Lecture 1 - Microeconomics

ENCI707: Engineering Demand and Policy Analysis

Outline

1. Microeconomic theory of demand
2. Quantitative analysis of consumer demand
   1. Concept of utility
   2. Direct/indirect utility
   3. Demand function
   4. Functional forms of demand function

Microeconomic Theory of Demand

Microeconomics

• Branch of economics dealing with behaviour of economic agents including:​
  • Consumers​
  • Labour force (workers)​
  • Firms​
  • Investors​
  • Markets: Interactions among all of the above​
• Demand-Supply:​
  • Demand function representing behavior of users/agents​
  • Supply function representing level-of-service, congestion, & behavior of service providers​
  • Market clearance = demand & supply equilibrium​

Demand-Supply Relationship

• Short-term demand-supply: Ex. Roadway link at an instant​
• Long-term demand-supply: Ex. Residential housing​
• Equilibrium

Shifting Curves

Comparative Statics

• Create a model of market behavior:​
  • Explain consumer & firm choices as functions of exogenous variables – e.g., income & government policy​
• Develop scenarios​
  • Changes in exogeneous variables​
• Derive changes in the endogenous variables

Comparative Statics Example​

• The market for taxi service:​
  • Supply function: \(Q_S\) = – 125 + 125P​
  • Demand function: \(Q_D\) = 1000 – 100P​
  • Where does the market clear?​
• What happens if demand shifts such that now \(Q_D\) = 1450 – 100P?​

Comparative Statics Example - SOLUTION

Quantitative Analysis of Consumer Demand

Concept of Utility​

• Utility: a measure of benefit or attractiveness of alternative courses of action​
  • Assumption that people choose action that maximizes utility (profit) and minimizes disutility (cost)​
• An abstract/latent concept that gives only ordinal ranking:​
  • No inherent meaning​
  • No generic unique function​
  • Unaffected by monotonic transformation​
• Utility measures:​
  • Direct utility: f(quantity of consumption)​
  • Indirect utility: f(utility of consumption)​

Consumption Set: Bundle of Choices​

• Derive demand function based on concept of utility maximization subject to budget and constraints​
• Consumption set [X]: a possible bundle of choices (goods: real or virtual)​
• Consumer has preferences over the consumption set​
• Properties of a consumption bundle (of the mathematical function defining bundle):​
  • Complete, reflexive, transitive, continuous, and convex​

Properties of a Consumption Bundle​

• Completeness: \(A \succsim B \vee A \precsim B\) (weak preference) - You have preferences
• Reflexivity: \(A \sim A\) - You are indifferent between copies of the same product
• Transitivity: \(A \succsim B \wedge B \succsim C -> A \succsim C\) - If you prefer A over B and B over C, then you must prefer A over callout
• Continuity: Preferences are continuous and can be represented by continuous utility functions
• Convex: For x, y, z \(\in\) X where \(y \succsim x\) and \(z \succsim x\) \(\forall \theta \in [0,1]\)

\(\theta y + (1-\theta)z \succsim x\) - For any two bundles that are each viewed as at least as good as a third bundle, a weighted average of the two bundles is viewed as being at least as good as the third bundle

Utility Function & Consumption Set

• Utility Function: Summary of a consumer’s behaviour by means of a mathemnatical function

  • A continuous function

  \[f(U(x)) \ge f(U(y)) \text{ iff } U(x) \ge U(y)\]

  • Utility is better presented in ordinal sense than in cardinal numbers

Utility Function: Diminishing Marginal Utility of Consumption

Consumption Demand​

• Consumers are utility maximizers: Consumers maximize utility subject to budget/income constraints​
  • Facing prices (p) of goods/services (consumption) and budget constraints (I=income)​
  • Consumer allocates income across goods/services to maximize utility (U)​
• Marshallian (Uncompensated) Demand: Demand for items that maximizes the utility of consumption subject to a constrained budget​
• Hicksian (Compensated) Demand: Demand for items that minimizes expenditure to attain a certain (fixed) level of consumption utility

Demand Functions​

• Functions explaining quantity demanded as function of price, income (or total available budget), utility, etc.​
  • Marshallian Demand Function: Utility maximization subject to optimal allocation of budget to consumption of goods​
    • Uses direct utility function​
    • Concept of production function (for firms)​
  • Hicksian Demand Function: Expenditure minimization subject to optimal level of utility of consumption bundle:​
    • Uses indirect utility function​
    • Concept of expenditure function (for firms)

