Lecture 5: Uncertainty and Linear Programs

Introduction

The speaker greets the audience and introduces the reading list for the lecture.

Reading List

• The lecture will finish two, three things that were deliberately not done at the end of the previous lecture on firms and production sets.

• There are other things on the reading list about the great Japanese earthquake.

Decision Making Under Uncertainty and Linear Programs

The speaker discusses decision making under uncertainty and linear programs.

Material Covered

• Decision making under uncertainty and linear programs is largely coming from Nicholson and Schneider.

• Several slides today have material on Medville from that book.

• Study guide questions are provided, including defining increasing, decreasing, and constant returns to scale in production.

Proving Maximization of Profits Homogeneous of Degree 1 in Prices p

The speaker asks a question about proving maximized profits are homogeneous of degree 1 in prices p.

Question & Answer

• How would you prove that maximized profits are homogeneous of degree 1 in prices p?

• Homogeneous of degree 1 means that if you increase every element of the price factor by a constant that maximized profits increased by that same constant.

• Profit function is equal to p.z, where z is the production vector.

• If we double the price p, then we also double the profit.

Production Possibilities and Input-Output Matrices

In this section, Robert Townsend discusses the concept of production possibilities and input-output matrices.

Doubling Prices

• Doubling the price of every component of the price vector does not change the slope of isoprofit lines. The tangency with the production possibilities frontier would be the same point.

Isoquant Definition

• An isoquant is a set of input boundaries which, given productivity, will output the same level of output y.

• All input boundaries produce the same level of output.

Leontief Input-Output Matrices

• Leontief's view of an economy's production possibilities involves three sectors: raw materials, services, and manufacturing.

• The input-output matrix summarizes how much is needed to produce final demand for raw materials, services, and manufactured goods.

• Final demands are given in a three-dimensional column vector that tells us how much we need to deliver to consumers or for exports in each sector.

Linear Equations and Matrix Algebra

The section discusses the use of matrix algebra to solve linear equations in control variables, specifically x. The solution is obtained by inverting the matrix, i minus a times dx.

Solving Linear Equations

• Output gets used as inputs, so we have to produce i minus a times X.

• Ax gives you how much of that x gets used up in producing services, raw materials, and manufacturing. I minus a times x should just equal demand.

• This is a linear system of linear equations. All we need to do is invert this matrix, i minus a.

• The x solution we want is i minus a times dx.

Google's PageRank Algorithm

The section discusses Google's PageRank algorithm and its similarity to Leontief's ranking of industries.

PageRank Algorithm

• Google's algorithm ranks search results based on their relevance to the user's query.

• The algorithm is based on the same ideas that underlie Leontief's ranking of industries.

• Search results are displayed in a network diagram with related items connected via pathways.

• Items are ranked by distance from the center and reordered for display purposes.

Great East Japanese Earthquake

The section briefly mentions the Great East Japanese earthquake and its aftermath.

Earthquake Aftermath

• A large earthquake followed by a tidal wave caused significant damage to structures along the coast.

Supply Chain Disruption

In this section, the speaker discusses how to identify industries that have been supplying to headquarters and how supply chain disruptions can affect production.

Identifying Industries

• The speaker highlights a small area on the screen and explains that they are looking at which industries had been supplying to headquarters.

• Input-output sense is used to determine which industries are linked to headquarters.

• Examples of input suppliers include tire manufacturers in Japan who supply General Motors in Detroit.

First Order Connection

• The first order connection refers to areas directly linked to the disruption.

• In this case, it includes all of Japan since there is either an immediate input supplier or output purchaser indirectly damaged in the sense of the first degree.

• This primary area of damage covers most of Japan and caused a drop in national level GDP by about 1.5%.

Decision Making Under Uncertainty

This section covers decision making under uncertainty and techniques such as programming and linear programming.

Lottery Prizes

• The speaker introduces a lottery with n distinct prizes, possibly zero or negative if it takes money away.

• X represents the amount of winnings or losings if that even turns up in the lottery.

• There are n possible things that can happen with probabilities pi for each outcome.

Variability and Dispersion

• Variability or variance is denoted as sigma squared X, which is just the dispersion of X around its mean mu X.

• Expectation means taking weighted averages with pi I probabilities as weights.

• Standard deviation divided by mean is called coefficient of variation.

Covariance Between Two Random Variables

• If there are two dimensions for a prize (X and Y), we can define covariance between two random variables x and y as expectation of product subtracting off means.

