Consumption Growth

Abstract

In this chapter, we intend to discuss the consumer behavior theory, the law of demand, the growth of consumption, and so on. We try to explain the consumer behavior theory based on the term “utility density.” If the utility density of different commodities can be compared, they must be the same kind of commodity. If not, they belong to different kinds.

The difference of utility density within the same kind of products, and the quality difference, forms the difference among commodities. In theory, the difference of commodity quality can be infinitely subdivided. It cannot be theoretically realized that every commodity with infinitely subdivided quality meets the needs of complete competition market. However, we can reduce the differences of the quality through effective combination, decrease the kinds of commodities with different qualities, and substitute the infinite levels of commodity quality differences which is theoretical for limited quality combinations, which can be realized in reality.

Quality difference results in monopoly, and the profit therein will lead the manufacturers to maintain and expand the advantages of monopoly. While the development of the production can raise people’s actual income level, consumers with raised income will consume more commodities with higher utility density (high quality), which makes the growth of consumption spontaneous.

Commodity with lower utility density (low quality) will see a downward trend in the consumption amount with increasing income. Even if its price is reduced, it cannot avoid the trend from a negative slope to a positive one of the demand curve. One of the manifestations of consumption increase is the change from the luxury products to the common ones, and from the common ones to the Giffen goods.

Annex

Hypothetical Model

• 1. Within a certain period of time, as one prolongs his/her consumption of certain product or service, the utility gained per unit of time decreases, such as decreasing marginal utility density.

• 2. For the same category of products, the higher the utility density, the higher the quality.

• 3. With time and budget constraints, consumers purchase various products with different qualities to realize maximized utility.

Variables

• Y: consumer income

• (i,j): product in category i and of quality j

• P _ij : the price of product (i,j)

• Q _ij : the quantity of product (i,j)

• T _ij : the time used to consume Q _ij

• ${\overline{P}}_{ij}$: the price to consume one unit of product (i,j)

• ${\overline{P}}_{ij}=\left({P}_{ij}\times {Q}_{ij}\right)/{T}_{ij}$

3.1.1 1. Optimal Equilibrium Model

Objective function: U = f(T ₁₁, T ₁₂, … T _MN )

$$ \mathrm{s}.\mathrm{t}.\kern3em Y={\displaystyle \sum_{i,j}{P}_{ij}\times {Q}_{ij}} $$

(A.1)

$$ \kern1.75em T={\displaystyle \sum_{ij}{T}_{ij}} $$

(A.2)

With regard to constraint (A.1), we have

$$ Y={\displaystyle \sum_{i,j}{P}_{ij}\times {Q}_{ij}}\iff Y={\displaystyle \sum_{i,j}{\overline{P}}_{ij}\times {T}_{ij}} $$

(A.3)

In the following, we consider equilibrium consumption under constraints (A.2) and (A.3).

• 1. Solution for consumer equilibrium condition under constraint (A.3)

  Applying the Lagrange multiplier method, we assume

  $$ L=f\left({T}_{11},{T}_{12},\dots {T}_{MN}\right)+\lambda \left(Y-{\displaystyle \sum_{ij}{\overline{P}}_{ij}\times {T}_{ij}}\right) $$
  $$ \frac{\partial L}{\partial {T}_{ij}}={f}_{ij}^{\prime}-\lambda {\overline{P}}_{ij}=0\Rightarrow \lambda {\overline{P}}_{ij}={f}_{ij}^{\prime}\overset{\Delta}{=}M{U}_{ij t} $$

  And to any two categories of products, we have

  $$ \frac{{\overline{P}}_{ij}}{{\overline{P}}_{mn}}=\frac{M{U}_{ij t}}{M{U}_{mn t}} $$

  (A.4)

• 2. Solution for consumer equilibrium condition under constraint (A.2)

  If $L=f\left({T}_{11},{T}_{12},\dots {T}_{MN}\right)+\lambda \left(T-{\displaystyle \sum_{ij}{T}_{ij}}\right)$

  $$ \frac{\partial L}{\partial {T}_{ij}}={f}_{ij}^{\prime}-\lambda =0\Rightarrow {f}_{ij}^{\prime}=\lambda $$

  Therefore,

  $$ \frac{M{U}_{ijt}}{M{U}_{mnt}}=1 $$

  (A.5)

• 3. Under constraints (A.2) and (A.3), the optimal consumer equilibrium solution is

  $$ \frac{{\overline{P}}_{ij}}{{\overline{P}}_{mn}}=\frac{M{U}_{ij t}}{M{U}_{mn t}}=1 $$

  (A.6)

3.1.2 2. Changes of Equilibrium

• 1. Optimal equilibrium in a market with indifferent products

  In a market with indifferent products, there is no quality difference among the products of the same category, such as

  $$ {\overline{P}}_{i1}={\overline{P}}_{i2}=\dots ={\overline{P}}_{iN}\overset{\Delta}{=}{\overline{P}}_i $$