Indirect Utility and Demand​

\[V(\mathbf{p},I) = \max U(\mathbf{x}) \text{ such that } \mathbf{px} = I\]

• Indirect Utility Function: The function \(V(\mathbf{p},I)\) that gives the maximum utility achievable at a given price vector (\(\mathbf{p}\)) and income \(I\) is called the indirect utility function
• Consumption Bundle: The value of \(\mathbf{x}\) that solves the maximization problem is called the demanded bundle
• Demand Function: The function that connects \(\mathbf{p}\) and \(I\) is called the demand function

Indirect Utility Function​

• Often useful to consider utility obtained by a consumer as an indirect measure - as a function of prices and income rather than actual consumed quantities
• Assume that \(V(p_1,p_2,I)\) is our indirect utility (value) function

\[ \begin{align*} V(p_1, p_2, I) &= U(x_1^*, x_2^*) \\ &= U(x_1(p_1, p_2, I), x_2(p_1, p_2, I)) \end{align*} \]

Marshallian Demand Function​

• When the indifference curves are strictly convex, the solution is unique

\[x_1^* = x_1(p_1, p_2, I)\]
\[x_2^* = x_2(p_1, p_2, I)\]

• If quantity demanded is expressed as a function of prices and income (or total available budget), called Marshallian demand function

Definition of Goods​

Ordinary Good

\(\frac{dx_1(p_1,p_2,I)}{dp_1} < 0\)

Giffen Good

\(\frac{dx_1(p_1,p_2,I)}{dp_1} > 0\)

Complementary Goods

\(\frac{dx_2(p_1,p_2,I)}{dp_1} < 0\)

Substitutional Goods

\(\frac{dx_2(p_1,p_2,I)}{dp_1} > 0\)

Definition of Goods

• Income elasticity of demand

\(\eta = \frac{dx_1}{dI}\frac{I}{x_1}\)

\(\frac{dx_1(p_1,p_2,I)}{dI} < 0\) -> Inferior Good \(\eta < 0\)

\(\frac{dx_1(p_1,p_2,I)}{dI} > 0\) -> Normal Good \(\eta > 0\)

\(\frac{dx_1(p_1,p_2,I)}{dI} > \frac{x_1}{I}\) -> Superior Good \(\eta > 1\)

Hicksian Demand Function ​

• When indifference curves are strictly convex, the solution is unique where

\[x_1^* = x_1(p_1,p_2,U) \text{ and } x_2^* = x_2(p_1,p_2,U)\]

• If quantity demanded is expressed as a function of prices and utility, called Hicksian demand function
• Hicksian demand tells us what consumption bundle achieves a target level of utility and minimizes total expenditure
  • As it is based on indirect utility, Hicksian demand is not directly observable

Marshallian & Hicksian Demand​

Can use Slutsky equation to obtain an analytic result

\[\frac{\partial x_1}{\partial p_2} = \underset{\text{Substitution Effect}}{\frac{\partial h_1(p_1,p_2,u)}{\partial p_2}} - \underset{\text{Income Effect}}{\frac{\partial x_1}{\partial I}x_2(p_1,p_2,I)}\]

where

• \(h_1(p_1,p_2,U)\) is the Hicksian demand for good 1
• \(U = v(p_1,p_2,I)\) is the desired indirect utility level
• \(x_1(p_1,p_2,I)\) is the Marshallian demand for good 1 (similar for good 2)

Indifference Curves​

• Let \(U(x_1,x_2,\dots)\) be a twice differentiable utility function. If we increase the quantity of goods, how does the consumer respond?
  • Indiference curve: shows combination of commodities for which total utility is constant
  • Changes in total utility are zero for any change in the consumption bundle on the same indifference curve

Marginal Rate of Substitution (MRS)​

Direct Utility Maximization​

A utility function that is a function of quantity consumed (i.e., indifference curves)

\[x^*(\mathbf{p},I)=\max U(x_1,x_2)\]

\[\text{Subject to } p_1x_1+p_2x_2 \le I\]

