Log Linear Demand and Supply
Consider the following “log-linear” supply and demand model:
Demand: ???????? = ???? + ?????????? + ??????????, ???? < ??, ???? > ??
(1)
Supply: ???????? = ???? + ?????????? + ?????????? ???? > ??, ???? < ??
(2)
where P and Q have their usual meaning, I denotes household income and W the
wage rate the firm must pay to hire workers.
(1A) (10 points) What is the own price elasticity of demand in this market? Prove your
answer, showing your work.
(1B) (5 points) Is this a normal good or inferior good? Prove your answer, showing your
work..
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(1C) (15 points) You are interested in knowing how the equilibrium price of this product
will vary with a change in the competitive wage (W). What is the equilibrium price
function? How will equilibrium price vary with a variation in W? Be as precise as you can
be, always showing your work.
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(2) (20 points) Deriving demand functions
David has a quasi-linear utility function in X and Y of the form:
?? = ??
1
2 + ??
Income is I and the two prices are ???? and ???? respectively.
(2A) (15 points) Derive algebraically his demand function for X. Show your work.
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(2B) (5 points) Is David’s demand for x is independent of his income?
Demonstrate your answer formally.
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(3) (30 points) Taxing away the benefits of a price changes
A consumer in a two good economy (??, ??) with fix prices has the following preference
function:
?? = ????
Income is I and the two prices are ???? and ???? respectively.
(3A) (15 points) Derive algebraically his demand functions for X. Show your
work.
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(3B) (15 points)
Continuing the problem in 3A, consider the following situation: Income I=120 and the two
prices are ???? = 4 and ???? = 1 respectively. Suddenly the price of X falls to 3 (???? =
3).
The government is tempted to capture all the gains the consumer gets from this price fall
to finance a worthy project. How much income can they take from the consumer and still
leave the consumer no worse off than he was before the price change?
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(4) (20 points) The Corn Market and Farmer Prosperity
You are a political planner for an incumbent president and you know that farmers tend to
vote for the party in power when times are good (or at least better) and for the challenger
if times are bad. Times are good if total farm income (farm revenue) increases. . Thanks
to the many favors your boss has done for interest groups, your boss has campaign
reserves for more advertising if campaign problems arise in corn country.
You check the fundamentals of the corn market. Agriculture Department economists
assure you the market is competitive and also report that the elasticity of demand is ????
and the elasticity of supply is ????. (They give you a number for each but we want the
general form so we use symbols for those numbers.)
Then you learn that growing conditions are fabulous and that the country will have a
bumper crop of corn. Should you commit your advertising reserves to the corn states or
not? Be as specific in your answer as you can