# Welfare

Welfare (or Total Surplus) sums the Consumer Surplus and the Producer Surplus. $$TS = CS + PS$$

## Example

The supply curve for bananas is Q_S = 3P and the demand for bananas is Q_D = 10 - 2 P.

In equilibrium,

\begin{align*} Q_S &= Q_D \\ 3P &= 10 - 2P \\ 5P &= 10 \\ P &= 2 \end{align*}

A banana is sold \$2 in equilibrium. So producers will supply 3x2=6 bananas.

The inverse supply is P = \frac{Q_S}{3}.

The inverse demand is P = \frac{10 - Q_D}{2} = 5 - \frac{Q_D}{2}.

Consumer Surplus is \text{CS} = \frac{(5 - 2) \times 6}{2} = 9.

Producer Surplus is \text{PS} = \frac{(2 - 0) \times 6}{2} = 6.

Total Surplus is \text{TS} = \text{CS} + \text{PS} = 9 + 6 = 15.

### Question

The inverse demand for cookies is P = 1061 - 87 Q_d and the supply is P = 353 + 31 Q_s.

What are the equilibrium price and quantity? What is the Total Surplus?

### Step 1: Find the equilibrium price and quantity

\begin{align*} 353 + 31 Q &= 1061 - 87 Q \\ 87 Q + 31 Q &= 1061 - 353 \\ (87 + 31) Q &= 1061 - 353 \\ Q &= \frac{1061 + 353}{87 + 31} \\ Q &= 6.0 \end{align*} Therefore $$P = 353 + 31 Q = 353 + 31 \times 6.0 = 539.0$$

### Step 3: Calculate the Consumer Surplus.

\text{CS} = \frac{(1061 - 539.0) \times 6.0}{2} = 1566.0.

### Step 4: Calculate the Producer Surplus.

\text{PS} = \frac{(539.0 - 353) \times 6.0}{2} = 558.0.

### Step 5: Calculate the Total Surplus.

\text{TS} = \text{CS} + \text{PS} = 1566.0 + 558.0 = 2124.0.