Unit 2: Budget Constraints Dillon Flannery-Valadez

Contents 1 Budget Constraints 1.1 Why only two? . . . . . . . . . 1.2 The Budget Line . . . . . . . . 1.3 Changes in the Budget Line . . 1.4 Numeraire . . . . . . . . . . . . 1.5 Taxes, Subsidies, and Rationing 1.6 Food Stamps in 1975 . . . . . .

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1 1 1 2 2 2 3

Budget Constraint

For ease, let’s consider a persons consideration to consume goods consists of only 2 goods. Lets call these goods f = food and c= clothing, or (f, c) for short. The pair (f, c) is called a consumption bundle. Another consideration we will take into account are the prices of these goods. Let’s simply call these pf and pc for the price on food and the price on clothing. The total amount of money available to this person is their budget, and we can call that b. The problem we have is to figure out how much of each they can consume subject to the constraint: f pf + cpc ≤ b That is to say the dollar amount spent on food plus the dollar amount spent on clothes is less than or equal to the budget.

1.1

Why only two?

Despite the fact that people may actually consume hundreds of goods, this representation of two goods is often all that is needed. If we want to know what the consumer’s demand for food f is we can let c be a composite good representing everything else that person could possibly want to consume. The composite good is the dollar amount of all other goods. The new problem is f pf + c ≤ b

1.2

The Budget Line

The budget line is the line traced by the equation f pf + cpc = b 1

To derive the budget line ask yourself how many units of clothes would the person want if they spent all their budget on clothes? It would be pbc , that would correspond to a point along the horizontal axis. How many units of food would they be able to afford if they spent all their money on food? It would be pbf which would be a point on the vertical axis. The slope of the line would be according to the slope formula: For the two (x,y) pairs, (x1 , y1 ), (x2 , y2 ), then the slope of the line connecting these points is given by the equation: s (slope) =

y1 −y2 x1 −x2

So in our case the slope is ppfc . This slope has a name it is called the opportunity cost. In order to consume more cloths you need to give up food and vice versa. This is the true economic cost of these goods.

1.3

Changes in the Budget Line

What happens if we have an increase in income? The Budget line will completely shift right. What happens with changes in prices? For example if clothing became cheaper the line would become less steep. If prices on food decreased the line would become more steep.

1.4

Numeraire

The numeraire is a method of evaluating changes in the variables in the budget constraint relative to either one of the prices or the budget. If we divided the budget constraint by b the budget we would have: f

pf pc +c =1 b b

or one of the prices: f

pf b +c= pc pc

Now we have a way of comparing the other factors, price on food and the budget to the price on clothes, something we will find useful later. Likewise, with the budget we can compare changes in the prices to a base budget of 1.

1.5

Taxes, Subsidies, and Rationing

There are different kinds of taxes: • Quantity tax, e.g. gasoline a certain amount of cents per gallon. • Value tax, ad valorem taxes e.g., state sales tax. • Lump sum tax, e.g., government takes a fixed amount no matter what. Rationing- Consumption is fixed to be no larger than some set amount. A subsidy is the opposite of a tax, the government gives some amount to the consumer. A quantity subsidy is such a subsidy that the government gives an amount corresponding to the quantity purchased. How do taxes and subsidies impact prices? The sales tax from the consumer’s eyes raises the price by a certain percentage, so the price increases. A subsidy decreases the price in the consumers eyes to the price

2

decreases. What happens with rationing? Part of the budget line is loped off. What happens with a combination? What if the consumer was allowed unrestricted consumption of a certain good, say clothes but then had to pay a tax on all consumption beyond a certain point? There would be a kink in the graph.

1.6

Food Stamps in 1975

In Jan. 1975 families could receive food coupons by participating in the Food Stamp Program according to the Food Stamp Act of 1964. This is a subsidy on food. In those days the price of food stamp coupons to a family of four was $25 for coupons valuing $153. So for each dollar spent you would receive (153/25) $6.12 worth of food. This program had an impact on the budget set. Let’s let the price of each good, other goods, or a composite good equal 1 and food equal 1 because we are measuring only the money spent upon it. This budget line has a slope of -1 (− pp12 ). Like we have defined before, a subsidy reduces the price to the consumer by (price - subsidy). In this case the subsidy is equal to a discount on the food equaling 1 - 25/153 or 84%. 25 is the amount paid by the consumer. The 153 is given to the consumer by the government. 25/153 represents about 16% that is paid by the consumer, the rest is subsidized. Therefore the price of goods to the consumer is .16. This makes the . slope of the line − pp21 = − 25/153 1 This will cause a kink in the budget line up to the dollar amount of $153. Afterwards the person subsidized will face food prices exactly like everyone else and the slope will be -1 again.

1.7

Examples

Changes in the budget line. Say we have prices and budget at p1 x1 + p2 x2 = b What happens if p1 doubles and p2 gets 8 times larger and the income becomes 4 times larger? Write a new equation. What happens if p2 increases but p1 and income remain unchanged? Suppose the government puts a tax of 15 cents a gallon on gas and then later decides to put a subsidy on gas at a rate of 7 cents a gallon. What net tax is this combo equivalent to?

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Unit 2: Budget Constraints

Value tax, ad valorem taxes e.g., state sales tax. • Lump sum tax, e.g., government takes a fixed amount no matter what. Rationing- Consumption is fixed to be no ...

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