# Understanding Subsidy Benefit, Cost, and Market Effect

## Mathematically speaking, a subsidy functions like a negative tax

Most of us know that a per-unit tax is an amount of money that the government takes from either producers or consumers for each unit of goods that is bought and sold. A per-unit subsidy, on the other hand, is an amount of money that the government pays to either producers or consumers for each unit of goods that is bought and sold. Mathematically speaking, a subsidy functions like a negative tax.

When a subsidy is in place, the total amount of money that the producer receives for selling goods is equal to the amount that the consumer pays plus the amount of the subsidy. Alternatively, one can say that the amount a consumer pays goods is equal to the amount that the producer receives minus the amount of the subsidy.

Here's how a subsidy affects market equilibrium:

## Market Equilibrium Definition and Equations

First, what is market equilibrium? Market equilibrium occurs when the quantity supplied of a good in a market (Qs in the equation here) is equal to the quantity demanded in a market (QD in the equation).

These equations provide enough information to locate the market equilibrium induced by a subsidy on a graph.

## Market Equilibrium With a Subsidy

To find the market equilibrium when a subsidy is put in place, a couple of things must be kept in mind.

First, the demand curve is a function of the price that the consumer pays out of pocket for a good (Pc), since this out-of-pocket cost influences consumers' consumption decisions.

Second, the supply curve is a function of the price that the producer receives for a good (Pp) since this amount affects a producer's production incentives.​

Since quantity supplied is equal to quantity demanded in a market equilibrium, the equilibrium under the subsidy can be found by locating the quantity where the vertical distance between the supply curve and the demand curve is equal to the amount of the subsidy. More specifically, the equilibrium with the subsidy is at the quantity where the corresponding price to the producer (given by the supply curve) is equal to the price that the consumer pays (given by the demand curve) plus the amount of the subsidy.

Because of the shape of the supply and demand curves, this quantity is going to be greater than the equilibrium quantity that prevailed without the subsidy. We can, therefore, conclude that subsidies increase the quantity bought and sold in a market.

## Welfare Impact of a Subsidy

When considering the economic impact of a subsidy, it's important not only to think about the effect on market prices and quantities but also to consider the direct effect on the welfare of consumers and producers in the market.

To do this, consider the regions on this diagram labeled A-H. In a free market, regions A and B together comprise consumer surplus, since they represent the extra benefits that consumers in a market receive from a good above and beyond the price that they pay for it.

Regions C and D together comprise producer surplus since they represent the extra benefits that producers in a market receive from a good above and beyond their marginal cost.

Together, the total surplus, or total economic value created by this market (sometimes referred to as social surplus), is equal to A + B + C + D.

## Consumer Impact of a Subsidy

When a subsidy is put in place, the consumer and producer surplus calculations get a bit more complicated, but the same rules apply.

Consumers get the area above the price that they pay (Pc) and below their valuation (which is given by the demand curve) for all the units that they buy in the market. This area is given by A + B + C + F + G on this diagram.

Therefore, consumers are made better off by the subsidy.

## Producer Impact of a Subsidy

Similarly, producers get the area between the price that they receive (Pp) and above their cost (which is given by the supply curve) for all the units that they sell in the market. This area is given by B + C + D + E on the diagram. Therefore, producers are made better off by the subsidy.

In general, consumers and producers share the benefits of a subsidy regardless of whether a subsidy is directly given to producers or consumers. In other words, a subsidy given directly to consumers is unlikely to all go to benefit consumers, and a subsidy given directly to producers is unlikely to all go to benefit producers.

Which party benefits more from a subsidy is determined by the relative elasticity of producers and consumers, with the more inelastic party seeing more of the benefit.

## Cost of a Subsidy

When a subsidy is put in place, it's important to consider not only the impact of the subsidy on consumers and producers but also the amount that the subsidy costs the government and, ultimately, taxpayers.

If the government provides a subsidy of S on each unit bought and sold, the total cost of the subsidy is equal to S times the equilibrium quantity in the market when the subsidy is put in place, as given by this equation.

## Graph of Cost of a Subsidy

Graphically, the total cost of the subsidy can be represented by a rectangle that has a height equal to the per-unit amount of the subsidy (S) and a width equal to the equilibrium quantity bought and sold under the subsidy. Such a rectangle is shown in this diagram and can also be represented by B + C + E + F + G + H.

Since revenue represents money that comes into an organization, it makes sense to think of money that an organization pays out as negative revenue. Revenue that a government collects from a tax is counted as a positive surplus, so it follows that costs that a government pays out via a subsidy are counted as negative surplus. As a result, the "government revenue" component of total surplus is given by -(B + C + E + F + G + H).

Adding up all the surplus components results in a total surplus under the subsidy in the amount of A + B + C + D - H.