Example 2: Marginal rate of substitution

U(x,y)=xy⁴ – utility function for the representative consumer

x, y – two goods

Calculate the MRS.

Please select the utility calibration point at x=y=1 as the reference quantity. In an equilibrium, final demand always equals endowments for both goods, because these are the only sources of demand and supply. If we set endowments for this model equal to the demand function calibration point, the model equilibrium price ratio equals the benchmark MRS.

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Example 3: Labor-Leisure decision

A single consumer is endowed with labor which is either supplied to the market or repurchased as leisure. The consumer utility function over market goods (x and y) and leisure (l) is Cobb-Douglass:

U(x,y,l)=ln(x)+ln(y)+ln(l).

Goods x and y may only be purchased using funds obtained from labor sales:

x+y= labor productivity*labor supply

where goods x and y both have a price of unity at base year. An increase in productivity is equivalent to a proportional decrease in the price of x and y. Evaluate the wage elasticity of labor supply.

Theoretical notes:

The elasticity of labor supply is an important parameter in labor market studies. It should be an input rather than an output of a general equilibrium models obtained from econometric estimations.

Cobb-Douglas function (s=1): Q=A*Y^d*X^1-d

CES function: Q=A*[d*Y^(s-1)/s+(1-d)* X^(s-1)/s]^s/(s-1)

The elasticity of substitution measures the curvature of an isoquant (i.e. the percentage change in the factor ratio divided by the percentage change in the marginal rate of technical substitution):s=[d(Y/X) / dMRTS] * [MRTS / (Y/X)]

The marginal rate of technical substitution measures the slope of an isoquant (i.e. how one of the inputs must adjust in order to keep output constant when another input changes): MRTS=-(dQ/dX)/(dQ/dY)=-MP_X/MP_Y