1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 5.1
MICROECONOMICS
Principles and Analysis
Frank Cowell
November 2006

2. Ex 5.1(1) Question
purpose: construct a simple model of household supply and examine how it works
method: build model up step-by-step through the question parts

3. Ex 5.1(1) Preference map x2
other goods
Shift the origin to (0, k)
Draw ICs homothetic to the shifted origin l
indifference curves are “shifted” Cobb-Douglas
k is min requirement of other goods.
a is share of budget of rice after an amount has been set aside to buy the min requirement l
Expanto l u1 l u0 k x1
rice

4. Ex 5.1(2) Question method: Work out the budget constraint.
Use the utility function to set out the Lagrangean
Find the FOCs for an interior solution
Find the demand functions
Use these to get household supply function

5. Ex 5.1(2) Budget constraint
Use good 2 as unit of value
price of rice (good 1) is p
price of all other goods (good 2) is 1
The consumer’s income is therefore:
y := pR1 + R2
The budget constraint is
px1 + x2  y
where y is given by the above

6. Ex 5.1(2) Lagrangean method
The Lagrangean is
a log(x1) + [1–a ] log(x2–k) + l [ y – px1 – x2 ]
The FOC for an interior maximum are a
— – pl = 0
x1*
1–a
—— – l = 0
x2*–k
y – px1* – x2* = 0

7. Ex 5.1(2) Demand functions From the FOC: a px1* = — l 1–a x2* = k + ——
Adding these and using the budget constraint, we have
y = k + 1/l
Eliminating l in the above:
x1* = — [y – k] p
x2* = ak + [1–a ] y

8. Ex 5.1(2) Supply function Supply of good 1 is given by
S(p) := R1 – x1*.
Substituting in for y, we have a
S(p) = [1–a ]R1 – — [R2 – k] p
Supply increases with price if R2 > k

9. Ex 5.1(2) Solution k S x2 Endowment other goods x2 Budget constraint
Optimal consumption
Supply of rice x* l
Expanto R l k p x1
rice S

10. Ex 5.1(3) Let c be the amount of the ration
If ak + [1–a ] y  c nothing changes from previous case
Otherwise
px1 + c = y
so that
R2 – c
x1 = R1 + ——— p
c – R2
S(p) = ———

11. Ex 5.1(3) Modified solution
other goods
Original solution x2
Generous ration
Severe ration c
Generous ration has no effect x* l
Expanto
Severe ration on other goods affects supply of rice c
x** l R l k x1
rice S

12. Ex 5.1: Points to remember
Use diagram to understand features of utility function
Model supply as mirror image of demand
Use diagram to see effect of ration