Question

3. Show whether the cost function C (q,+q14-(qiq) exhibits both economies of scope and economies of scale b Derive the bDerive the conditions under which the cost function C(qi.q) 1+(q + g2) exhibits economies of scope. 4 Consider a market in which all firms have the following total cost function C(g)-150+q+0.5q What level of output corresponds to the minimum average total b) cost (minimum efficient scale)? If firms are perfectly competitive, what is the equation of an individual firm's supply function? Assume that market demand is given by Q 300-2p In the long run, what will be the equilibrium level of output and prices if the market is perfectly competitive? c) d) In the long run, how many firms will exit in the market? (Assume once again that market is perfectly competitive.) e Compute total industry profit in the long run. 5. Consider the production function given by Q-la-ka, α > 0, 1-labour and k = For which values of a does this production technology exhibits IRS, CRS, and DRS? 6 Consider the cost function given by Q-F + c Q, where F, c > 0. a) A what output level is the average cost minimized? b) Infer whether this technology exhibits IRS, CRS, or DRS. Explain.

Answer 1

Q.3.

a)

If we have a multi‐product product cost function C(Q1 ,Q2), there exists economies of scope if there exist some Q1 Q2 such that 1,

$\small C(Q_1, Q_2) < C(Q_1, 0) &plus; C(Q_2, 0)$

In our example,

$\small C(q_1, 0) = q_1^{\frac{1}{4}} \hspace{5em} C(0, q_2) = q_2^{\frac{1}{4}}$

and $\small C(q_1, q_2) = q_1^{\frac{1}{4}} &plus; q_2^{\frac{1}{4}} - (q_1q_2)^{\frac{1}{4}} < C(q_1,0) &plus; C(0, q_2)$

Thus, the firm has economies of scope.

To determine if the firm has economies of scale, let's calculate the Ray Average cost function.

Ray Average Costs (RAC(q)) for the composite quantity q:

$\small q: \frac{C(\lambda_1 q, \lambda_2 q)}{q}$ which considers production along rays s.t. $\small \sum_{i=1}^{n} \lambda_i = 1$

• Slope of RAC(q) curve determines economies of scale for a multiproduct firm.

$\small RAC(q): \frac{C(\lambda_1 q, \lambda_2 q)}{q}$

If $\small \lambda_1 = \lambda_2 = 0.5$

$\small RAC(q) = \frac{( q^{\frac{1}{4}} &plus;q^{\frac{1}{4}} - q^{\frac{2}{4}})}{q} = \frac{2q^{\frac{1}{4} } - q^\frac{1}{2}}{q} = 2q^{-\frac{3}{4}} - q^{-\frac{1}{2}}$

Slope of RAC(q) =

$\small \frac{\partial RAC(q)}{\partial q} = -\frac{3}{2}q^{-\frac{7}{4}} &plus; \frac{1}{2}q^{-\frac{3}{2}} < 0$

Hence the firm has economies of scale.

b)

If we have a multi‐product product cost function C(Q1 ,Q2), there exists economies of scope if there exist some Q1 Q2 such that 1,

$\small C(Q_1, Q_2) < C(Q_1, 0) &plus; C(Q_2, 0)$

$\small \Rightarrow 1 &plus; (q_1&plus;q_2)^2 < 1 &plus; q_1^2 &plus; 1 &plus; q_2^2$

$\small \Rightarrow q_1^2 &plus; 2q_1q_2 &plus; q_2^2 < q_1^2 &plus; 1 &plus; q_2^2 \Rightarrow 2q_1q_2 < 1\Rightarrow q_1q_2 < \frac{1}{2}$