Mathematical derivation of the Production Possibility Frontier

Question

What are the mathematical basics of production possibility frontier? How can I derivate it? Can I have an example for it?

Answer 1

The question is broad, but I believe there is plenty of literature that defines this concept in similarly broad terms. The following is adapted from the Wikipedia on Pareto Efficiency, which is the mathematical basis of the Production Possibilities Frontier.

There may be better definitions out there, but this one should probably work in a lot of cases:

The Production Possibilities Frontier, $P(Y)$, may be more formally described as follows. Consider a system with function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, where $X$ is a compact space of feasible decisions (including allocations of time and endowment goods) in the metric space $\mathbb{R}^n$, and $Y$ is the feasible set of criterion vectors (say, final goods and services) in $\mathbb{R}^m$, such that $Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}$.

We assume that the preferred directions of criteria values are known so that more of any good in $Y$ is better. A point $y^{\prime\prime} \in \mathbb{R}^m$ strictly dominates another point $y^{\prime} \in \mathbb{R}^m$, written as $y^{\prime\prime} > y^{\prime}$, means that for each element index $i$, $y''_i \geq y'_i$ and there is at least one element $j$ such that $y_j'' > y_j'$. The Pareto frontier is thus written as:

$P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} > y^\prime, y^{\prime\prime} \neq y^\prime \; \} = \emptyset \}. $

Answer 2

Notice that it is not always possible to find a algebraic expression for the PPF.

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A context where the PPF is usually found is in a 2x2 model, where there are two sectors or goods ($x$ and $y$) and two factors of production ($K$ and $L$). The shape of the PPF depends on the relative intensities in which each sector/good uses those factors.

For example, assume each sector $j$ has a CRS production function $F_j(K_j,L_j)$, with different technological parameters. Let us also assume a fixed endowment of factors of production, complete factor mobility across sectors, and competitive markets, such that payment of factors of production freely adjust in order to allocate factors into sectors. Also, assume the price of the goods fixed, e.g. as if the economy is small and open to international markets.

For a given set of relative factor prices, $r/w$, common to both sectors, optimal relative factors in each sector are given by:

$$ \dfrac{\frac{\partial F_x}{\partial K_x}}{\frac{\partial F_x}{\partial L_x}} = \frac{r}{w} = \dfrac{\frac{\partial F_y}{\partial K_y}}{\frac{\partial F_y}{\partial L_y}} $$

The first fraction above is a function of $\frac{K_x}{L_x}$, whereas the last fraction is a function of $\frac{K_y}{L_y}$. Since by definition $L_x+L_y=L$ and $K_x+K_y=K$, we have a system of three equations with four unknowns. This means that we can reduce the system to one equation with two unknowns. Whether this equation yields a closed form solution for on variable in terms of the other (e.g. $L_x=f(K_x$) depends on the functional form of $F_j$. Apparently, this is possible only in the case of trivial production functions like Cobb-Douglas or Leontief (for an example of the former, see here).

Assuming we can find such equation $L_x=f(K_x$), the algebraic solution to the PPF follows immediately. This is because you have reduced all four endowment variable sin terms of just one (for instance, $K_x$). Therefore, to find the PPF you can evaluate $F_x$ and $F_y$ for every possible value of $K_x$, and draw the map. Alternatively, you can solve the homogeneous equation $x-F_x(K_x)=0$ for $K_x$ as a function of $x$, and then replace this $K_x^*(x)$ into $F_y$, from where you get $y=f(x)$, having in mind the restrictions on the variable space (i.e. $x,y>0$). Notice again that solving the above homogeneous equation is not always possible. For an example where it is possible, see link above. The PPF might look like:

To close the model, the way that the actual equilibrium is found depend on international prices, where the MRS is tangent to the PPF. Alternatively, in a closed economy, the isoquant of consumer preferences provides the MRS.

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To complement @jmbejara comment about the relation between the PPF and Pareto optimality, notice that the condition that MRTS is equal to relative factor prices is exactly the definition of Pareto optimality in a production context. We can see this in the Edgeworth box. For standard production functions, the isoquants for each good are convex functions in the $\{K,L\}$ space. We can see the Edgeworth box of this economy:

The example above is clearly inefficient, as relative wages can adjust until the MRTS is the same across goods. In effect, the optimal allocation is:

Answer 3

Assume there are only two goods. The price of Y defined by X amount and the price of Y is defined by X respectively. Then by assuming a fixed amount of production resources (for example labor and capital), the PPF could be obtained. In this context, the cost functions should be defined as a function of other commodities. Here we could use X as a base commodity, overall production possibility function could be written as $ Q=c(x,y) $ and $ y=f(x) $ then, a total differentiation could solve the problem. $$ dQ=MC_x dx + MC_y { \frac{dy}{dx}} dx $$ $dQ=0$ therefore, $$ \frac{dy}{dx} =- \frac{MC_x}{MC_y} $$

Details are here .