Key concepts and results for consumer theory (Part 1)
Preferences
Consumers have preferences over a set of goods which can be represented by a utility
function U ( X 1 , X 2 )that maps consumption bundles to a total utility level.
Marginal utility is the increase in the utility level when the consumer consumes an additional
unit of one good, leaving everything else constant.
δ U ( X 1 , X 2) δU ( X 1 , X 2 )
MgUx 1= MgUx 2=
δ X1 δ X2
o Goods have positive marginal utility.
o Bad have negative marginal utility.
Marginal Rate of Substitution is the number of units of X 2 that a consumer is willing to
sacrifice in exchange for one additional unit of X 1 at a given bundle.
−MgUx 1
MRS=
MgUx 2
Utility functions are monotonic whenever both X 1 and X 2 are goods. Monotonicity implies
that the consumer “always prefers more to less”.
Utility functions are convex whenever the indifference curves are actually curves
(differentiable) that give its back to the origin. You can confirm convexity by taking the
derivative of the MRS with respect to X1, the result must be negative.
Budget
Consumers face scarcity of resources (income), and their budget constraint is defined by:
P x1 X 1 + Px 2 X 2=m
−Px 1
The slope of the budget constraint is: , the relative prices. This is the rate at which the
Px 2
market allows the consumer to sacrifice units of X 2 in exchange for one unit of X 1 .
Consumers always choose a point on the budget constraint when optimizing consumption
due to monotonicity. However, the “budget set” includes all points on the constraint, and
below it, because those are the points that consumption can afford.
Optimization
Consumers solve the following problem:
max U ( X 1 , X 2 ) s.t. P x1 X 1 + Px 2 X 2=m
Choosing quantities of X 1 and X 2
The first order condition results in:
Px 1
MRS=
Px 2
Which is called the tangency condition, for this equation represents all the collection of
points in which the slope of the budget constraint equates the slope of the indifference
curves.
1
, To be sure we are on the budget constraint, we substitute the first order condition into the
budget constraint equation and solve for X 1 . Then we substitute X 1 into the first order
condition to find X 2 .
The result of consumption optimization are the ordinary demands for X 1 and X 2 , which
depend on the prices and income.
Cobb Douglas
α β
Utility function has the form: U ( X 1 , X 2 )= X 1 X 2 with α >0 and β >0. “Well behaved”,
that is, convex and monotonic.
Optimizing consumption always leads to a bundle with positive demanded quantities for
both goods given by ordinary demands:
X1= ( α +α β ) Pm
x1
X2= ( α +β β ) Pm
x2
α β
Percentage of income spent on each good is fixed and given by: for X 1 and
α+β α+β
for X 2 . However, expenditure (that is P x1 X 1 for X 1 and P x2 X 2 for X 2 ), varies whenever
income or any of the prices change.
Elasticities for the ordinary demands:
E x1 , px 1=−1 E x 1, px2=0 E x1 , m=1
E x 2, px 2=−1 E x 1, px2=0 E x1 , m=1
Implications:
o Both X 1 and X 2 have “unitary demands”, whenever the own prices changes in y
%, consumption changes in -y%.
o X 1 and X 2 are independent of each other, that is, when the price of one good
changes, consumption of the other good remains unchanged.
o X 1 and X 2 are normal goods, that is, their consumption increases with increases
in income (and viceversa). As a result, the Engel curve of both goods has positive
slope.
Quasilinear
Utility function has the form: U ( X 1 , X 2 )=f ( X 1) + α X 1 with α >0 and f is a non-linear
function. “Well behaved”, that is, convex and monotonic.
Optimizing consumption may lead to consume both goods in positive quantities (when
income is “high enough”) or to consume only the non-linear good -in this case, X 1 -
(when income is below certain threshold).
Ordinary demands:
o For X 1 , it always comes directly from the tangency condition (MRS=relative
prices), by solving this equation for X 1 .
o For X 2 , it is necessary to substitute the demand for X 1 into the budget
constraint and solve for X 2 .
Percentage of income spent and expenditure on each good varies and most be
computed for each optimal bundle.
