Exam 4: Demand

Consider a consumer with the Cobb-Douglas utility function U(q₁,q₂)= ,where q₁ and q₂ are the quantities of goods 1 and 2 consumed,respectively.This consumer has an income denoted by Y which is devoted to goods 1 and 2.The prices of goods 1 and 2 are denoted p₁ and p₂. a.What is this consumer's MRS as functions of q₁ and q₂? b.Write out the Lagrangian for the consumer's utility maximization problem. c.Using the Lagrangian method,derive the consumer's demand equations for both goods as functions of the variables p₁,p₂,and Y.

Why can't all goods be inferior?

A Californian student consumes Internet services (I)and books (B).Her preferences are represented by a Cobb-Douglas utility function of the type: U(I,B)= I.⁵B.⁵.Initially Y = 100,PI = PB = 1.Lately,however,because of the electricity shortage,the price of the Internet services has increased to 2.The government has decided to give a transfer to the student so that she can recover her initial welfare.In order to determine the transfer the government has hired three consultants who have made the following suggestions: Consultant A: The transfer should allow the student to buy her initial bundle. Consultant B: The transfer should allow the student to get her initial level of utility. Consultant C: The government should give her a transfer of 20. a.Using the expenditure function,find the amount of the transfer implied by consultant A. b.Find the amount of the transfer implied by consultant B. c.Determine whether the consumer is better or worse off from Consultant C's suggestion than before the price increases.

Jose gets utility from the goods he consumes and also the income he earns.His utility function is given by U(q₁,q₂)= q₁q₂Y Derive Jose's demand equations.Does the fact that he derives utility directly from income affect his consumption in this case?

A Consumer Price Index adjustment overcompensates for inflation because it ignores

Show that the following preferences are not consistent for a rational individual. i.An individual when confronted with prices of p₁ = $4 and p₂ = $8 chooses q₁ = 1 and q₂ = 5. ii.The same individual facing prices of p₁ = $6 and p₂ = $9 chooses q₁ = 5 and q₂ = 3.

A movement upward along an upward-sloping Engel curve corresponds to

Many elderly people have Social Security payments as their sole source of income.Because of this,there have been attempts to adjust these payments so as to keep up with changing prices.This process is called "indexing"; this question will lead you through the process. Suppose that in the year 2000,a typical Social Security recipient consumed only Food and Housing.The price of housing was $15/unit and the price of food was $5/unit.Denote the quantities of food and housing per month by F and H respectively.This consumer received $150/month and consumed five units of housing and 15 units of food. a.Write down the algebraic representation of this consumer's budget constraint. b.Show that the bundle (5,15)is just affordable.Identify another bundle on the budget constraint where both goods are consumed in positive amounts.Suppose that in 2004 the price of food rose to $10/unit and housing rose to $20/unit. c.How much additional income is required such that the original bundle is just affordable at the new prices. d.On the graph with Housing on the horizontal axis and food on the vertical axis,depict the original budget constraint and the new budget constraint after income is compensated according to c.Include the indifference curve going through the point (5,15). f.Is the consumer better,worse,or equally well off in 2004 relative to 2000 following the compensation? Explain.What will happen to the consumer's choice of food and housing consumption? How can you tell? g.The Laspeyres Price Index,as we have been describing above,measures inflation by looking at how the cost of a particular bundle of goods a consumer purchases in a previous year changes.Critics claim that this measure of inflation overstates changes in the cost of living.Do you agree or disagree? Explain.

Median household income is $50,000 per year.The typical household spends about $125 per year on milk,which has an income elasticity of about 0.07.From this information,we can conclude that

-Even though Mary's income is very low,she makes sure that she purchases enough milk for her family to drink.As her income rises,she does buy more milk.Which graph in the above figure best represents Mary's Engel curve for milk?

For the following, please answer "True" or "False" and explain why. -If the price-consumption curve is upward sloping when the price of the good measured on the horizontal axis changes,then the demand curve for that good will be upward sloping.

Richard receives government transfer payments and currently consumes five guns and six goose livers.Assume the price of guns decreases by 10% and the price of goose liver increases by 20%.The government raises Richard's transfer payments so he can still afford five guns and six goose livers.Does this constitute a true cost-of-living adjustment?

Why would you expect the demand for diamond jewelry to fall faster than plastic,costume jewelry when all incomes fall?

For the following, please answer "True" or "False" and explain why. -An increase in income (all else equal)will ALWAYS lead to a parallel shift of the budget line.

Rachel spends her income,Y,on Rock Shows (R)and Sunglasses (S)with prices pR and pS.Rachel's preferences are given by the Cobb-Douglas utility function U(X,Y)= R.⁸S.² a.Write out the Lagrangian for Rachel's utility-maximization problem. b.Use the Lagrangian to derive Rachel's optimal choice,(R*,S*). c.For a given utility level,U₀,derive Rachel's Expenditure function E(pR,pS,U₀). d.Use the Expenditure function to derive Rachel's compensated demand for Rock Shows.

Suppose a graph is drawn to show a consumer's preferences for football tickets and basketball tickets.The quantity of football tickets is measured on the horizontal axis.If the price-consumption curve is horizontal when the price of football tickets changes,then

-The above figure shows a consumer's indifference curves for soda and all other goods.Assuming a budget of $100,derive the consumer's demand for soda for prices of $4 and $10 per case of soda.Estimate the price elasticity of demand for soda.

Sylvia allocates her monthly income between Food and Housing.Her budget share spent on food in a given month is always 30%,and for Sylvia,food is a "necessity" (income elasticity between zero and one).Derive the maximum and minimum values for the income elasticity of demand for housing.

For each of the following statements,define all of the underlined terms.Then,explain why the statement is true or false. a.If a consumer views two goods as perfect substitutes then their optimal choice will be a corner solution. b.The substitution effect from a price increase states that the consumer will always choose a smaller amount of that good to consume.However,the income effect states that consumption can move in either direction. c.Suppose Alf and Bo have convex indifference curves.Alf likes units of "X" more than units of "Y" but Bo likes units of "Y" much more than units of "X." Then,in the optimum,Alf's marginal rate of substitution will be different from Bo's even if they face the same prices. d.All Giffen goods are normal goods,but not all normal goods are Giffen goods. e.Economists assume that preferences are ordinal.This implies that given two utility functions and one is a monotonic transformation of the other,then they represent the same preferences over bundles of goods.

Jill consumes nothing but soup and cola.It is possible that both soup and cola are necessities for Jill.