In: Economics

- A firm has the production function Q = 4LK. The marginal products are given by MPL = 4K and MPK = 4L. Suppose that w = r = 25. Write out the expression for the long run total cost curve. Does this firm experience economies or diseconomies of scale? Does this firm experience increasing, decreasing or constant returns to scale?

**TC = 25Q ^{0.5} ;
Economies of Scale; IRS**

Solution:

We are given the following:

Production function: Q = 4*L*K ; marginal product of labor, MPL = 4K ; marginal product of capital, MPK = 4L

With wage rate, w = $25 and rental rate, r = $25

Total cost = wage rate*labor quantity + rental rate*capital quantity

TC = w*L + r*K

The optimal cost function can be found by tangency condition that is where slope of iso-cost function (w/r) equals the slope of iso-quant function (marginal rate of technical substitution, MRTS = MPL/MPK

MPL/MPK = 4K/(4L) = (K/L)

So, it must be: K/L = 25/25 which gievs us, K/L = 1 or K = L

Then, substituting this in the production function, we get: Q = 4*L*L

So, Q = 4*L2

We get L = (Q/4)1/2 = Q1/2/2

So, K (= L) = Q1/2/2

We get the total cost function as: TC = w*L + r*K

TC(Q) = 25*Q1/2/2 + 25*Q1/2/2

TC(Q) = 25*Q1/2

Economies of scale and diseconomies of scale can be found by long run average cost: If the long run average cost decreases with increase in quantity, then the firm experiences economies of scale. Similarly, if the long run average cost increases with increase in quantity, then the firm experiences diseconomies of scale.

Average total cost, ATC(Q) = TC(Q)/Q = 25/Q1/2

Clearly, ATC has quantity in denominator, meaning ATC decreases with increasing quantity. So, this firm experiences economies of scale.

Lastly, we can find returns to scale as follows:

If changing all inputs by a common factor changes the total output by a greater factor, same factor, and lower factor, then firm experiences an increasing returns to scale, constant returns to scale, and decreasing returns to scale, respectively.

So, production function: Q(K, L) = 4*L*K

Q(tK, tL) = 4*tL*tK = t2*(4LK) = t2Q(L, K)

Thus, we can say that the firm experiences increasing returns to scale.

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