Suppose a production function is q=K1/2L1/3q=K1/2L1/3 and in the short run capital (KK) is fixed at 100. If the wage is $10 and the rental rate on capital is $20, the short-run average cost is

a. SAC = 2000q+q21002000q+q2100.

In the short run, capital is fixed and labor is variable. So the total variable cost is equal to the price of labor (w) times the quantity of labor (L) while the total fixed cost is equal to the cost of capital (r) times the quantity of capital K.

TC=VC+FCTC=wL+rKTC=10L+(20)(100)TC=10L+2000TC=VC+FCTC=wL+rKTC=10L+(20)(100)TC=10L+2000

Total cost is in terms of Q (not L), so the next step is to find a way to replace L with Q. This can be done by solving the production function for L in terms of Q (keeping in mind that capital is fixed at 100 units). By substituting this into the equation above, the total cost equation can then be found.

q=K1/2L1/3q=1001/2L1/3q=10L1/3L1/3=q10L=(q10)3L=q31000soTC=10L+2000TC=10(q31000)+2000TC=q3100+2000q=K1/2L1/3q=1001/2L1/3q=10L1/3L1/3=q10L=(q10)3L=q31000soTC=10L+2000TC=10(q31000)+2000TC=q3100+2000

Finally, the average cost is equal to the total cost divided by the quantity.

AC=TCQTC=q3100+2000qAC=q2100+2000qAC=TCQTC=q3100+2000qAC=q2100+2000q

Total Cost of Production:

The total cost of production is the amount of money that must be spent in order to produce specific units of output. The total cost is equal to the fixed cost plus the variable cost.