budget sets

8/10/2019 Budget Sets

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Scarcity (V.2 and L.3)

Outline

1 Scarcity (V.2 and L.3)

The Budget Set

Changes in Parameters

Policies and the Budget Set

Scarcity and the Budget Set


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The Budget Set

Outline

1 Scarcity (V.2 and L.3)

The Budget Set





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The Budget Set

The Budget Set

Can’t consume negative quantities, x ≥ 0 ad y ≥ 0.

Consumers are price takers and purchase goods in a perfectly

competitive market at prices px and py.They are endowed with monetary income, m.

His expenditure on consumption bundle (x, y) has to satisfy his budget

constraint,

pxx + pyy ≤ m. (1)



S i (V 2 d L 3)

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The Budget Set

The Budget Set

The budget set is the set of all affordable consumption bundles.

Figure: Budget Sets

x

y

m

py

m

px

pxx+ pyy ≤ m



S it (V 2 d L 3)

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The Budget Set

Numeraires

Equation (1) is expressed in absolute prices, the amount of money a

consumer has to forego to obtain one more unit of that good.

px is measured in dollars per unit of good x.

We can also have a real measure of prices. Measure the price and incomein terms of units of another good.

Suppose x is the numeraire good, and px the numeraire price (the

reference price to measure all other prices).

For example the budget constraint can be expressed in relative prices,

x + py

pxy ≤

m

px



Scarcity (V 2 and L 3)

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The Budget Set

Numeraires

For example the budget constraint can be expressed ,

x + py

pxy ≤

m

px

pypx

is the relative prices and is measured in (dollars/units of y)(dollars/units of x)

= units of xunits of y

.

How much consumption of x the consumer has to give up to consume a

unit of y.

mpx

is the consumer’s real income measured in units of x,dollars

dollars/units of x = units of x.




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The Budget Set

The Budget Set with numeraire

The budget set is the same when it is written in terms of relative prices

Figure: Budget Sets

x

y

m

py

m

px

x+ py

pxy ≤

m

px




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The Budget Set

Opportunity Cost

Opportunity cost of one unit of good x is the relative price of good x in

terms of good y.

The value of a resource in its best alternative use.

If consumer buys a unit of x the opportunity cost is the quantity of good y

he must “sacrifice” to obtain that unit.




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S y ( 3)

The Budget Set

Opportunity Cost

We would like to know how consumption of y changes when the agent

decides to consume x and neither the income nor the prices change.

We can totally differentiate the binding budget constraint with respect tox and y,

pxdx + pydy = 0 ⇒ dy

dx = −

px

py(2)

Equation (2) indicates the rate at which the market allows the consumer

to exchange good x for good y.




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y ( )

The Budget Set

Opportunity Cost and Slope of the Budget Constraint

Solving the budget constraint for y,

y = − px py

x + m py

.

The opportunity cost of consumption of good x is not money but the

forgone consumption of good y.

The relative price dydx = −

pxpy

measures this opportunity cost.

Negative sign indicated tradeoff for the consumer, and increase in the

consumption of good x implies a decrease in consumption of good y.




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Outline

1 Scarcity (V.2 and L.3)The Budget Set






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Changes in Income

Suppose income increases to m′ > m. Then the budget constraint shifts

parallel to the right.

Figure: Increase in Income

x

y

m

py

m

px

m′

px

m′

py




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Changes in Prices

Suppose price of good y increases to p′

y > py. The budget constraint pivots on

the x intersect.

Figure: Increase in py

x

y

m

p′

y

m

px

m

py




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Outline

1 Scarcity (V.2 and L.3)The Budget Set






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Taxes and subsidies

Quantity tax: a tax charged per each unit bought or sold.

If there is a quantity tax of t > 0 dollars per unit of good x, the price for

the consumer increases to px +

t.

Ad valorem tax : (Value tax) a tax on the value of a good bought or sold.

If there is a value tax of τ ∈ (0, 1) on good x, the price for the consumer

increase to px(1 + τ ).

Lump sum tax: An amount of money charged to consumers or producers.

If there is a lump sum tax to the consumer of T dollars , the consumers

income would now be m− T .




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Rationing

The government could also impose rationing constraints.

Suppose good x is ration such that no more than x̄.

Figure: Increase in py

x

y

m

py

m

pxx̄



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