Leibniz

4.4.1 Altruistic preferences: Finding the optimal distribution

Anil has won the lottery and must decide what to do with his 10,000 rupees. He has altruistic preferences: while he is pleased to receive the money, he also cares about his neighbour Bala who did not win anything. We can use the technique of constrained optimization to model his decision.

In Leibniz 3.5.1 we used constrained optimization to solve Alexei’s problem: choosing daily hours of free time $t$ and his exam grade $y$ so as to maximize his utility $U(t,\ y)$, subject to what is feasible given his exam grade production function. Alexei’s problem can be expressed as:

$$\text{choose } t \text{ and } y \text{ to maximize } U(t,\ y) \text{ subject to the constraint } y=f(24- t)$$

where $f$ is his production function.

At Alexei’s optimal point, the rate at which he is able to exchange hours of free time for grade points is equal to the rate at which he is willing to exchange them. In other words, the marginal rate of transformation (MRT) equals the marginal rate of substitution (MRS). We referred to this as Alexei’s first-order condition for optimization.

Anil’s problem can be written very similarly. He also wants to maximize his utility, which depends on two goods, money for himself and money for Bala. And he is faced by a constraint: he has only 10,000 rupees to divide between himself and his neighbour. If we denote Anil’s money by $x$, Bala’s by $y$ and Anil’s utility function by $U(x,\ y)$, then Anil’s problem is to:

$$\text{choose } x \text{ and } y \text{ to maximize }U(x,\ y) \text{ subject to the constraint } x + y = 10,000$$

The equation $x + y = 10,000$ describes the feasible frontier along which Anil can split his lottery prize if none of the money is lost or taxed.

We have seen two methods for solving constrained optimization problems (see Leibniz 3.5.1). One is the substitution method, in which we start by substituting from the constraint into the objective function. The other method, which we use here, is to apply the first-order condition:

$$\text{MRT} = \text{MRS}$$

You can see this in Figure 4.5 of the text, reproduced as Figure 1: the optimal allocation $B$ lies at the point of tangency of Anil’s indifference curve and the constraint (feasible frontier).

If we knew Anil’s preferences (his utility function), we could solve the constrained optimization problem to determine the point $B$ precisely. Let’s suppose he has a Cobb-Douglas utility function of the same form as Alexei’s in Leibniz 3.5.1:

$$U(x,\ y) = x^\alpha y^\beta$$

where $\alpha$ and $\beta$ are positive constants. Anil’s marginal utilities are found as usual by partial differentiation:

$$\frac{\partial U}{\partial x} = \alpha x^{\alpha -1} y^\beta = \frac{\alpha U}{x}, \quad \frac{\partial U}{\partial y} = \beta x^\alpha y^{\beta -1} = \frac{\beta U}{y}$$

His marginal rate of substitution (the absolute value of the slope of the indifference curve) is the ratio of the marginal utilities:

$$\text{MRS }= \left.\frac{\partial U}{\partial x}\middle/ \frac{\partial U}{\partial y} \right. = \left. \frac{\alpha U}{x} \middle/ \frac{\beta U}{y}\right. = \frac{\alpha y}{\beta x}$$

The marginal rate of transformation is the absolute value of the slope of the feasible frontier, $x + y = 10,000$. Writing this as $y = 10,000 - x$, we see that the slope is $-1$, so:

$$\text{MRT} = 1$$

In other words, Anil can transform his money into money for Bala at the rate of one for one. Equating the MRS with the MRT gives us the first-order condition for Anil’s optimal choice:

$$\alpha y = \beta x$$

There are infinitely many points in the $xy$-plane satisfying this condition—all the points where the slope of the indifference curve is $-1$, which lie on a straight line through the origin. But we want the one on the feasible frontier. Thus Anil’s optimum point is found by solving the pair of simultaneous equations:

$$x+y=10,000, \quad \beta x - \alpha y = 0$$

You can check (for example, by using the first equation to substitute for $y$ in the second) that the solution is:

$$x = \frac{10,000 \, \alpha}{\alpha +\beta}, \quad y = \frac{10,000 \, \beta}{\alpha +\beta}$$

For example, if Anil’s preferences are such that $\alpha =0.7$ and $\beta=0.3$, these expressions reduce to $x =7,000$ and $y=3,000$ as in the text: Anil gives 3,000 rupees to his neighbour Bala, and keeps 7,000 rupees for himself.

We can write the solution to Anil’s problem in terms of the shares of the lottery prize that should go to Anil and Bala respectively:

$$\frac{x}{10,000} = \frac{\alpha}{\alpha +\beta}, \quad \frac{y}{10,000} = \frac{\beta}{\alpha +\beta}$$

A quick check of the algebra above should convince you that the optimal shares remain $\alpha/(\alpha +\beta)$ and $\beta/(\alpha +\beta)$ if $10,000$ is replaced by any other positive number: the proportions of the prize allocated to Anil and Bala are independent of its size. Notice also that this answer enables us to do something we have not done up to now, namely to give some interpretation of the parameters $\alpha$ and $\beta$ of Anil’s utility function. The higher $\alpha$ is relative to $\beta$, the more Anil cares about his own money relative to Bala’s.

Another property that you can observe here is that only the ratio of $\alpha$ to $\beta$ is relevant for Anil’s optimal choice. He would make the same choice if $\alpha =70$ and $\beta=30$ because his indifference curves would have exactly the same shapes, although the scale on which utility is measured would be different.

These features of the solution are consequences of Anil having a Cobb-Douglas utility function. With a different type of utility function he might split the money in different proportions depending on the size of the prize.

Read more: Sections 15.1, 17.1, 17.3 of Malcolm Pemberton and Nicholas Rau. 2015. Mathematics for economists: An introductory textbook, 4th ed. Manchester: Manchester University Press.