Weird Leontief production function I am solving some micro related exercises and I came across this weird Leontief production function: $$Q =\left(\min\{K, L\} \right)^b$$  

I am not sure how to solve it. I have to find the inputs' demand, cost function etc.... Should I solve it as $K^b=L^B=Q$ and then proceed as usual with the standard Leontief?
This is not a weird case, but a Leontief production function which is not homogeneous of degree one, but homogeneous of degree $b$. You can see this if you use the connection between a C.E.S. production function and the Leontief one.

Consider 
$$Q_b=[a K^{-\rho} +(1-a) L^{-\rho}  ]^{-\frac{b}{\rho}},\;\; b&gt;0$$

$$\Rightarrow Q_b = \frac 1{[a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{b}{\rho}}}$$

Take the limit when $\rho \rightarrow \infty$. Since we are interested in the limit when $\rho\rightarrow \infty$ we can ignore the interval for which $\rho \leq0$, and treat $\rho$ as strictly positive.

Without loss of generality, assume $K\geq L \Rightarrow (1/K^{\rho})\leq (1/L^{\rho})$. We also have $K, L &gt;0$. Then we verify that the following inequality holds:

$$(1-a)^{b/\rho}(1/L^{b})\leq  Q_b^{-1} \leq (1/L^{b}) $$

$$\rightarrow (1-a)^{b/\rho}(1/L^{b})\leq  [a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{b}{\rho}} \leq (1/L^{b}) \tag{1}$$

by raising throughout to the $\rho/b$ power to get

$$(1-a)(1/L^{\rho}) \leq a (1/K^{\rho}) +(1-a) (1/L^{\rho})  \leq (1/L^{\rho}) \tag {2}$$
which indeed holds, obviously, given the assumptions. Then go back to the first row of $(1)$ and

$$\lim_{\rho\rightarrow \infty} (1-a)^{b/\rho}(1/L^{b}) =(1/L^{b})$$

which sandwiches the middle term in $(1)$ to $(1/L^{b})$ , so 

$$\lim_{\rho\rightarrow \infty}Q_b = \frac {1}{1/L^b} = L^b = \big[\min\{K,L\}\big]^{b} \tag{3}$$

For more, see this answer.