Using Lagrangian

\[L = U(x_1,x_2) + \lambda(I - p_1x_1 - p_2x_2)\]

First order conditions for optimum consumption

\[\frac{\partial L}{\partial x_1} = 0 \text{ and } \frac{\partial L}{\partial x_2} = 0\]

\[\frac{\partial U(x_1,x_2)}{\partial x_1} = \lambda p_1 \text{ and } \frac{\partial U(x_1,x_2)}{\partial x_2} = \lambda p_2\]

Direct Utility Maximization​

Langrangian Multipler (\(\lambda\)) can be eliminated by taking a ratio of partial derivatives

\[\frac{\frac{\partial U(x_1,x_2)}{\partial x_2}}{\frac{\partial U(x_1,x_2)}{\partial x_1}} = \frac{\lambda p_2}{\lambda p_1} = \frac{p_2}{p_1}\]

• Ratio of price = Marginal Rate of Substitution (MRS) = slope of indifference curve
• \(p_1x_1 + p_2x_2 = I\) is the budget line
• Point of tangency between indifference curve and budget line = point of optimal consumption

Envelope Theorem​

• Utility (value) function forms an upper envelope of the graphs of the parameterized family of function \({U(x,\cdot)}_{x \in X}\)
• Given an exogenous parameter (\(a\)) that affects \(U(x_1,x_2,a)\), we can obtain the following useful relationship

\[\frac{\partial U(x_1,x_2,a)}{\partial a} = \frac{\partial L(x_1^*,x_2^*,a)}{\partial a}\]

Marginal Utility of Income (Application of Envelope Theorem)​

• Income is an exogenous input from the perspective of optimization

\[\frac{\partial L(x_1^*,x_2^*)}{\partial I} = \frac{\partial U(x_1^*,x_2^*)}{\partial I} = \lambda\]

• \(\lambda\) is the marginal utility of income

Indirect Utility Function​

• Indirect utility optimization is the dual of primal direct utility optimization (expenditure function) - minimize expenditure to maintain utility \(\bar{U}\) \[V(p_1,p_2,I) = \min_{p_1,p_2} p_1x_1 + p_2x_2 - \lambda (\bar{U} - U(x_1,x_2))\] \[\text{s.t. } U(x_1,x_2) \ge \bar{U}\]

• Equivalently, by duality (indirect utility function) - maximize utility at prices \(p_1,p_2\) \[V(p_1,p_2,I) = \max_{p_1,p_2} U(x_1^*,x_2^*) + \lambda (I - p_1x_1 + p_2x_2)\] \[\text{s.t. } p_1x_1 + p_2x_2 \le I\]

Properties of Indirect Utility Function​

• Property 1: \(V(\mathbf{p},I)\) is non-increasing in prices \(\mathbf{p}\) and non-decreasing in income \(I\)
  • Any increase in prices or decrease in income contracts the affordable set of commodities - nothing new is available to the consumer, so utility cannot increase
• Property 2: \(V(\mathbf{p},I)\) is homogeneous degree zero
  • No change in the affordable set or preferences if both prices and income are increased or decreased by a constant

Roy’s Identity (Another Application of Envelope Theorem)​

• Connects Marshallian demand function and indirect utility function
• Similarly, Sheppard’s lemma connects Hicksian demand function and expenditure function

Roy’s Identity (Another Application of Envelope Theorem)​

\[V(p_1,p_2,I) = U(x_1^*,x_2^*) + \lambda (I - p_1x_1 - p_2x_2)\] \[\frac{\partial V(p_1,p_2,I)}{\partial p_1} = - \lambda x_1(p_1,p_2,I)\] \[\frac{\partial V(p_1,p_2,I)}{\partial I} = \lambda\] \[x_1(p_1,p_2,I) = - \frac{\frac{\partial V(p_1,p_2,I)}{\partial p_1}}{\frac{\partial V(p_1,p_2,I)}{\partial I}} \text{(direct utility from indirect utility)}\]

Continuous Demand​

Functional Forms for Demand/Production/Expenditure​

• Cobb-Douglas function​
• Transcendental logarithm (translog) function​
• Constant elasticity of substitution (CES) function​
• Almost ideal demand system (AIDS)

