You are the owner of a competitive firm. Prices of labor and capital are \$1 each. The production function is $$F(L,K)=L^{0.25}k^{0.25}$$
- calculate how many units of K and L you need to produce 100 units of product optimally (i.e. while minimizing costs), and what is your expenditure on inputs
- Suppose now that your (total) cost function is given by c(q)=2q (where q is quantity produced) and the demand function is p=14-3q. Calculate marginal cost,the optimal quantity produced, and profits.
For #1 I said K=L. L*K = 100 so L and K = 10/each. Is this right?
For #2 I calculated 14-3q = 2. q = 4. Then marginal cost = 14-3(4) = 2. Is this also right?
Any help would be appreciated. Thanks in advance.
Answer for $1.$
The cost function is $ C(K,L)=K+L$. This has to be minimized under the constraint $100=L^{0.25}\cdot K ^{0.25}$
Solving the constraint for $K$
$K ^{0.25}=\frac{100}{L^{0.25}}$
$K=\frac{100^4}{L}$
Inserting the term for K in $C(K,L)$
$C(L)=\frac{100^4}{L}+L$
Derivative w.r.t L
$\frac{dC}{dL}=-\frac{100^4}{L^2}+1=0 \qquad |+\frac{100^4}{L^2}$
$1=\frac{100^4}{L^2} \qquad |\cdot L^2$
$L^2=100^4\Rightarrow L^*=100^2=10,000$ since $L\geq 0$
Inserting the value of L into the constraint to calculate K
$100=\left(100^2 \right)^{0.25}\cdot K ^{0.25}$
The solution is $K^*=100^2=10,000$