# Robinson Crusoe Model

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- Additional Contributors: None
Categories: CategoryModel CategoryMicro CategoryGeneralEquilibrium

- Related Models: None
Created: 2008-08-06

Suggested Citation: See license page.

Contents

## Introduction

The Robinson Crusoe model illustrates the basic ideas behind General Equilibrium models using the simple framework of a one-producer one-consumer economy. The main concept is that of a **Walrasian** (competitive) equilibrium under which two conditions are satisfied:

- Optimality - agents maximise given prices
- Feasibility - all markets clear

## Model

The consumer maximises their utility by consuming leisure $$ z $$ at the opportunity cost $$ w $$ and a good $$ c $$ at price $$ p $$ . There is a trade-off between the two commodities because $$ c $$ is produced using labour $$ L = H - z $$ (where $$ H $$ is the total time endowment) .

The firm produces $$ q $$ units of the consumption good such that $$ q=f(L) $$ and maximises profits $$ \pi = pq - wL $$ where $$ w $$ is the wage rate. The firm is owned by the consumer, so the consumer's budget constraint becomes $$ pc \le \pi + w(H-z) $$ .

Both the consumer and producer act as price-takers, i.e. take the market prices $$ p $$ and $$ w $$ as given. Consumption and production decisions are made separately subject to these prices, with the Walrasain equilibrium characterised by the price vector $$ (p^*,w^*) $$ at which the markets for leisure $$ z $$ and the consumption good $$ c $$ clear:

$$ c^*(p^*,w^*) = q^*(p^*,w^*) $$ (assuming non-satiation)

$$ H-z^*(p^*,w^*) = L^*(p^*,w^*) $$

## Solving

### Consumers

Lagrangian = $$ u(c,z) + \lambda[\pi + w(H-z) - pc] $$

FOCs:

- $$ MU_c = \frac{\partialu(c,z)}{\partialc} = \lambdap $$
- $$ MU_z = \frac{\partialu(c,z)}{\partialz} = \lambdaw $$

$$ MRS_{c,z} = \frac{MU_z}{MU_c} = \frac{w}{p} $$

### Producers

$$ \max_{L} \pi = pf(L) - wL $$ $$ \Rightarrow $$ $$ \frac{\partialf(L)}{\partialL} = MRT = \frac{w}{p} $$

### Equilibrium

By choosing the point where their indifference curve (production function) is tangental to the relative price line, consumers (producers) are acting optimally. However, this choice is not necessarily feasible, since unless these points coincide there will be excess demand or supply in both the leisure/labour and consumption good markets. Therefore, the competitive equilibrium is defined by the price vector $$ (p^*,w^*) $$ that clears both markets, in this case such that $$ MRS = \frac{w}{p} = MRT $$ .

The Walrasian equilibrium is the same allocation as if there was a social planner maximising consumer welfare subject to endowment and technological constraints.