Markets under Perfect Competition

Aggregate Demand

Consider the market for a single commodity, the demand of commodity comes from various consumers. Each consumer will decide his demand $D_i(i=1,\dots n)$ according to his utility maximizing condition. So if $P$ is the price of the commodity. Then demand, $D_i=D_i(P)$ is a function of price. So, aggregate demand is a sum of all the individual demands.

$$D=\sum _{i=1}^n{D}_i(P)$$

Aggregate Supply

The supply of a commodity comes from individual firms producing their commodity. Suppose there are $m$ firms in the industry, the supply comes from each of these firms, is according to its profit maximizing condition, $M{C}_i=P$¹. If price is more than the minimum average variable cost AVC, then only the firm will supply the quantity $S_i$², which is a solution of $M{C}_i=P$ otherwise the firm is not going to supply anything, since the firm will not be able to meet the AVC.

Suppose there are 100 firms in the industry, the cost function $C_i=0.1q_i^3-2q_i^2+15q_i+10$. Find the individual supply and the market supply.

• Know that $0.1q_i^3-2q_i^2+15q_i$ is the AVC and $10$ is the fixed cost.
• $M{C}_i$ = $0.3q_i^2-4q_i+15$.
• So each of the firms will try to maximize their profits by deciding their supply using $M{C}_i=P$.

We first derive the supply function of this individual firm. We get a quadratic equation with a = 0.3, b = -4, c = (15 - P). On solving which we get the roots as,

$$\begin{array}{rl}{q}_i& =\frac{4\pm \sqrt{16-4\times 0.3\times (15-P)}}{0.6}\\ q_i& =\frac{4\pm \sqrt{16-1.2(15-P)}}{0.6}\\ q_i& =\boxed{\frac{4\pm \sqrt{1.2P-2}}{0.6}}\end{array}$$

16 - 1.2*15

Now we find the $q_i$ that minimizes the AVC Now, we would like to find the optimal value of AVC, which is $\frac{VC}{q}=\frac{\partial \text{AVC}}{\partial q_i}$.

$$\begin{array}{rl}\frac{\partial \text{AVC}}{\partial q_i}& =\frac{\partial (0.1q_i^2-2q_i+15)}{\partial q_i}\\ & =0.2q_i-2=0\\ & \\ ⟹q_i& =10\end{array}$$

Footnotes

1. Transclude of Market-under-Perfect-Competition-2024-09-30-13.49.53.excalidraw

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2. $i^{th}$ firm supplies $S_i$ ↩