Cobb-Douglas Direct Utility Function​

• Simple example \[\max U(x_1,x_2) = x_1^{\alpha}x_2^{\beta}\] \[\text{s.t. } p_1x_1+p_2x_2 \le I\]

• Langrangian \(L = x_1^{\alpha}x_2^{\beta} + \lambda(I - p_1x_1 - p_2x_2)\)

Cobb-Douglas Direct Utility Function​

• Applying first order conditions (FOC) for optimal utility \[\frac{\partial L}{\partial x_1} = 0 \text{->} \alpha x_1^{\alpha - 1}x_2^{\beta} = \lambda p_1\] \[\frac{\alpha U}{x_1} - \lambda p_1 = 0 \therefore U = \frac{\lambda p_1x_1}{\alpha}\] Similarly, \[U = \frac{\lambda p_2x_2}{\beta} \therefore x_1 = x_2\frac{\alpha p_2}{\beta p_1}\]

Cobb-Douglas Direct Utility Function​

• Translating \(x_1\) and \(x_2\) into the budget equation, the optimal quantity demands (utility maximizing) are \[x_1^*= \frac{I}{p_1}\left(\frac{\alpha}{\alpha + \beta}\right) \text{ and } x_2^*= \frac{I}{p_2}\left(\frac{\beta}{\alpha + \beta}\right)\]

• With Cobb-Douglas utility, the consumer spends a fixed proportion of income/budget on each commodity

Cobb-Douglas Direct Utility Function​

• Marginal rate of substitution (MRS) \[\frac{\frac{\partial U(x_1,x_2)}{\partial x_1}}{\frac{\partial U(x_1,x_2)}{\partial x_2}} = - \frac{x_2}{p_1}\frac{\alpha}{\beta} = - \frac{p_1}{p_2}\]

Elasticity of Substitution​

• MRS: Measures the slope of the indifference (isoquant) curve

• Elasticity of substitution (\(\sigma\)): Measures the curvature of the indifference (isoquant) curve \[\sigma = \frac{\Delta \left(\frac{x_2}{x_1} \right)/\frac{x_2}{x_1}}{\Delta MRS/MRS} = \frac{\partial \left(\frac{x_2}{x_1} \right)}{\partial MRS}\frac{MRS}{\left(x_2/x_1\right)}\] \[\frac{\partial \ln\left(\frac{x_2}{x_1}\right)}{\partial ln (|MRS|)} \text{(think dln(x)/dx = 1/x and re-arrange)}\]

• Absolute value of MRS to ensure logarithm of non-negative values

Cobb-Douglas Utility Function​

• Elasticity of substitution (\(\sigma\)) \[|MRS| = \frac{x_2}{x_1}\frac{\alpha}{\beta}\] \[\frac{x_2}{x_1} = \frac{\beta}{\alpha}|MRS|\] \[ln\left(\frac{x_2}{x_1}\right) + \ln\left(\frac{\alpha}{\beta}\right) = \ln(|MRS|) \text{take derivatives wrt ln(MRS)}\] \[\sigma = d\ln\left(\frac{x_2}{x_1}\right)/ln(|MRS|) = 1 \]
• Unit elasticity of substitution: unit curvature of the indifference curve

Cobb-Douglas Utility Function​

• Substituting the values of \(x_1^*\) and \(x_2^*\) into the Cobb-Douglas function gives \[U = \left(\frac{\alpha I}{p_1(\alpha + \beta)}\right)^{\alpha}\left(\frac{\beta I}{p_2(\alpha + \beta)}\right)^{\beta}\] \[U = U_0 I^{\alpha + \beta}\left(p_1^{-\alpha}+p_2^{-\beta}\right)\] Where \(U_0\) is a constant
• Here U is not a function of quantity consumed, therefore can be referred to as an indirect utility function
• One must use Roy’s Identity

Roy’s Identity & Cobb-Douglas Function​

\[U(p_1,p_2,I) = U_0I^{\alpha + \beta}p_1^{-\alpha}p_2^{-\beta}\] - According to Roy’s Identity \[x_1^*(p_1,p_2,I) = \left(\frac{\partial U(p_1,p2_,I)}{\partial p_1} \right)/\left(\frac{\partial U(p_1,p2_,I)}{\partial I} \right)\] \[\frac{\partial U(p_1,p_2,I)}{\partial p_1} = -\alpha U_0I^{\alpha + \beta}p_1^{-\alpha-1}p_2^{-\beta}\] \[\frac{\partial U(p_1,p_2,I)}{\partial I} = (\alpha + \beta)U_0I^{\alpha + \beta - 1}p_1^{-\alpha}p_2^{-\beta}\] \[x_1^*(p_1,p_2,I) = \frac{\alpha I}{p_1 (\alpha + \beta)} \text{(similar result for )} x_2^*(p_1,p_2,I)\] - If we refer to the previous derivations of \(x_1^*\) and \(x_2^*\), Roy’s identity works!

