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Chapter 5: Problem 8

Show that the share of income spent on a good \(x\) is \(s_{x}=\frac{d \ln E}{d \ln p_{x}},\) where \(E\) is total expenditure

Short Answer

Expert verified
Question: Show that the share of income spent on a good x (denoted as \(s_x\)) can be represented as the derivative of the logarithm of total expenditure (\(\ln E\)) with respect to the logarithm of the price of the good (\(\ln p_x\)). Answer: \(s_x = \frac{d \ln E}{d \ln p_x}\)

Step by step solution

01

Find the share of income spent on good \(x\)

Given that \(x\) is the purchased quantity of good \(x\) and \(p_x\) is its price, the total expenditure on good \(x\) can be represented as \(x \times p_x\). To find the share of income spent on good \(x\), we can divide the total expenditure on good \(x\) by total expenditure \(E\): \(s_x = \frac{x \times p_x}{E}\).
02

Logarithmic differentiation

Now, we need to find the derivative of the logarithm of total expenditure (\(\ln E\)) with respect to the logarithm of the price of the good (\(\ln p_x\)). Using the chain rule, we have: \(\frac{d \ln E}{d \ln p_x} = \frac{d \ln E}{dE} \times \frac{dE}{dx \times p_x} \times \frac{dx \times p_x}{d p_x}\)
03

Differentiate \(\ln E\) with respect to \(E\)

Using the rules of differentiation for logarithmic functions, we have: \(\frac{d \ln E}{dE} = \frac{1}{E}\)
04

Differentiate \(E\) with respect to \(x \times p_x\)

In this case, we need to differentiate the total expenditure \(E\) with respect to the product \(x \times p_x\). Since the total expenditure is composed of different goods, the derivative of \(E\) with respect to \(x \times p_x\) will be equal to the share of the income spent on good \(x\). \(\frac{dE}{dx \times p_x} = s_x\)
05

Differentiate \(x \times p_x\) with respect to \(p_x\)

This is a simple differentiation using the product rule: \(\frac{dx \times p_x}{d p_x} = x + p_x \times \frac{dx}{d p_x}\) However, we see that good x is being bought, meaning for an increase in the price of good x (a small increase in \(dp_x\)), good x (x) will still be bought. Because of this, we know that the increase of good x (x) in response to the price change is zero: \(\frac{dx}{d p_x} = 0\) Therefore, the derivative simplifies to: \(\frac{dx \times p_x}{d p_x} = x\)
06

Assemble the final result

Now, let's plug the results from Steps 3-5 back into the formula from Step 2: \(\frac{d \ln E}{d \ln p_x} = \frac{1}{E} \times s_x \times x\) Since we know that \(s_x = \frac{x \times p_x}{E}\), we can make a substitution: \(\frac{d \ln E}{d \ln p_x} = \frac{1}{E} \times \frac{x \times p_x}{E} \times x\) This simplifies to: \(\frac{d \ln E}{d \ln p_x} = s_x\) Thus, we have successfully shown that \(s_x = \frac{d \ln E}{d \ln p_x}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Differentiation
Logarithmic differentiation is a mathematical technique used to simplify the process of finding the derivative of complicated functions. It involves taking the natural logarithm (ln) of both sides of an equation before differentiating. This is especially useful when dealing with products, quotients, or powers of variables.

When applying logarithmic differentiation, we make use of several properties of logarithms, such as:
  • The logarithm of a product is the sum of the logarithms: \(\text{If } y = uv, \text{ then } \ln(y) = \ln(u) + \ln(v)\).
  • The logarithm of a quotient is the difference of the logarithms: \(\text{If } y = \frac{u}{v}, \text{then} \ln(y) = \ln(u) - \ln(v)\).
  • The logarithm of a power is the exponent times the logarithm of the base: \(\text{If } y = u^v, \text{then } \ln(y) = v \cdot \ln(u)\).
In this exercise, logarithmic differentiation plays a crucial role in demonstrating how the share of income spent on a good relates to the price elasticity of demand. We take the natural logarithm of total expenditure (\(E\)) and differentiate it with respect to the natural logarithm of the price of the good (\(p_x\)), applying the chain rule to decompose the differentiation process into manageable steps.
Total Expenditure
Total expenditure is an economic term that describes the total amount of money spent by consumers on goods and services. It is a critical element in analyzing consumer behavior and in understanding the dynamics of the market economy. In this context, total expenditure on a particular good (\(x\)) is calculated as the product of the quantity purchased (\(x\)) and the price of that good (\(p_x\)):

\(E_x = x \times p_x\)

In microeconomic theory, the total expenditure is often used to examine how the amount spent on different goods and services changes in response to various factors, such as price changes. By analyzing the relationship between total expenditure and these factors, economists can derive insights into the income elasticity of demand, which measures how the quantity demanded of a good changes with consumer income. Understanding how total expenditure on a good interacts with its price is essential for determining the share of income consumers are willing to allocate to that good, a concept that is central to the given exercise.
Price Elasticity of Demand
Price elasticity of demand is a measure that economists use to understand how the quantity demanded of a good or service responds to a change in its price. More specifically, it reflects the percentage change in quantity demanded resulting from a one percent change in price. The formula for price elasticity of demand (\(E_d\)) is:

\(E_d = \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}}\)

When the value of price elasticity is greater than 1, the demand is considered elastic, meaning consumers are highly responsive to price changes. When it is less than 1, demand is inelastic, and consumers are less responsive to price changes. For the total expenditure method, if the product of price and quantity demanded (total expenditure) increases with a price increase, the good is said to have inelastic demand. Conversely, if total expenditure decreases with a price increase, demand is elastic.