  With optimal equilibrium condition (A.6), $\frac{{\overline{P}}_i}{{\overline{P}}_m}=\frac{M{U}_{it}}{M{U}_{mt}}=1$, which means ${\overline{P}}_i={\overline{P}}_m$

  And $\overline{P}=\left(P\times Q\right)/T=P\times V$

  Therefore, P _i × V _i = P _m × V _m

  $$ \frac{V_i}{V_m}=\frac{P_m}{P_i} $$

  (A.7)

  Equation (A.7) demonstrates that the time used to consume different products is the same for all consumers; so optimal equilibrium in a market with indifferent products is a chance phenomenon.

• 2. Optimal equilibrium in a market of differentiated products

  In a differentiated market, consumers can alter their choices of qualities by changing ${\overline{P}}_{ij}$. In this way, consumers could reach equilibrium condition (A.4) by choosing the appropriate qualities of products.

• 3. Demand Function

  Here we consider the demand functions for each category of products. Assume each category of products is consumed at a certain speed, and then the demand condition of each category equals the distribution of time on each category.

With the objective function and equilibrium condition (A.4) derived from constraint (A.3), the consumption of products in the same category and of different qualities meets:

$$ \frac{{\overline{P}}_{ij}}{{\overline{P}}_{ik}}=\frac{M{U}_{ij t}}{M{U}_{ik t}} $$

(A.8)

With income constraint (A.3), for the products of category i, it stands

$$ Y={\displaystyle \sum_{i,j}{\overline{P}}_{ij}}\times {T}_{ij}={\displaystyle \sum_{j\ne i}{\overline{P}}_{jl}}\times {T}_{jl}+{\displaystyle \sum_i{\overline{P}}_{ik}\times {T}_{ik}} $$

(A.9)

Assume $Y-{\displaystyle \sum_{j\ne i}{\overline{P}}_{jl}\times {T}_{jl}={Y}_i}$ is the consumer’s expenditure on the products of category i. Applying (A.8) and (A.9) and to product (i, m) we have

$$ {Y}_i={\displaystyle \sum_i{\overline{P}}_{ik}\times {T}_{ik}}={\overline{P}}_{im}\times {T}_{im}+{\displaystyle \sum_{k\ne m}{\overline{P}}_{ik}\times {T}_{ik}={\overline{P}}_{im}}\times {T}_{im}+{\displaystyle \sum_{k\ne m}{\overline{P}}_{im}}\frac{M{U}_{ik t}}{M{U}_{im t}}\times {T}_{ik} $$

And the demand for product (i, m) can be expressed as

$$ {T}_{im}=\frac{Y_i}{{\overline{P}}_{im}}-{\displaystyle \sum_{k\ne m}\frac{M{U}_{ik t}}{M{U}_{im t}}}\times {T}_{ik} $$

(A.10)

In order to specify different demand conditions for the products of different qualities, we assume that in terms of quality, (i, m) ≤ (i, n), where m ≤ n. In particular, (i, 1) has the worst quality.

To simplify, we use (i, 1) as a reference. A is the discrimination factor of other products within category i relative to (i, 1) in quality, and ${\overline{P}}_{im}=A{\overline{P}}_{i1}$.

Apparently A ≥ 1, and the larger A becomes, the higher the quality of product is. Demand function (A.10) can also be expressed as

$$ {T}_{im}=\frac{Y_i}{A{\overline{P}}_{i1}}-{\displaystyle \sum_{k\ne m}\frac{M{U}_{ik t}}{M{U}_{im t}}}\times {T}_{ik} $$

(A.11)

Next we discuss the demand function of products under different conditions.

• 1. If $T>{\displaystyle \sum_{ij}{T}_{ij}}$, when there is a change in the prices of products, consumers have adequate time to adjust their consumption on various products. So we have the following Eq. (A.10):

  $$ {P}_{im}\downarrow \to {\overline{P}}_{im}\downarrow \to {T}_{im}\uparrow \to {Q}_{im}\uparrow $$

• 2. If $T={\displaystyle \sum_{ij}{T}_{ij}}$, then consumers lack time. When the prices change, consumers can only reach equilibrium by adjusting consumption portfolio, not the time of consumption. Here we discuss the impact on the consumption of the price changes of products with different qualities:

  • (a) A = 1, P _i1 ↓ → A ↑ → Q _i1 ↓; for instance, the demand for product of the lowest quality drops with price.

  • (b) $A>1,\kern1em {\overline{P}}_{im}\downarrow \to A\uparrow$; for example, the demand for higher quality products rises as prices go down, the demand for relatively low-quality products falls, and the demand for medium-quality products remains unchanged. The demand for the highest-quality product will monotonically increase.