# 2.6: Subsidies

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The policy objective of a subsidy is to help producers, or encourage the use of a good. The seller’s price is higher than the buyer’s price by the amount of the subsidy $$(s)$$.

$P_s = P_b + s\nonumber$

The subsidy is the vertical distance between the seller’s price and the buyer’s price, as shown in Figure $$\PageIndex{1}$$.

## Welfare Analysis of a Subsidy

The welfare analysis of the subsidy compares the initial market equilibrium with the post-subsidy equilibrium.

$ΔCS = + C + D + E,\nonumber$

$ΔPS = + A + B,\nonumber$

$ΔG = – A – B – C – D – E – F,\nonumber$

$ΔSW = – F,\nonumber$

and

$DWL = F.\nonumber$

Both consumers and producers gain from the subsidy, but at a large cost to tax payers (the government).

## Quantitative Welfare Analysis of a Subsidy

Suppose that the inverse demand and supply of corn are given by:

$P_b = 12 – 2Q^d,\nonumber$

and

$P_s = 2 + 2Q^s,\nonumber$

Where $$P$$ is the price of corn in USD/bu, and $$Q$$ is the quantity of corn in billion bushels. Market equilibrium is found where supply equals demand: $$Q^e = 2.5$$ billion bu of corn and $$P^e = P_b = P_s = 7$$ USD/bu of corn (Figure $$\PageIndex{2}$$).

With the subsidy, the price relationship is given by:

$P_s = P_b + s.\nonumber$

Assume that the government sets the corn subsidy equal to 2 USD/bu. Substitution of the inverse supply and demand equations into the price equation yields:

$2 + 2Q^s = 12 – 2Q^d + 2\nonumber$

Since $$Q^d = Q^s = Q’$$ after the tax:

\begin{align*} 4Q’ &= 12\\[4pt] Q’ &= 3 \text{ billion bushels of corn.}\end{align*}

The quantity can be substituted into the inverse supply and demand equations to find the buyer’s and seller’s prices.

$P_b = 6 \text{ USD/bu},\nonumber$

and

$P_s = 8 \text{ USD/bu}.\nonumber$

These prices are shown in Figure $$\PageIndex{1}$$. The welfare analysis is:

\begin{align*}ΔCS &= + C + D + E = + 2.75 \text{ USD billion}\\[4pt] ΔPS &= + A + B = + 2.75 \text{ USD billion}\\[4pt] ΔG &= – A – B – C – D – E – F = – 6 \text{ USD billion}\\[4pt] ΔSW &= – F = – 0.5\text{ USD billion}\\[4pt] DWL &= F = + 0.5 \text{ USD billion}\end{align*}

Note again that the change in social welfare equals the sum of the welfare changes due to the tax: $$ΔSW = ΔCS + ΔPS + ΔG$$. Although the deadweight loss is not large, the government cost is large, making subsidies effective in helping producers and encouraging consumption of the good, but expensive for society.

This page titled 2.6: Subsidies is shared under a CC BY-NC license and was authored, remixed, and/or curated by Andrew Barkley (New Prairie Press/Kansas State University Libraries) .