Suppose that a firm's production function is given by: q = 10K0

As given, the cost curves will be derived when the value of K and L are derived. The equilibrium values of K and L are derived when we minimize the cost with the given production function.

MinTC=wL+rK(Given)q=10K0.5L0.5(Lagrange)l=10L+20K+λ{q−10K0.5L0.5}dldK=20−λ[10(0.5)L0.5K0.5]MinTC=wL+rK(Given)q=10K0.5L0.5(Lagrange)l=10L+20K+λ{q−10K0.5L0.5}dldK=20−λ[10(0.5)L0.5K0.5]

dldL=10−λ[10(0.5)K0.5L0.5]dldL,dldK=010[10(0.5)K0.5L0.5]=20[10(0.5)L0.5K0.5]K=12LdldL=10−λ[10(0.5)K0.5L0.5]dldL,dldK=010[10(0.5)K0.5L0.5]=20[10(0.5)L0.5K0.5]K=12L

Putting the values of K and L in the production function,

q=10K0.5L0.5q=10(12L)0.5L0.5q=10√2LL=√210qq=10K0.5L0.5q=10(12L)0.5L0.5q=102LL=210q

Keeping K Fixed

TC=10(√210K)+20¯¯¯¯¯KTC=√2q+20¯¯¯¯¯KMC=dTCdqMC=√2TC=10(210K)+20K¯TC=2q+20K¯MC=dTCdqMC=2

TVC=√2qAVC=qAC=TCqAC=√2+20¯¯¯¯¯Kq

please see the attached file for the complete solution.