Constant Elasticity of Substitution (CES) Direct Utility Function​

• General specification: \(U = \left(\sum_i \alpha_i x_i^{\rho}\right)^{1/\rho}\)
• For an example with two alternatives: \(U = \left(\alpha_1 x_1^{\rho} + \alpha_2 x_2^{\rho} \right)^{1/\rho}\)
• A flexible function that can represent various forms of indifference curves (demand functions) based on the value of \(\rho\)

CES Direct Utility Function​

\[MRS = - \frac{\alpha_1}{\alpha_2}\left(\frac{x_2}{x_1}\right)^{\rho-1}\] - Elasticity of subsitution \[\ln(|MRS|) = \ln\left(\frac{\alpha_1}{\alpha_2}\right) + (\rho-1)\ln\left(\frac{x_2}{x_1}\right)\] \[\ln\left(\frac{x_2}{x_1}\right) = \frac{1}{\rho-1}\ln(|MRS|) + \frac{1}{\rho-1}\ln\left(\frac{\alpha_2}{\alpha_1}\right)\] \[\sigma = \frac{d \ln\left(\frac{x_2}{x_1}\right)}{d \ln(|MRS|)} = \frac{1}{\rho-1}\]

CES Direct Utility Maximization​

• Consider a simple two alternative example \[U = (\alpha_1 x_1^{\rho} + \alpha_2 x_2^{\rho})^{\frac{1}{\rho}}\]

• Using Lagrangian function and FOC for \(x_1\) and \(x_2\) \[x_1 = x_2\left(\frac{p_1/\alpha_1}{p_2/\alpha_2}\right)^{\frac{1}{\rho-1}} \text{ and } x_2 = x_1\left(\frac{p_2/\alpha_2}{p_1/\alpha_1}\right)^{\frac{1}{\rho-1}}\]

• Substituting either result into the budget constraint, optimal demands are

\[x_1^* = \frac{I(p_1/\alpha_1)^{\frac{1}{\rho-1}}}{p_1(\frac{p_1}{\alpha_1})^{\frac{1}{\rho-1}}+p_2(\frac{p_2}{\alpha_2})^{\frac{1}{\rho-1}}} \text{ and } x_2^* = \frac{I(p_2/\alpha_2)^{\frac{1}{\rho-1}}}{p_1(\frac{p_1}{\alpha_1})^{\frac{1}{\rho-1}}+p_2(\frac{p_2}{\alpha_2})^{\frac{1}{\rho-1}}}\]

CES Indirect Utility Maximization​

• For general case, optimal demand is

\[x_j^* = \frac{I(p_j/\alpha_j)^{\frac{1}{\rho-1}}}{\sum_k p_k (\frac{p_k}{\alpha_k})^{\frac{1}{\rho-1}}}\]

• Indirect utility function at optimal direct utility level \[V = \left(\sum_k \alpha_k \left(\frac{I(\frac{p_k}{\alpha_k})^{\frac{1}{\rho-1}}}{\sum_k p_k (\frac{p_k}{\alpha_k})^{\frac{1}{\rho-1}}}\right)^{\rho}\right)^{\frac{1}{\rho}} = \frac{I}{\sum_k p_k (\frac{p_k}{\alpha_k})^{\frac{1}{\rho-1}}}\left(\sum_k(\alpha_k^{\frac{1}{\rho}}(\frac{p_k}{\alpha_k})^{\frac{1}{\rho-1}})^{\rho}\right)^{\frac{1}{\rho}}\]

