically this involves pivoting the budget line. We can think of connecting together the optimal points to construct the price offer curve as illustrated in Figure left side. This curve represents the bundles that would be demanded at different prices for good 1.

The Price offer curve depicts as insufficiently, so what its complicated looks likes as for linking the dots {x₁*(p₁') → x₁*(p₁'') → x₁*(p₁'''') IN THE Left side of Figure.

So, The curve containing all the utility- maximizing bundles traced out as p₁ changes, with p₂ and y constant, is the p₁- price offer curve. The plot of the x₁-coordinate of the p₁- price offer curve against p₁ is the ordinary demand curve for commodity 1.

* coordinate 조정하다

We can depict this same information in a different way. Again, hold the price of good 2 and money income fixed, and for each different value of p₁ plot the optimal level of consumption of good 1. The result is the demand curve depicted in Figure right side. The demand curve is a plot of the demand function, x₁ (p₁ , p₂ , m), holding p₂ and m fixed at some predetermined values.

(1) Then new Question arise. 
What does a p₁ price-offer curve look like for Cobb-Douglas preferences?

Take U(x₁,x₂)=x₁ªx₂ᵇ , Then the ordinary demand functions for commodities 1 and 2 are

x₁*(p₁,p₂,y) = a/(a+b) × y/p₁
and
x₂*(p₁,p₂,y) = b/(a+b) × y/p₂

Notice that x₂* does not vary with p₁ so the p₁ price offer curve is flat.

p₁- price offer curve & demand curve

(2) Second Question. 

What does a p₁ price-offer curve look like for a perfect-complements utility function?

U(x₁,x₂) = min{x₁,x₂}

Then the ordinary demand functions for commodities 1 and 2 are
x₁*(p₁,p₂,y) = x₂*(p₁,p₂,y) = y / p₁ + p₂

With p₂ and y fixed, higher p₁ causes smaller x₁* and x₂*
As p₁ → 0, x₁* = x₂* → y / p₂
As p₁ → ∞, x₁* = x₂* → 0

p₁- price offer curve & demand curve

(3) Third Question,
What does a p₁ price-offer curve look like for a perfect-substitutes utility function?

U(x₁,x₂) = x₁ + x₂

Then the ordinary demand functions for commodities 1 and 2 are

x₁*(p₁,p₂,y) = 0 if p₁ > p₂  //  y/p₁ if p₁ < p₂

x₂*(p₁,p₂,y) = 0 if p₁ < p₂  //  y/p₂ if p₁ > p₂

p₁- price offer curve & demand curve

Own-Price Changes (→ Ordinary & Giffen Goods) 

Usually we ask “Given the price for commodity 1 what is the quantity demanded of commodity 1?"
But we could also ask the inverse question “At what price for commodity 1 would a given quantity of commodity 1 be demanded?"

For example,
Given p₁’, what quantity is demanded of commodity 1? Answer: x₁’ units
The inverse question is: Given x₁’ units are demanded, what is the price of commodity 1? Answer: p₁’

x₁*(p₁,p₂,y)         → (reverse) →　   (x₁*)⁻¹ = P(x₁*)

demand to quantity        //                quantity to demand

Exampling)

A Cobb-Douglas example Case)

x₁*(p₁,p₂,y) = ay / (a+b)p₁
is the ordinary demand function and

p₁ = ay / (a+b)x₁*
is the inverse demand function.

A perfect-complements example Case)

x₁* = y / p₁ + p₂
is the ordinary demand function and

p₁ = y/x₁* - p₂
is the inverse demand function.

6.2 Income Offer Curves and Engel Curves

Income Changes (→ Income Offer and Engel Curves) 

How does the value of x₁*(p₁,p₂,y) change as y changes, holding both p₁ and p₂ constant?

We have seen that an increase in income corresponds to shifting the budget line outward in a parallel manner. We can connect together the demanded bundles that we get as we shift the budget line outward to construct the income offer curve. This curve illustrates the bundles of goods that are demanded at the different levels of income, as depicted in Figure below. The income offer curve is also known as the income expansion path. If both goods are normal goods, then the income expansion path will have a positive slope, as depicted in Figure below.

A plot of quantity demanded against income is called an Engel curve.

Exampling)

(1) Income Changes and Cobb Douglas Preferences 

An example of computing the equations of Engel curves; the Cobb-Douglas case.
U(x₁,x₂)=x₁ªx₂ᵇ 

Then the ordinary demand functions for commodities 1 and 2 are

x₁* = ay / (a+b)p₁
and
x₂* = ay / (a+b)p₂

Rearranged to isolate y, these are:

y = (a+b)p₁x₁* / a　(Engel curve for good 1)
and
y = (a+b)p₂x₂* / b　(Engel curve for good 2)

(2) Income Changes and Perfectly Complementary Preferences

Another example of computing the equations of Engel curves; the perfectly- complementary case.

U(x₁,x₂) = min{x₁,x₂}

The ordinary demand equations are
x₁* = x₂* = y / (p₁ + p₂)

Rearranged to isolate y, these are:
y = (p₁ + p₂) x₁*        (Engel curve for good 1)
y = (p₁ + p₂) x₂*        (Engel curve for good 2)

(3) Income Changes and Perfectly Substitutable Preferences

Another example of computing the equations of Engel curves; the perfectly- substitution case.

