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 = 123 - 47 Q_d` and the supply is `P = 13 + 63 Q_s`.

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

Step 1: Find the equilibrium price and quantity

$$ \begin{align*} 13 + 63 Q &= 123 - 47 Q \\ 47 Q + 63 Q &= 123 - 13 \\ (47 + 63) Q &= 123 - 13 \\ Q &= \frac{123 + 13}{47 + 63} \\ Q &= 1.0 \end{align*} $$ Therefore $$ P = 13 + 63 Q = 13 + 63 \times 1.0 = 76.0 $$

Step 2: Draw the graph

Step 3: Calculate the Consumer Surplus.

`\text{CS} = \frac{(123 - 76.0) \times 1.0}{2} = 23.5`.

Step 4: Calculate the Producer Surplus.

`\text{PS} = \frac{(76.0 - 13) \times 1.0}{2} = 31.5`.

Step 5: Calculate the Total Surplus.

`\text{TS} = \text{CS} + \text{PS} = 23.5 + 31.5 = 55.0`.

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