Chapter 1 Section 7

Excel Companion Chapter 1 section 7

Open the file NonlinSD.xls and activate the first worksheet.

Notice that the first function, f, is a demand function and the second function, g, is a supply function. The third function, h, is the difference function ‘demand minus supply’. Detecting a sign change in the difference function h(x) = f(x) - g(x) is easier than scanning a long table looking for points where f(x) and g(x) are about the same. The quantity at which supply equals demand can be found by solving the equation f(x) = g(x) which is algebraically equivalent to solving f(x) - g(x) = 0.

SITUATION: Suppose a demand function is defined as and a supply function is defined as where x is in in thousands. What is the equilibrium point? If we set f(x) - g(x) = 0 and transform the resulting equation we eventually get the equation . Exact solutions to this cubic equation are not readily obtained. How then can we obtain good approximations to the actual root(s) of this equation?

NonlinSD.xls - table f g h

There are several approaches that can be used to determine values of the independent variable for which supply price is approximately equal to demand price. Start by entering expressions in B11, C11, and D11 for the defining formulas of f(x), g(x), and h(x) respectively, where h(x) is the difference function f(x) - g(x).

An inspection of the table or graph reveals that the values on the supply and demand functions are close to each other when x is near 15. When the new_h macro button is pressed, the defining formula for the difference function, h(x), is copied to a target cell with a variable name "guess" instead of "x". Apply goal seeking on worksheet NonlinSD.xls with x = 15 as your initial guess for the root of h(x). You should discover that an approximate root is 15.252. We can interpret this to mean that the equilibrium point is about 15,252 units. Substituting this value in cell C7 (or in cell C8) we see that the equilibrium price is approximately $9.88.

QUESTIONS:

1.	Keep f(x) the same and modify the supply function so that . The formula for h(x) must be changed.
	a) What is the new equilibrium point?
	b) What is the new equilibrium price?
2.	Find the equilibrium quantity and equilibrium price for:
	a demand function given by:
	a supply function given by:

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