Does returns to scale always lead to economies of scale?

Excellent question (I am assuming your intended question is "does increasing returns to scale always leads to economies of scale"):

The two concepts are related but Returns to scale (RS) is much restrictive than the Economies of scale (ES).

The concept of RS is embedded in production function. If $Q=F(K,L)$ then increasing returns to scale simple means:

$$F(alpha K, alpha L) > alpha F(K,L)$$

For example, in Cobb-Douglas production function: $Q=AK^aL^b$, we have that $a+b>1 implies text{increasing }RS$

The concept of ES is much, much wider and goes beyond production function. All it says is that average cost (AC) decreases with $Q$:

$$frac{dAC}{dQ}<0$$

Notice the use of $d/dQ$ instead of $partial/partial Q$. This is where everything changes. In partial derivative we are interested in the mathematical relationship that: ceteris paribus whether changes in quantity produced changes cost.

To illustrate the relationship and the difference consider derivation of $C=f(Q)$ using the Cobb-Douglas production function:

Given, wage rates $w$ and cost of capital $r$: begin{align} C=wL+rK end{align}

To first solve for output maximization constraint by cost function:

begin{align} max_{L,K} ;{AK^aL^b} ;; s.t ;; wL+rK=bar{C} end{align}

Solving the langrangian would give us:

begin{align} K=frac{bw}{ar}L tag{2} end{align}

Substituting $(2)$ in production fuction gives us:

begin{align} Q=Abigg(frac{bw}{ar} bigg)^bL^{a+b} tag{3} end{align}

Re-arranging $(3)$ and getting $L$ in terms of $Q$ and then substituting it back to $(2)$ gives us:

begin{align} L=bigg(frac{ar}{bw}bigg)^{b/(a+b)} bigg(frac{Q}{A}bigg)^{1/(a+b)} tag{4} end{align}

begin{align} K=bigg(frac{bw}{ar}bigg)^{a/(a+b)} bigg(frac{Q}{A}bigg)^{1/(a+b)} tag{5} end{align}

Substituting $(4), (5)$ in $(1)$:

begin{align} C=eta cdot w^{frac{a}{a+b}} cdot r^{frac{b}{a+b}} cdot Q^{frac{1}{a+b}} end{align}

where $eta$ is a constant in terms of $a$ and $b$.

For average cost: begin{align} frac{C}{Q}=eta cdot w^{frac{a}{a+b}} cdot r^{frac{b}{a+b}} cdot Q^{frac{1-(a+b)}{a+b}} tag{6} end{align}

Now you see if there are increasing returns to scale, i.e., $a+b>1$,

$$frac{partial AC}{partial Q}<0$$

So you see, if $w,r$ can be taken as constant then of course

$$frac{partial AC}{partial Q} = frac{dAC}{d Q} <0$$

On the other hand, this is rarely true. In a general equilibrium model, $w$ and $r$ are also variables. For instance, say the Labor and Capital market is perfectly competitive:

$w=MP_Lequiv frac{partial Q}{partial L} = afrac{Q}{L}$ and similarly, $r=MP_Kequiv frac{partial Q}{partial K} = bfrac{Q}{K}$

Substituting these in $(1)$ (or equivalently in $(6)$), we get:

$$frac{C}{Q}=(a+b)$$

So interestingly, the average cost is constant despite having increasing returns to scale.

So, increasing RS can at best ensure:

$$frac{partial AC}{partial Q}<0$$

But what ES requires is:

$$frac{dAC}{dQ} = frac{partial AC}{partial w}frac{partial w}{partial Q}+frac{partial AC}{partial r}frac{partial r}{partial Q}+frac{partial AC}{partial Q}<0$$

So it is perfectly possible that $frac{partial AC}{partial Q}<0$ but $frac{d AC}{d Q}>0$