In this problem, we will play economist. Consider four variables, price (P), demand (Q), income (I), and wages (W). More specifically, where Q is the quantity of household demand for a product A, P is the unit price of product A, I is household income, W is the wage rate for producing product A. Assume that income and wages are modeled separately (are independent). This example comes from Pearl (2000). 

a. Using your background knowledge of which variables influence each other when changed, express this problem as a causal network. Include terms for unmodeled variables. 

b. Express this problem a system of structural equations. 

c. Suppose we now change the price of the item, we do(p = x). What is the new joint distribution, P(I, Q, W | do(p = x))?