3.1 Solution Functions and Comparative Statics

One kind of function that we’ll encounter a lot in Econ 50 is a solution function. That is, we’ll solve some problem — an optimization problem, or a characterization of equilibrium — but the solution will be a function of the underlying parameters of the model.

For example, one task from Econ 1 or High School econ might be to find the equilibrium price and quantity in a market. If you’re given specific supply and demand functions, you can solve for the equilibrium price by equating supply and demand. For example, suppose I were to give you the supply and demand functions \(\begin{aligned}S(p) &= 4p\\ D(p) &= {1600 \over p}\end{aligned}\) If we set these two equal to one another, we get \(\begin{aligned}S(p) &= D(p)\\ 4p &= {1600 \over p}\\ p^2 &= 400\\ p^\star &= 20\end{aligned}\) Plugging this back into either the demand or supply function gives us the quantity in the market will be \(Q^\star = S(20) = D(20) = 80\) In this class, however, we’ll derive supply and demand from first principles. We’ll see how consumers’ preferences (captured by a variable $\alpha$) and income ($m$) affect their demand, and how firms’ level of capital ($\overline K$) and the wage rate they have to pay workers ($w$) affects their supply; and we’ll multiply the individual demand and supply curves by the number of consumers ($N_C$) and the number of firms $(N_F)$ respectively; so by week 8 or 9 we’ll have supply and demand functions that look like this: \(\begin{aligned}S(p) &= N_F \times {\overline K p \over 2w}\\ D(p) &= N_C \times {\alpha m \over p}\end{aligned}\) If we set these two equal to one another, we get the equilibrium price and quantity as functions of the underlying parameters $\alpha$, $m$, $\overline K$, $w$, $N_C$, and $N_F$: \(\begin{aligned}S(p) &= D(p)\\ N_F \times {\overline K p \over 2w} &= N_C \times {\alpha m \over p}\\ p^2 &= {N_C \over N_F} \times {2 \alpha m w \over \overline K}\\ p^\star(\alpha, m, \overline K, w, N_C, N_F) &= \sqrt{\frac{N_C}{N_F} \times {2 \alpha m w \over \overline K}}\end{aligned}\) Plugging this back into either the demand or supply function gives us the quantity in the market will be \(Q^\star(\alpha, m, \overline K, w, N_C, N_F) = \sqrt{N_FN_C\overline K \alpha m \over 2w}\) This looks like a lot of variables! But in fact, what we’re seeing is that we can do all our usual Econ 1 comparative statics just from these expressions:

• If more consumers enter the market $(\uparrow N_C)$, or if consumers desire this good more $(\uparrow \alpha)$, or if consumers get more money $(\uparrow m)$ the demand curve shifts to the right; the equilibrium price and quantity both rise.
• If more firms enter the market $(\uparrow N_F)$ or if firms invest more capital in producing this good $(\uparrow \overline K)$, the supply curve shifts to the right; the equilibrium price falls, and the equilibrium quantity rises.
• If the wage rate increases $(\uparrow w)$, the supply curve shifts to the left; the equilibrium price rises, and the equilibrium quantity falls.

You can see how each of these effects (and their opposites) play out in the following diagrams. The right-hand diagram shows market supply and demand; the left diagram shows individual demand, and the middle diagram shows individual supply.

[ See interactive graph online at https://www.econgraphs.org/graphs/competition/equilibrium/individual_market_supply_demand ]

In short, the equilibrium price and quantity are now multivariate solution functions; and we can use these functions to help us understand how a range of factors affect this particular market.