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[sohini]

There are 100 competitive firms in an industry, each with a short run cost curve C(q)=q^2+4q+K, where K is a positive constant. The industry dd curve is q=100-10p. Find the short run equilibrium for this industry. What limit must K satisfy for the firms to survive, all other things remaining the same?  

[Khushboo]

In a competitive firm, P=MC
MC= 2q+4
P= 2q+4 is the firm supply curve
Hence the industry supply curve is Q= 100q = 100/2 (P-4) = 50P-200.
Industry demand is Q=100-10P
By equating demand and supply we get, P=5. So Q=50 and q=1/2

For the firms to survive, TR - TC >_ 0
                                5/2 - [ (1/2)(1/2) + 4(1/2) +K ] >_ 0
                              so,  K <_ 1/4

[sohini]

you have taken 2q+4=MC as the ss curve, this means the entire MC is the ss curve, why is it so? as the ss curve lies above the min AVC

[PRANAV JAIN]

FOR THE FIRM TO SURVIVE,AC<_P AND NOT THE ONE KHUSBU SAID...

[Amit Goyal]

Hi Pranav, AC ≤ P (what you said) is equivalent to TC ≤ TR (what Khushboo said).

[priyanka p]

 Suppose that prices of all variable factors and output double.what will be the effect on short run equilibrium output of the competitive firm? examine whether short run profit of firm would double?

[Amit Goyal]

Hi Priyanka,

Short-run Objective of the firm can be written as:
Max (with respect to L) pf(L, K) -  wL - rK
where f(L, K) is the production function, L and K are variable and fixed inputs, respectively.
First order condition can be used to solve the problem (provided production function exhibits decreasing marginal returns to variable input L)
FOC is p(∂f/∂L) = w, we can solve this for optimal L.
Notice that when p and w doubles the FOC is still the same:
FOC is 2p(∂f/∂L) = 2w is equivalent to p(∂f/∂L) = w and hence the optimal L is same as above.
Now K is unchanged and from our analysis we figured out that L is also unchanged, hence optimal output will remain the same. Let Q* be the optimal output and L* be the equilibrium employment. Then Short run profit (gross of fixed cost) = pQ* - wL* for the former case and 2pQ* - 2wL* for the latter.
Hence, Optimal short-run profit (gross of fixed cost) will double.
Short-run profit (net of fixed cost) will more than double:
2pQ* - 2wL* - rK > 2pQ* - 2wL* - 2rK = 2(pQ* - wL* - rK)