Statistical Review and Lottery

In this section, the speaker discusses statistical review and lottery. The speaker explains how to plot variables in a cloud to determine covariance and correlation. They also define random variables and lotteries.

Covariance and Correlation

• High x means low y, the covariance would be negative.

• Correlations are by construction going to be between minus 1 and 1.

• These concepts are defined for random variables.

Lottery

• The underlying ingredients of a lottery are x1 and x2, but they can only take on a finite number of values.

• Each state corresponds to a particular consumption bundle in R2, and we can call pi S the probability of being in state S.

• A lottery puts a non-negative probability on any particular state.

• The set of all possible lotteries is called capital Pi or simplex. It's where each element is non-negative, and all the elements add up to 1.

Preferences with Expected Utility

• Preferences can now be defined with squiggly script inequality over the set of possible lotteries.

• Von Neumann Morgenstern expected utility is used to rank all possible lotteries by their expected utility criterion.

• This expected utility criterion is linear in the choice objects which are the lotteries.

Axioms for Ranking Lotteries

• Three axioms need to be satisfied: completeness, transitivity, independence.

• Completeness says that all lotteries can be ranked and give us a number.

• Transitivity is satisfied because if A is preferred to B, lottery pi is preferred to lottery pi prime.

• Independence is the idea that the objective function is linear in the probabilities.

Axioms of Independence and Continuity

In this section, the speaker discusses the axioms of independence and continuity in expected utility theory.

Independence Axiom

• The alpha-weighted sum of lotteries pi and pi prime is weakly preferred to pi prime if alpha equals 1.

• If alpha is less than 1, then some weight is put on an extraneous third lottery, pi double prime.

• The notion of independence means that whatever this other lottery looks like, it doesn't matter for fixed alpha when it comes to weighting pi and pi double prime.

• Continuity is similar but not identical. It says that if we have three lotteries (pi, pi prime, and pi triple prime) rank ordered such that pi is preferred to pi prime which is preferred to pi double prime, then there exist numbers alpha and beta such that a weighted sum of pi and pi double prime weighted by alpha is strictly preferred to pi prime which is strictly preferred by the beta weighted sum of pi and pi double prime.

Cardinality in Choice Under Uncertainty

• Expected utility becomes cardinal rather than ordinal in choice under uncertainty.

• We care about the degree of curvature in the orange line mapping expected utility outcome as we vary input x.

• Income moves around with a mean expected value of x with endpoints at lowest value x0 and highest value x1.

• The lottery creating the random variable produces a weighted average of x0 and x1 with weights producing exactly this expected value.

• The actual utility of the lottery can be achieved by taking a weighted average of the utilities at extreme points.

Risk Aversion and Utility Functions

In this section, the speaker discusses risk aversion and utility functions. They explain how households choose between a lottery and a deterministic value, and how increasing the curvature of the function can lower the expected utility of the lottery. The speaker also introduces standard measures of risk aversion, including absolute risk aversion and relative risk aversion.

Absolute vs Relative Risk Aversion

• Households may choose a deterministic value over a lottery as long as that value is not less than the risk premium. Anything below this threshold, they would choose.

• Increasing the curvature of the function lowers the expected utility of the lottery because there is more risk associated with it.

• Standard measures of risk aversion include absolute risk aversion (how marginal utility changes as consumption changes) and relative risk aversion (how much utility changes as percentage of wealth changes).

Constant Absolute vs Relative Risk Aversion

• Constant absolute risk aversion refers to willingness to take up gambles like starting with $100 and adding or subtracting 10.

• Constant relative risk aversion has to do with proportional gambles where you start with say 100 or 1,000 and have 10% gains or losses.

• Under constant relative risk aversion, all you care about are those percentage differences from your initial wealth.

• Attitudes toward risks depend on these functions.

Lotteries and Uncertainty Over Time

In this section, the speaker discusses lotteries and uncertainty over time. They introduce a diagram that shows how time evolves from day one to day three, reflecting every possible realization and path to get there. The speaker explains that when some things happen in this diagram, other things don't happen.

Uncertainty Over Time

• The diagram shows how time evolves from day one to day three, reflecting every possible realization and path to get there.

• When some things happen in the diagram, other things don't happen. For example, when you move left from a certain point, you've eliminated the other two branches of the tree.

• Expected utility is taken into account from day one onward.