U(x₁,x₂) = x₁ + x₂

The ordinary demand equations are

x₁*(p₁,p₂,y) = 0 if p₁ > p₂  //  y/p₁ if p₁ < p₂

x₂*(p₁,p₂,y) = 0 if p₁ < p₂  //  y/p₂ if p₁ > p₂

Suppose p₁ < p₂. Then x₁*=y/p₁ & x₂*=0
Suppose p₁ > p₂. Then x₁*=0 & x₂*=y/p₂

(4) Income Changes: Quasilinear Utility

In every example so far the Engel curves have all been straight lines. Is this true in general?
No.

Another kind of preferences that generates a special form of income offer curves and Engel curves is the case of quasilinear preferences. Recall the definition of quasilinear preferences given in Chapter 4. This is the case where all indifference curves are “shifted” versions of one indifference curve.

Traits :
(1) Each curve is a vertically shifted copy of the others.
(2) Each curve intersects both axes.

Quasilinear preferences:
U(x₁,x₂) = f(x₁) + x₂

For example, 
U(x₁,x₂) = √x₁ + x₂

right side Panel is Engel curve for good 1

upon Figure is the Engel curve for good 2

6.1 Normal and Inferior Goods 

Income Changes (→ Normal & Inferior Goods) 

We would normally think that the demand for each good would increase when income increases, as shown in Figure 6.1. Economists, with a singular lack of imagination, call such goods normal goods. If good 1 is a normal good, then the demand for it increases when income increases, and de- creases when income decreases. 

If something is called normal, you can be sure that there must be a possibility of being abnormal. And indeed there is. Figure 6.2 presents an example of nice, well-behaved indifference curves where an increase of income results in a reduction in the consumption of one of the goods. Such a good is called an inferior good. This may be “abnormal,” but when you think about it, inferior goods aren’t all that unusual. There are many goods for which demand decreases as income increases; examples might include gruel, bologna, shacks, or nearly any kind of low-quality good.

For a normal good the quantity demanded always changes in the same way as income changes:

Δx₁ / Δm  > 0

A good for which quantity demanded rises with income is called normal. Therefore a normal good’s Engel curve is positively sloped( + ).

A good for which quantity demanded falls as income increases is called income inferior. Therefore an income inferior good’s Engel curve is negatively sloped( - ).

Income Changes (→ Good 2 Is Normal, Good 1 becomes Income Inferior)

In the Engel curve for good 2(normal), if x₂* increase than y(preference) also increase.

But, In the Engel curve for good 1(inferior), if x₁* increase than y(preference) but, It cannot move after some point. 

@6.4 Ordinary Goods and Giffen Goods

A good is called ordinary if the quantity demanded of it always increases as its own price decreases.
Ordinary good: Ordinarily, the demand for a good increases when its price decreases, as is the case here.

Downward-sloping demand curve → Good 1 is ordinary

It is logically possible to find well- behaved preferences for which a decrease in the price of good 1 leads to a reduction in the demand for good 1. Such a good is called a Giffen good, after the nineteenth-century economist who first noted the possibility. An example is illustrated in Figure below.

If, for some values of its own price, the quantity demanded of a good rises as its own-price increases then the good is called Giffen. 
Giffen good: since the demand for it decreases when its price decreases.

Demand curve has a positively sloped part → Good 1 is giffen

Income Changes (→ Cross-Price Effects)

If an increase in p₂ increases demand for commodity 1 then commodity 1 is a gross substitute for commodity 2.
If an increase in p₂ reduces demand for commodity 1 then commodity 1 is a gross complement for commodity 2.

Example)
A problem set as "perfect-complements" case:

x₁* = y / (p₁ + p₂)

∂x₁*/∂p₂ = - y / (p₁ + p₂)² < 0

Therefore commodity 2 is a gross complement for commodity 1.

**So we could know that 

∂x₁*/∂p₁ > 0 then Giffen

∂x₁*/∂p₁ < 0 then Ordinary

∂x₁*/∂y > 0 then Normal

∂x₁*/∂y < 0 then Inferior

	∂p₁	∂y
∂x₁*	> 0, Giffen	> 0, Normal
∂x₁*	< 0, Ordinary	< 0, Inferior

Summary

1. The consumer’s demand function for a good will in general depend on the prices of all goods and income.

2. A normal good is one for which the demand increases when income increases. An inferior good is one for which the demand decreases when income increases.

3. An ordinary good is one for which the demand decreases when its price increases. A Giffen good is one for which the demand increases when its price increases.

4. If the demand for good 1 increases when the price of good 2 increases, then good 1 is a substitute for good 2. If the demand for good 1 decreases in this situation, then it is a complement for good 2.

5. The inverse demand function measures the price at which a given quan- tity will be demanded. The height of the demand curve at a given level of consumption measures the marginal willingness to pay for an additional unit of the good at that consumption level.

[Reference]

[1] Hal R. Varian - Intermediate Microeconomics_ A Modern Approach, 8th Edition  -W.W. Norton & Co. (2010)