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Eco 329 Urban Economics
Spring 2016
Assignment Set A
Assignment #1
Consider determining the algebraic solution for a separating equilibrium. Rewrite the whole special operational
equations for both a household and a firm in a form where the coefficients of the “u”-terms are collected. Write these
equations out with enough room to put a bracket below each set of collected coefficients of the u-terms, and write below
each of these brackets a distinct letter to label it. Now write out a separate set of equations giving the immediately implied
equations (that is, before any manipulation or substituting or solving) that follow from utility maximization. Each of these
equations should be labeled (to the left of each of them) by the parenthesized code labeling the same equation in the
Theory Notes followed by the letter labeling the collected set of coefficients which are used to deduce the equation and
then finally followed by a colon before the equation you have written.
Assignment #2
Using your results of Assignment #1, clearly go through the steps and clearly give a short written justification for each
step showing how the equations giving algebraic expressions for the slopes of the bid-rent and wage are determined.
Clearly indicate for each final expression the label of that equation in the Theory Notes.
Assignment #3
Using the collected equations you wrote in Assignment #1 and the implied equations laid out in that assignment, explain
step by step how to get further implications from these operational equations beyond those produced in Assignment #2.
Clearly indicate for each of the final implications the label of that equation in the Theory Notes.
Assignment #4
Remember that “the solution” to our model for the separating equilibrium can be expressed by a set of 3 functions and
two values consisting of the household bid-rent function, the firm bid-rent function, the wage function, the firm-household
boundary, and the household-agricultural boundary. First, write out now the functions and values to give the price/location
expression of such a solution, expressing the coefficients and constants in the equations and the boundary values only by
their appropriate single symbol (whose values are still unknown). These unknown varibles are what we are looking for in
solving our model. Now count the number of unknowns that need to be determined to write such a solution.
You are then going to build a set of equations where the unknowns are the unknown coefficients in the equations and
the unknown boundary values which together are needed to express the equilibrium solution. This set of equations has to
have the same number of equations as the number of unknowns to determine. The set of equations should be written out in
a vertical list, one equation per line) so that each particular unknown occurs in a column each one below/above that same
unknown in the equation above/below and the constant expressions, that is, expressions with no unknown in it, occur to
the right of each of the equal signs in the equations. The equal signs of the equations should also be in a column.
This set of equations will be consist of the immediate equations built from the appropriate collected coefficients in the
special operational equations set equal to zero, the simplified special operational equations, the definitional boundary
equations, and the demand-supply condition. Remember you are not solving equations as you do this, you are just writing
out all these equations. After you write them all out, you have to collect the coefficients of all the unknowns in each of the
equations. You can then arrange this set of equations as described above. Finally, count the number of equations and
unknowns in this system of equations. Those two numbers should be equal. You should also check that all the expressions
giving the coefficients of each of the unknowns in each of the equations contain only paraneters known before any
computation begins.
This is the system of equations describing the model which was referred to right at the beginning of the Theory Notes.
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The Theory Notes is essentially solving this system of equations step by step using the “substitution method”. This
method works here because the model was set up to be very simple. Another more general solution method, the “matrix
inversion method” from linear algebra, is available for systems of equations that are not so simple, but that technique is
beyond what we will try in this class.
Assignment #5
Consider a slightly more general model, one where we add a second kind of household, so that we now have a type 1
and type 2 household. The utility parameters (the givens in the utility function and conceptual equation) are different for
each type of household. Each type of household also supplies a different type of labor. There is still only one type of firm,
but each firm now requires a specific amount of each type of labor in its fixed proportion production function.
You are to do two tasks.
Your first task is to write out how a solution to this could be expressed analogous to the way a solution to the 1-type-of household
model was expressed in price/location form at the beginning of Assigment #4. You may have to add new
parameters, functions, and variables. Count how many unknowns there are in this solution.
Now write out the system of equations that need to be solved for this new model in the same way as you did for the 1-
type-of-household model in Assignment #4.