Question

The marginal cost of producing a cherry soda is $2 per pack. The cost function of this bottling plant is expressed as C(q)=2q. The competition produces apple soda, and their cost function for their bottling plant is C(q)=5q. a. Does the apple soda bottling plant have a higher marginal cost than the bottling plant for Cherry soda? What is the intuition for this answer? b. We did account for fixed costs of production. The true cost function of cherry soda is C(q)=400+2q. Is the elasticity of cherry soda increasing or decreasing in quantity (q)?

Answer 1

Answer

a. Yes, the apple soda bottling plant have a higher marginal cost than the bottling plant for cherry soda.

In the competitive market, the existing firms always enjoy the cost advantage over the newly entrant firms. In the example, the existing firm is cherry soda bottling plant. To compete with cherry soda, the apple soda plant enters the market. As a new producer, its cost is higher than the cherry soda plant. So, the apple soda bottling plant have a higher marginal cost than the bottling plant for cherry soda.

The cost function of cherry soda bottling plant is;  C(q) = 2q

$\therefore$Marginal cost(MC) of cherry soda bottling plant = dC(q) / dq = 2 , i.e., $2

The cost function of apple soda bottling plant is; C(q) = 5q

$\therefore$Marginal cost(MC) of apple soda bottling plant = dC(q) / dq = 5 , i.e., $5

So, we see that MC of apple soda bottling plant is higher than the MC of cherry soda bottling plant.

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b. The cost elasticity of cherry soda is decreasing in quantity.

The actual cost function of cherry soda is, C(q) or C = 400 + 2q ..... (1)

The output(q) elasticity of cost function shows the responsiveness of the change in total cost(C) due to change in output. It is measured as the ratio of the percentage change in total cost and percentage change in output.

$\therefore$ Cost elasticity =($\Delta$C/C) / ($\Delta$q / q)

Or, Cost elasticity = ($\Delta$C/$\Delta$q) * (q/C) , where '$\Delta$' stands for 'change'

Let us differentiate equation(1) with respect to q,

dC(q) / dq = 0 + 2*(dq / dq) ; [ the fixed cost doesn't change with the change in output. Here fixed cost is $400]

Or, dC(q) / dq = 2

Or, $\Delta$C / $\Delta$q = 2 ; where 'dC(q)' = $\Delta$C , 'dq' = $\Delta$q

$\therefore$ Cost elasticity = 2 *(q / 400 + 2q) ; [as , C = 400 + 2q]

Or, Cost elasticity = 2q / (400 + 2q)

Or, Cost elasticity = 2q / 2(200 + q)

Or, Cost elasticity = q /(200 + q) $<$1 ; [as, 200 + q $>$ q ;  $\therefore$ q /(200 + q) $<$1]

From the above cost elasticity, we see that the percentage change in total cost is less than the percentage change in output. So, the cost elasticity of cherry soda is decreasing in quantity; and the firm experiences economies of scale.

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