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:

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.