This concept is intertwined with the exercise in question, as understanding how the share of income spent on a good changes with respect to its price provides insight into the good’s elasticity of demand. Using logarithmic differentiation, we've shown that the share of income spent on a good is proportional to the derivative of the natural logarithm of total expenditure with respect to the natural logarithm of the price, which is a fundamental aspect of price elasticity of demand analysis.

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Most popular questions from this chapter

Over a 3 -year period, an individual exhibits the following consumption behavior: $$\begin{array}{lcccc} & p_{x} & p_{y} & x & y \\ \hline \text {Year 1} & 3 & 3 & 7 & 4 \\ \text {Year 2} & 4 & 2 & 6 & 6 \\ \text {Year 3} & 5 & 1 & 7 & 3 \\ \hline \end{array}$$ Is this behavior consistent with the principles of revealed preference theory?

The general form for the expenditure function of the almost ideal demand system (AIDS) is given by \\[ \begin{aligned} \ln E\left(p_{1}, \ldots, p_{n}, U\right)=& a_{0}+\sum_{i=1}^{n} \alpha_{i} \ln p_{i}+\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \gamma_{i j} \ln p_{i} \ln p_{j} \\ &+U \beta_{0} \prod_{i=1}^{k} p_{k}^{\beta_{i}} \end{aligned} \\] For analytical ease, assume that the following restrictions apply: \\[ \gamma_{i j}=\gamma_{j i}, \quad \sum_{i=1}^{n} \alpha_{i}=1, \quad \text { and } \quad \sum_{j=1}^{n} \gamma_{i j}=\sum_{k=1}^{n} \beta_{k}=0 \\] a. Derive the AIDS functional form for a two-goods case. b. Given the previous restrictions, show that this expenditure function is homogeneous of degree 1 in all prices. This, along with the fact that this function resembles closely the actual data, makes it an "ideal" function. c. Using the fact that \(s_{x}=\frac{d \ln E}{d \ln p_{x}}\) (see Problem 5.8 ), calculate the income share of each of the two goods.

Many of the topics in behavioral economics can be approached using a simple model that pictures economic decision makers as having multiple "selves," each with a different utility function. Here we examine two versions of this model. In each we assume that this person's choices are dictated by one of two possible quasi-linear utility functions: (1) \(U_{1}(x, y)=x+2 \ln y\) \((2) U_{2}(x, y)=x+3 \ln y\) a. Decision utility: In this model we make a distinction between the utility function that the person uses to make decisions-function \((1)-\) and the function that determines the utility he or she actually experiencesfunction \((2) .\) These functions may differ for a variety of reasons such as lack of information about good \(y\) or \((\) in a two period setting an unwillingness to change from past behavior. Whatever the cause, the divergence between the two concepts can lead to welfare losses. To see this, assume that \(p_{x}=p_{y}=1\) and \(I=10\) i. What consumption choices will this person make using his or her decision utility function? ii. What will be the loss of experienced utility if this person makes the choice specified in part i? iii. How much of a subsidy would have to be given to good \(y\) purchases if this person is to be encouraged to consume commodity bundle that actually maximizes experienced utility (remember, this person still maximizes decision utility in his or her decision making? iv. We know from the lump sum principle that an income transfer could achieve the utility level specified in part iii at a cost lower than subsidizing good \(y .\) Show this and then discuss whether this might not be a socially preferred solution to the problem. b. Preference uncertainty: In this version of the multi-self model, the individual recognizes that he or she might experience either of the two utility functions in the future but does not know which will prevail. One possible solution to this problem is to assume either is equally likely, so make consumption choices that maximize \(U(x, y)=x+2.5 \ln y\) i. What commodity bundle will this person choose? ii. Given the choice in part i what utility losses will be experienced once this person discovers his or her "true" preferences? iii. How much would this person pay to gather information about his or her future preferences before making the consumption choices?

As defined in Chapter \(3,\) a utility function is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope: The \(M R S\) depends on the ratio \(y / x\) a. Prove that, in this case, \(\partial x / \partial I\) is constant. b. Prove that if an individual's tastes can be represented by a homothetic indifference map then price and quantity must move in opposite directions; that is, prove that Giffen's paradox cannot occur.

Thirsty Ed drinks only pure spring water, but he can purchase it in two different-sized containers: 0.75 liter and 2 liter. Because the water itself is identical, he regards these two "goods" as perfect substitutes. a. Assuming Ed's utility depends only on the quantity of water consumed and that the containers themselves yield no utility, express this utility function in terms of quantities of 0.75 -liter containers \((x)\) and 2 -liter containers \((y)\) b. State Ed's demand function for \(x\) in terms of \(p_{x}, p_{y},\) and \(I\) c. Graph the demand curve for \(x\), holding \(I\) and \(p_{y}\) constant. d. How do changes in \(I\) and \(p_{y}\) shift the demand curve for \(x ?\) e. What would the compensated demand curve for \(x\) look like in this situation?
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