Translog Demand/Cost Function​

• Translog is a quadratic, logarithmic specification of an indirect utility function written in terms of expenditure-normalized prices
  • Normalizing each price by dividing by total expenditure (income) imposes homogeneity
• Logarithmic indirect utility (or cost) function is \[\ln(V) = \alpha_0 + \sum_j \alpha_j \ln\left(\frac{p_j}{I}\right) + \frac{1}{2}\sum_j\sum_k \beta_{jk} \ln\left(\frac{p_j}{I}\right) \ln\left(\frac{p_k}{I}\right)\]
• Optimum quantity demands are derived using a logarthmic version of Roy’s Identity

Almost Ideal Demand System (AIDS)​

• AIDS is a combination of Cobb-Douglas and translog demand functions, describing an expenditure function necessary to attain a specific utility level at a given price
• Typical form is

\[\ln(V) = \alpha_0 + \sum_j \alpha_j \ln(p_j) + \frac{1}{2}\sum_j \alpha_j \ln(p_j) + \frac{1}{2}\sum_j\sum_k \gamma_{jk} \ln(p_j) \ln(p_k) + \beta_o \prod_j p_k^{\beta_k}\]

Discrete Demand

Microeconomic Theory of Discrete Goods​

The consumer

• Selects the quantities of continuous goods: \(Q = (q_1,\dots,q_L)\)
• Chooses an alternative in a discrete choice set \(i = 1,\dots,j,\dots,J\)
• Discrete decision vector \((y_1,\dots,y_J), y_j \in {0,1}, \sum_j y_j = 1\)

Note

• In theory, an alternative describes the combination of all possible choices made by a consumer
• In practice, the choice set will be restricted for tractability

Example​

Choices

• Home location: discrete choice
• Car type: discrete choice
• Number of km driven per year: continuous choice

Utility Maximization​

\[U(Q,y,\tilde{z}^Ty,\tilde{z}^T,\theta)\]

• Q are quantities of continuous good
• y is the discrete choice
• \(\tilde{z}\) are the K attributes of the J discrete alternatives
• \(\tilde{z}y\) are the attributes of the chosen alternative
• \(\theta\) is a vector of parameters

Optimization Problem​

\[\max_{Q,y} U(Q,y,\tilde{z}^Ty)\]

Suject to

\[p^T Q + c^T y \le I\]

\[\sum y_j = 1\]

\[y_j \in {0,1} \forall j\]

Why is it impossible to derive a direct demand function?

Solution to Optimization Problem​

• In a mixed integer programming problem, need to condition on the discrete variables to obtain continuous demand functions​
• Continuous demand functions can be differentiated to obtain optimality conditions​
• Various solution algorithms exist: genetic algorithm, branch and bound, etc.

Solution to Optimization Problem​

Solution to Optimization Problem​

Solution to Optimization Problem​

Application to Demand Analysis

Demand Modelling: Application of Utility Theory​

• Demand models are descriptive, not prescriptive​
• Observed (revealed or stated) demand used to develop demand models:​
  • Observed demand is optimal demand.​
  • Utility theory used to specify the demand model that is supposed to predict the observed (optimal) demand​
  • Observed information contains measurement (epistemic) and random (aleatoric) uncertainties, so probability theory must be used to specify a stochastic model​
• Econometric model: application of statistics to behavioral/economic data based on a sample of observations​

Observing/Modelling Demands​

• Measurement of demands are through specification of random variables​
• Type of demand measurement (variable) defines the nature of corresponding econometric model​
• Underlying theory of microeconomics allows for meaningful interpretation of model parameters and results​
• For example:​
  • Cobb-Douglas theory, CES, translog, AIDS are used in aggregate travel demand models​
  • CES indirect utility specification forms the basis for many discrete choice models (e.g., GEV models), time/resource allocation models, etc.​

Observing Demands​

• Variable types/measurements of travel demand:​
  • Cardinal numbers or ordinal measurements​
  • Quantitative or qualitative​
  • Discrete, continuous, count, or ordinal

Modelling Infrastructure Demands​

• Data (revealed/stated preferences): data do not provide answers themselves​
  • We need models and visualizations to give context and structure​
• Identify underling theory to specify model of interest​
• Two types of models:​
  • Aggregate demand models​
  • Disaggregate demand/choice models​
• In either case, application of econometric techniques is crucial to ensure evidence-based analysis​
• Estimation of appropriate model parameters is the critical roadblock!​