State Contingent Commodities

In this section, the speaker discusses state contingent commodities and how they can be used to represent financial securities.

State Contingent Commodities

• A state contingent commodity is a commodity whose payoff depends on the state of the world.

• Financial securities can be represented as linear combinations of primitive Arrow-Debreu securities that only pay off in the event of one well-delineated event.

• All possible securities can be represented as probabilistic weighted averages over all possible events.

Financial Securities and Risk

In this section, the speaker discusses how financial securities are weighted averages over all possible events and how some securities are riskier than others.

Weighted Averages and Risk

• Any financial security is a weighted average over all possible events.

• Some securities are riskier than others because they only pay off under certain conditions or have higher coefficients of variation.

Primitive Arrow-Debreu Securities

In this section, the speaker explains what primitive Arrow-Debreu securities are and how they can be used to represent financial assets.

Primitive Arrow-Debreu Securities

• Primitive Arrow-Debreu securities are building blocks that only pay off in the event of one well-delineated event.

• All possible securities can be represented as linear combinations of these primitive Arrow-Debreu securities.

Diversification of Portfolios

In this section, the speaker discusses diversification of portfolios using an example from land holdings in England in 1719.

Land Holdings Example

• Mr. Darlaston's long, narrow strips of land holdings in Elford, Stratfordshire can be thought of as a portfolio of land holdings.

• The holding of land yields a dividend, which is the harvest.

• Scattered plots with high coefficients of variation can lead to disastrous events such as starvation.

• Consolidating land holdings can reduce costs and problems.

Bishop of Winchester's Holdings

In this section, the speaker discusses the Bishop of Winchester's holdings in England in 1719 and how they kept records.

Bishop of Winchester's Holdings

• The church was wealthy and kept records of all domains and crops being grown.

• The coefficient of variation denotes the variance divided by the mean or standard deviation divided by the mean.

Bishop of Winchester Estates

In this section, the Bishop of Winchester estates are discussed. The yields and harvest data for ecclesiastical estates are plotted on a graph to show the correlation between villages that are near and far from each other.

Correlation between Villages

• A statistical concept called "covariate" is being plotted on a graph to show the correlation between villages that are relatively near to one another, and villages that are relatively far away.

• The correlation drops when comparing yields from villages that are far away from each other.

• The location of the Bishop's estates may have been chosen strategically to ensure regular grain harvest for consumption and feeding horses.

Linear Programming

This section introduces linear programming as a technique used in solving optimization problems.

Linear Programming Problem

• Linear programming is a problem where x is the control variable, C represents weights on various x values, and there is a set of constraints represented by matrix A times x less than or equal to b.

• Leontief input-output matrix was an early version of linear programming problem without an explicit objective function.

• Consumer choice problem in competitive environment with budget constraint can be solved using linear programming.

Budget Problem

• Lottery choices can be made based on expected utility criterion with non-negative lotteries adding up to 1.

• Prices p depending on s (underlying discrete bundle) and budget in dollars can be used to buy a lottery. The price of getting that bundle s back is scaled by pi.

• Actuarial value of the payoff is used to pay for non-degenerate lotteries.

Firm Production

• A firm can produce n possible goods with a fixed supply of inputs. The technology tells us how much input i is needed to produce good j.

• Revenue, price times quantity, summing over all the j goods is the natural objective function.

Linear Programming Problems

In this section, the speaker discusses three types of linear programming problems: ordinary production, choice under uncertainty, and transportation.

Ordinary Production

• Ordinary production is a type of linear programming problem where you have to decide how much of each product to produce in order to maximize profit.

• The decision variables are the quantities produced for each product.

• The objective function is the profit function.

• There are constraints on resources such as labor and materials.

Choice Under Uncertainty

• Choice under uncertainty is a type of linear programming problem where you have to make decisions based on uncertain outcomes.

• The decision variables are the actions taken in response to different possible outcomes.

• The objective function is the expected value of some measure of performance.

• There are constraints that reflect the fact that only one outcome will occur.

Transportation Problem

• A transportation problem involves shipping goods from warehouses to retail outlets while minimizing costs.

• The decision problem is determining shipments from all warehouses to all potential outlets that minimizes total cost while meeting demand at all outlets and respecting inventory levels at all warehouses.

• Constraints include inventory levels at warehouses and demand at retail outlets.

Overall, these three types of problems use common tools such as linear programming techniques.