Real Income Trap

byStefanos Stavrianos •
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Real income remains how much money an individual or entity makes after accounting for inflation and is sometimes called real wage when referring to an individual's income. Overall, real income is an estimate of an individual’s purchasing power since the formula for calculating real income uses a broad collection of goods that may or may not closely match the categories an investor spends within. Purchasing power refers to what you are able to purchase, and it changes when your real income changes.

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     `RI = W - W * Ir`

`=> RI = (1 - Ir)*W`

where 

  • RI = Real Income
  • W = Nominal Wage 
  • Ir = Inflation Rate

Income Effect

The income effect is a change in the demand for a good or service due to a change in a consumer’s purchasing power, which is, in turn, due to a change in their real income. It’s part of the consumer choice economic theory that relates to how wealthy consumers feel. For example, when the price goes up the consumer is not able to buy as many products as he could purchase before. Can be easily understood that both firms and customers want real income to remain stable. Let `RI_1`, `RI_2` the real income, `W_1`, `W_2` the nominal wages and `Ir_1,\ Ir_2` the inflation rate at the time period `t_1` and `t_2`

        `RI_1 = RI_2`  

`=> (1 - Ir_1)*W_1 = (1 - Ir_2)* W_2`
 
`=> W_2 / W_1 = (1 - Ir_1)/(1 - Ir_2)`
 
`=> W_2  = (1 - Ir_1)/(1 - Ir_2) * W_1`

Also, let `k = (1 - Ir_1)/(1 - Ir_2) != 0`. Thus, the final equation is transformed to `W_2 = k * W_1`. 

The real meaning of this equation has 3 dimensions:

    First, `k = 1 <=> (1 - Ir_1) = (1 - Ir_2)` `<=> Ir_1 = Ir_2 <=> W_2 = W_1`

    Second, `k < 1 <=> (1 - Ir_1) < (1 - Ir_2)` `<=> Ir_1 > Ir_2 <=> W_2 < W_1`

    Third, `k > 1<=> (1 - Ir_1) > (1 - Ir_2)` `<=> Ir_1 < Ir_2<=> W_2 > W_1`

In other words, in a time period, in which the economy has high inflation the real income shrinks. The main solution to this bad situation is an increase in nominal wage. But is it possible? Every firm has two choices; increase the nominal wage, which can help the real income be stable, or not, which in the long run will shrink the demand and the economy. The cost of the increase for the firm i (i = 1, 2, 3,..., n) is `c_i` and the total cost is C = `\sum_(i=1)^n c_i`, which equals the sum of extra earnings for the employees - consumers.

Game Theory Tools


Game theory deals with interactive situations where two or more individuals, called players, make decisions that jointly determine the final outcome. Let's consider the following game: During an inflation time period, there are only 3 firms in the economy. Each firm prefers to get maximum utility with the minimum cost. Maximum utility (`U_(max)`) is achieved, when in the long run demand remains at a high level. But for this achievement, employees' real income has to be stable over time. 

Thus, `U_(max) <=> C = \sum_(i=1)^3 c_i `

Also, `C \approx c_1 + c_2`  and  `C \approx c_2 + c_3`  and  `C \approx c_1 + c_3`

And, `C != c_1`  or  `C != c_2`   or  `C != c_3`

In simple words, the economy will not be shrinking if on average real income is stable. That means, that is not necessary for all firms to increase the nominal wage, but enough of them have to.


1. Game-frame in strategic form:
  • Players (I) =  {Firm 1, Firm 2. Firm 3}

  • Strategies (`S_i`): {Yes (increase nominal wage, with cost `c_i`), No (stable nominal wage, with no extra cost)}; (`S_1`, `S_2`, `S_3`) = ({`c_1`, 0}, {`c_2`, 0}, {`c_3`, 0})

  • Strategy Profiles (s) = {(`c_1`, `c_2`, `c_3`), (`c_1`, `c_2`, 0), (`c_1`,0 ,`c_3`), (0, `c_2`, `c_3`), (`c_1`, 0, 0), (0, 0, `c_3`), (0, `c_2`, 0), (0, 0, 0)}

  • Outcomes (O): {C (high, stable demand in long run), N (shrinking demand in long run)}; outcomes are listed in Figure 2

  • f : S `\rightarrow` O is a function that associates with every strategy profile s an outcome, f(s)`\in` O

  • Utility: For every Player `U_(max)` = U(C) > U(N)

2. The graphic representation of the game is:
Figure 1: Firm's Game

Figure 1: Firm's Game [Own Processing]


Figure 2: Firm's Game Pay-offs

Figure 2: Firm's Game Outcomes [Own Processing]

In real life, every firm is selfish and greedy. It will make the decision, which will bring the desired utility at the lowest cost. In this firm's game, outcome C is strictly Pareto superior to N or, in terms of strategy profiles, ({`c_1`, `c_2`, `c_3`}, {`c_1`, `c_2`, 0}, {`c_1`,0 ,`c_3`}, {0, `c_2`, `c_3`}) are strictly Pareto superior to ({`c_1`, 0, 0}, {0, 0, `c_3`}, {0, `c_2`, 0}, {0, 0, 0}).

When a player has a strictly dominant strategy, it would be irrational to choose any other strategy, no matter what the other players do, because he would be guaranteed a lower payoff in every possible situation. If Firm 1 excepts that other firms will choose to increase nominal wages, it will choose to not, because in this strategy profile {0, `c_2`, `c_3`} the total Utility is `U_(max)` = U(C), without extra cost (`c_1`) for Firm 1. On the other hand, if Firm 1 excepts that other firms will not increase nominal wages will have not to increase it, because it is only a waste of money (U({`c_1`, 0, 0} = U(N) < U(C) = `U_(max)`). All firms follow the same logic.

As well, for every firm i, individual rationality leads to {0, 0, 0} despite the fact that both players would be better off if they both choose {`c_1`, `c_2`, `c_3`}. It is obvious that if the firms could reach a binding agreement to increase nominal wage then they would do so; however, agreements are not possible, because of the huge number of firms in an economic sector. Also, any non-binding agreement will be a disaster, because if one firm expects the other firms to stick to the agreement, then it will cheat. 

In conclusion, for every firm i, "No" strictly dominates "Yes" and s = {0,0,0} is a strictly dominant-strategy equilibrium, but in long run it will lead to an economic catastrophy.  In order to avoid this trap, an external body, like the state, has to force firms to increase nominal wages by at least `Ir_t - Ir_(t-1)` (`k = 1 <=> W_2 = W_1`).

Selected References

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Thank you for visiting my blog! I am Stefanos Stavrianos, a PhD Candidate in Computational Finance at the University of Patras. I hold an Integrated Master’s degree in Agricultural Economics from the Agricultural University of Athens and have specializations in Quantitative Finance from the National Research University of Moscow, Python 3 Programming from the University of Michigan, and Econometrics from Queen Mary University of London. My academic interests encompass economic theory, quantitative finance, risk management, data analysis and econometrics.

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