Learn Basic EconomicsLesson 5: Budget constraints

In the following examples we shall assume that income is equal to budget,

this means that we can ignore savings.

Income (Y) can be allocated between Coffee(C) or Donuts(D)

PC = the price per coffee

PD = the price per donut

Y = C*PC + D*PD

Price change

• Gradient = −PDPC =

−Price of donuts

Price of coffee• If the budget is £9.60, PD = £0.80 and PC

= £1.60, then this produces budget

constraint 1.

• This means you could either have 12

donuts or 6 coffees, if the whole budget

was spent on one good.

• The gradient is equal to -0.5, the gradient

of a budget constraint is called the

marginal rate of transformation (the rate

at which you can transform coffee into

donuts, in this case).

• [Remember opportunity cost is the value

of the foregone alternative. This means a

donut has an opportunity cost of half a

coffee.]

BC1

1.6C + 0.8D = 9.6

Price change

• Budget constraint 2 shows a scenario

where Coffee costs £2.40.

• This causes the gradient, and thus the

marginal rate of transformation to be −1

3.

• Opportunity set is the number of choices

you can make given a set budget.

• By the cost of Coffee increasing the

opportunity set has been restricted.

• Because the opportunity set has been

restricted this makes the consumer worse

off.

BC2

2.4C + 0.8D = 9.6

Decrease in income

• If there is reduction in the amount of

income, the gradient (marginal rate of

transformation) is unchanged as price is

unchanged.

• But the opportunity set is restricted so the

consumer is worse off.

• BC1 shows a budget/income of £9.60.

• BC2 shows a budget/income of £6.40.

(Coffee costs £1.60 and Donuts cost £0.80)

BC1

1.6C + 0.8D = 9.6

BC2

1.6C + 0.8D = 6.4

Constrained choice

• What is the highest achievable utility given

the budget constraint?

• Utility = 𝐶 ∗ 𝐷

• The budget constraint = 1.6C + 0.8D = 9.6

• A, B and C are on the utility curve 10

• D is on the utility curve 18

• E is on the utility curve 32(8,4)

(6,3)

(10,1)

(2,5)

(4.5,2.2)

Constrained choice

• This means that point D is the optimal

solution to maximise utility.

• It is at a higher utility than points A, C

and B.

• But it is within the budget constraint

unlike point E.

• Another way to see this is the optimal

point is by tangency.

• Where the gradient of the indifference

curve is equal to the gradient of the

budget constraint.

(8,4)

(6,3)

(10,1)

(2,5)

(4.5,2.2)

If the optimal point is where the gradient of the indifference curve is equal to the gradient

of the budget constraint, then the marginal rate of substitution must equal the marginal rate

of transformation.

Preference represented by the marginal rate of substitution = −Marginal Utility (for donnuts)

Marginal Utility (for coffee)=

−MUDMUC

Budget constraint represented by the marginal rate of transformation = −PDPC

From these equations we can deduce at the optimal point: MUD

MUDP𝐷

= MUC

P𝑐

• For example point A, C = 5 and D =2.

• Utility = 10• MRS = -2.5 (you would give up 2.5

coffees for 1 more donut)

• However the MRT is -0.5 (transform 0.5

coffees for a donut) so therefore point A is

an inefficient outcome as possible trades

are not being made. (8,4)

(6,3)

(10,1)

(2,5)

(4.5,2.2)

• At point C, on the other hand, the

marginal rate of substitution is 0.5 and so

the MRT = the MRS.

• But as point C is not on the budget

constraint line this means all the budget

has not been used up so this cannot be a

optimal solution, “non-satiation” means

the consumer always prefers to have more

which is why D is the only fully optimised

solution.

(8,4)

(6,3)

(10,1)

(2,5)

(4.5,2.2)

If there is a constant rate of substitution between Coffee and Donuts then this will cause a corner solution.In this case there is a corner solution at 6 coffees and 0 Donuts.

Here there is a corner solution at 12 Donuts and 0 coffees.

Thank you for viewing this lesson of learn basic economics.