2
Preferences
Consumers have preferences over a set of goods which can be represented by a utility
function U ( X 1 , X 2 )that maps consumption bundles to a total utility level.
Marginal utility is the increase in the utility level when the consumer consumes an additional
unit of one good, leaving everything else constant.
δ U ( X 1 , X 2) δU ( X 1 , X 2 )
MgUx 1= MgUx 2=
δ X1 δ X2
o Goods have positive marginal utility.
o Bad have negative marginal utility.
Marginal Rate of Substitution is the number of units of X 2 that a consumer is willing to
sacrifice in exchange for one additional unit of X 1 at a given bundle.
−MgUx 1
MRS=
MgUx 2
Utility functions are monotonic whenever both X 1 and X 2 are goods. Monotonicity implies
that the consumer “always prefers more to less”.
Utility functions are convex whenever the indifference curves are actually curves
(differentiable) that give its back to the origin. You can confirm convexity by taking the
derivative of the MRS with respect to X1, the result must be negative.
Budget
Consumers face scarcity of resources (income), and their budget constraint is defined by:
P x1 X 1 + Px 2 X 2=m
−Px 1
The slope of the budget constraint is: , the relative prices. This is the rate at which the
Px 2
market allows the consumer to sacrifice units of X 2 in exchange for one unit of X 1 .
Consumers always choose a point on the budget constraint when optimizing consumption
due to monotonicity. However, the “budget set” includes all points on the constraint, and
below it, because those are the points that consumption can afford.
Optimization
Consumers solve the following problem:
max U ( X 1 , X 2 ) s.t. P x1 X 1 + Px 2 X 2=m
Choosing quantities of X 1 and X 2
The first order condition results in:
Px 1
MRS=
Px 2
Which is called the tangency condition, for this equation represents all the collection of
points in which the slope of the budget constraint equates the slope of the indifference
curves.
1
, To be sure we are on the budget constraint, we substitute the first order condition into the
budget constraint equation and solve for X 1 . Then we substitute X 1 into the first order
condition to find X 2 .
The result of consumption optimization are the ordinary demands for X 1 and X 2 , which
depend on the prices and income.
Cobb Douglas
α β
Utility function has the form: U ( X 1 , X 2 )= X 1 X 2 with α >0 and β >0. “Well behaved”,
that is, convex and monotonic.
Optimizing consumption always leads to a bundle with positive demanded quantities for
both goods given by ordinary demands:
X1= ( α +α β ) Pm
x1
X2= ( α +β β ) Pm
x2
α β
Percentage of income spent on each good is fixed and given by: for X 1 and
α+β α+β
for X 2 . However, expenditure (that is P x1 X 1 for X 1 and P x2 X 2 for X 2 ), varies whenever
income or any of the prices change.
Elasticities for the ordinary demands:
E x1 , px 1=−1 E x 1, px2=0 E x1 , m=1
E x 2, px 2=−1 E x 1, px2=0 E x1 , m=1
Implications:
o Both X 1 and X 2 have “unitary demands”, whenever the own prices changes in y
%, consumption changes in -y%.
o X 1 and X 2 are independent of each other, that is, when the price of one good
changes, consumption of the other good remains unchanged.
o X 1 and X 2 are normal goods, that is, their consumption increases with increases
in income (and viceversa). As a result, the Engel curve of both goods has positive
slope.
Quasilinear
Utility function has the form: U ( X 1 , X 2 )=f ( X 1) + α X 1 with α >0 and f is a non-linear
function. “Well behaved”, that is, convex and monotonic.
Optimizing consumption may lead to consume both goods in positive quantities (when
income is “high enough”) or to consume only the non-linear good -in this case, X 1 -
(when income is below certain threshold).
Ordinary demands:
o For X 1 , it always comes directly from the tangency condition (MRS=relative
prices), by solving this equation for X 1 .
o For X 2 , it is necessary to substitute the demand for X 1 into the budget
constraint and solve for X 2 .
Percentage of income spent and expenditure on each good varies and most be
computed for each optimal bundle.
2
