Does a Robinson Crusoe economy have a substitution effect and an income effect?

Question

Consider a Robinson Crusoe economy. Let $c$ be consumption and let $l$ be leisure. Our utility function is defined as

$$U(c,l)$$

Geometrically, I think we can say his indifference curve should be concave up, so therefore $\frac{dc}{dl} >0$ and $\frac{d^2c}{dl^2} >0$.

Typically we have a production function $$y = f(\ell)$$ where $\ell$ is labor, $f'>0$,$f''<0$ and $c \leq y$. Also, I think $l = 1 - \ell$.

Also, note

$$MRS = MPL$$

WHY INCREASING PRODUCTIVITY MIGHT INCREASE LABOR

Suppose $f(\ell)=A \sqrt{\ell}$. Then $$MPL = f'(\ell) = \frac{A}{2\sqrt{\ell}}$$ So then if $A \rightarrow \lambda A$, then for fixed $MPL$ we must have $$MPL = \frac{\lambda A}{2\sqrt{\lambda^2 \ell}}$$

In other words, if $A$ increases, Crusoe works harder if we assume $MPL$ is fixed.

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WHY INCREASING PRODUCTIVITY MIGHT DECREASE LABOR

But increasing $A$ would also mean essentially that Crusoe gets more production per unit of labor consumed. This gives him more $c$ available. This will mean his $MU_c$ goes down.

Recall $$MRS = \frac {U_l} {U_c}$$ If we assume $MRS$ remains constant, then $MU_l$ must go down to offset this change. This implies $l$ goes up and therefore, $\ell$ goes down.

So one could also argue changing $A$ results in $\ell$ going down.

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MY QUESTION

What is the income effect and substitution effect in the case of changing $A$?

According to what my instructors have said (and I don't think they explain it well at all), the income effect is the latter change I described due to $A$ (i.e. decrease in $\ell$) whereas the substitution effect is due to the former change (increase in $\ell$).

But these don't match the income and substitution effects I learned about in my class on Marshallian and Hicksian demand functions. It seems very confused. Can someone define these explicitly or explain why my instructor has misused these two terms?

Answer 1

I don't see why equilibrium $MPL(\ell)$ would be fixed.

I think Robinson's income is measured by what he can get using his total time $t = l + \ell$.

Assume that given the original $A$ and total time $t$ his optimal choice was $(c_1,l_1)$. If the technology improves to $A'>A$ he can reach $(c_1,l_1)$
(or $U(c_1,l_1)$ if you take the Hicksian approach)
by only using total time $t' < t$.

Let us denote the optimal consumption bundle given technology $A'$ and time $t'$ by $(c_2,l_2)$. This is the intermediate step separating the income and substitution effects. The difference between $l_2$ and $l_1$ would be the substition effect. The 'income' in the two situations is considered equal as $(c_1,l_1)$
(or $U(c_1,l_1)$ if you take the Hicksian approach)
is barely feasible in both cases. Any difference in leisure consumption is due to the different transformation rate between leisure and consumption.

Let us denote the optimal consumption bundle given technology $A'$ and time $t$ by $(c_3,l_3)$. The difference between $l_3$ and $l_2$ is the income effect. The technology is identical in both cases, so any difference in consumption is due to the difference in 'income'.

Perhaps your instructors are trying to say something about the backward bending labor supply curve. It is difficult to tell from the available information.

Answer 2

Consider an increase in $A$. This is, for the same number of hours, output of production is larger.

The income effect is such that the individual redistributes resources toward the activity which became more productive. This is, the individual decides to work longer (a fall in leisure $l$). This is because the $MPL$ increased in that activity, and so the return of that activity increases. Recall that, because of the "First welfare theorem", the equilibrium in a centralised economy (like Robinson's) is the same than in a decentralised (i.e. market) economy. As such, implicitly, you can think of Robinson paying himself a wage. As $MPL$ increases, ceteris paribus, the equilibrium wage increases, and therefore he works more.

The substitution effect is such that the individual redistributes part of the windfall of greater productivity toward other activities which provide him utility (in this case, leisure). Since the optimisation of utility is a process involving "relative quantities", whenever each component faces decreasing marginal utility, the individual will benefit from "restoring the balance" by substituting resources (labour) away from the improved activity (production) towards that which did not improved (leisure).

Notice that the substitution effect depends on the nature of the utility function. If the marginal utility of their components are constant (for example, in $U = C + l$), then there is no substitution effect.

Here is an example of a utility function and production function where the two effects cancel-out:

$$U= log C - 2L^2$$

$$C=Y=AL^\alpha$$

where $L$ is labour.

You can easily show that the optimal hours supplied is $L^*=\frac{\sqrt\alpha}{2}$, which is independent of $A$. In other words, the income and substitution effect balance out.