# substitution effect

when ı study about link between leisure time and gdp my lecturer said that when people gdp as rise people want to work hard and he say that it is substitution effect. Also when ı research the substitution effect "The substitution effect is the decrease in sales for a product that can be attributed to consumers switching to cheaper alternatives when its price rises". But there are contrast between my teacher says and research. So my question what is substitution effect and what are the difference between income affect ?

• Hi @Burak. It is probably useful to evaluate matters in terms of their 'opportunity cost.' If for example, real GDP per capita increased, the opportunity cost of leisure is greater (i.e. it is costly to forego work if perceived earnings are high). In such circumstances, it might be perceived that individuals should substitute leisure for work. Yet, with consideration towards the income effect, opposite forces are in play. It may be favourable for an individual beyond a certain income threshold to substitute work for leisure. Nov 16 at 10:33

Assume that an individual can choose between consumption $$c$$ and leisure $$\ell$$. Let the wage be $$w$$ and assume that total available time is $$T$$.

Then the utility maximisation problem may look like this: $$\max u(c, \ell) \text{ s.t. } c + w \ell = w T.$$ In order to understand the budget constraint. Note that if the individual takes $$\ell$$ hours of leisure, then she works $$T - \ell$$ hours. This generates an income $$w (T - \ell)$$ that can be consumed so: $$c = w(T - \ell) \iff c + w \ell = w T.$$

If we solve the utility maximisation problem, then the optimal amount of leisure will depend on both the wage $$w$$ and the potential income $$wT$$, which we can write as: $$\ell(w, w T).$$ If the wage $$w$$ increases, this has two effects as it enters both the first and second argument of the leisure demand function. Taking derivatives, we get: $$\underbrace{\frac{\partial \ell(w, wT)}{\partial w}}_{SE} + \underbrace{\frac{\partial \ell(w, wT)}{\partial (wT)}T}_{IE}.$$ First for the substitution effect (SE), increasing $$w$$ makes leisure more expensive compared to consumption. If leisure is a normal good, this means that if $$w$$ increases, then the consumer will subsitute leisure for consumption. So leisure goes down and hours worked $$(T - \ell)$$ increases.

Next, there is also an income effect. If $$w$$ increases, then total potential income also increases. If leisure is a normal good, then this income effect is positive. More income will then lead to taking more leisure.

As such, we see that if $$\ell$$ is a normal good, then the SE and IE will be opposite to each other. So leisure can either increase or decrease if $$w$$ increases depending on whether the (absolute value of the) SE is greater or smaller than the IE.

These opposing forces can generate the so called backward bending labour supply curve. This occurs if for low levels of $$w$$ the SE is greater than the IE. So if wage increases then leisure decreases which means that labour supply increases. Once the wage is sufficiently high, the IE is greater than the SE, so if wage still increases then leisure increases and labour supply decreases.

Remark: Notice that although I call these income and substitution effects, they are not the same as the ones used for the Slutsky equation.