Question

Derive the Walras law for the Robinson Crusoe economy and discuss the policy implications of the law.

Answer 1

The walras law for this economy that would be derived with it as:

x^d(p_{x *} p_y, w^*) = x^s(p_{x *} p_y, w^*)

y^d(p_{x *} p_y, w^*) = y^s(p_{x *} p_y, w^*) and

L^x(p_{x *} p_y, w^*) + L^y(p_{x *} p_y, w^*) = L.

So

Solving this we get the answer i.e. zero

The production function is concave in two dimensions and quasi-convex in three dimensions. This means that the longer Robinson works, the more coconuts he will be able to gather. But due to diminishing marginal returns of labour, the additional number of coconuts he gets from every additional hour of labour is declining.

The point at which Crusoe will reach an equilibrium between the number of hours he works and relaxes can be found out when the highest indifference curve is tangent to the production function. This will be Crusoe's most preferred point provided the technology constraint is given and cannot be changed. At this equilibrium point, the slope of the highest indifference curve must equal the slope of the production function.

Recall that the Marginal rate of substitution is the rate at which a consumer is ready to give up one good in exchange for another good while maintaining the same level of utility. Additionally, an input's marginal product is the extra output that can be produced by using one more unit of the input, assuming that the quantities of no other inputs to production change. Then,

MP_L = MRS_{Leisure, Coconuts}

where,

MP_L = Marginal Product of Labour, and

MRS_{Leisure, Coconuts} = Marginal rate of substitution between Leisure and Coconuts

Crusoe's multifaceted role

Suppose Crusoe decides to stop being a producer and consumer simultaneously. He decides he will produce one day and consume the next. His two roles of consumer and producer are being split up and studied separately to understand the elementary form of consumer theory and producer theory in microeconomics. For dividing his time between being a consumer and producer, he must set up two collectively exhaustive markets, the coconut market and the labour market. He also sets up a firm, of which he becomes the sole shareholder. The firm will want to maximise profits by deciding how much labour to hire and how many coconuts to produce according to their prices. As a worker of the firm, Crusoe will collect wages, as a shareholder, he will collect profits and as a consumer, he will decide how much of the firm's output to purchase according to his income and the prevailing market prices. Let's assume that a currency called "Dollars" has been created by Robinson to manage his finances. For simplicity, assume that Price_Coconuts = $1.00. This assumption is made to make the calculations in the numerical example easy because the inclusion of prices will not alter the result of the analysis. For more details, refer to Numéraire commodities.

As a consumer, Crusoe will have to decide how much to work (or indulge in leisure) and hence consume. He can choose to not work at all, since he has an endowment of ? dollars from being a shareholder. Let us instead consider the more realistic case of him deciding to work for a few hours. His labour consumption choice can be illustrated in figure 4:

Note that labour is assumed to be a 'bad', i.e., a commodity that a consumer doesn't like. Its presence in his consumption basket lowers the utility he derives.^[1] On the other hand, coconuts are goods. This is why the indifference curves are positively sloped. The maximum amount of labour is indicated by L'. The distance from L' to the chosen supply of labour (L*) gives Crusoe's demand for leisure.

Notice Crusoe's budget line. It has a slope of w and passes through the point (0,?). This point is his endowment level i.e., even when he supplies 0 amount of labour, he has ? amount of coconuts (dollars) to consume. Given the wage rate, Crusoe will choose how much to work and how much to consume at that point where,