Question

Nonlinear Comparative Statics: Consider the supply and demand system represented by these two “structural” equations:

Q = S(P, W)

Q = D(P, Ps)

Here Q is quantity, P is market price, W is the wage, and Ps is the price of a substitute good. The endogenous variables are P and Q: determine how they respond to the exogenous variables Ps and W. You should totally differentiate this structural form in order to derive the partial derivatives of the implicit reduced form. Provide intuitve economic reasoning to sign the partial derivatives of the structural model functions, and use this information to sign the complete comparative statics results. Specifically:

- Totally differentiate the two equations. Provide an intuitive verbal interpretation of the differentials.

- Prepare for the comparative statics experiments by setting up a matrix representation of this system.

-Produce the inverse of the coefficient matrix. Prove by matrix multiplication that you actually have an inverse.

-Use your inverse matrix to solve for dP and dQ in terms of the exogenous changes dPs and dW.

- Provide extra detail for one comparative-statics experiment: find and sign ∂Q/∂W. Explain how you get from the previous step to this result.

- Provide a graphical illustration of this comparative-statics experiment.

Answer 1