Wednesday, May 7, 2014

Equilibrium in a two-good market


Nick Rowe put up a reference post about capital and interest. I thought I'd follow along with the information transfer model. Rowe initially sets up two markets: apples and bananas. If you'd like to start from the beginning with this, here's my original post on information theory in a two good market.

If we solve the equations (8a,b) and (9a,b) in this post for the price in terms of a change in quantity supplied $\Delta Q^{s}$ or demanded $\Delta Q^{d}$, we have

$$
P = p_{0} e^{\Delta Q^{s}/c_{s}} = p_{0} e^{- \Delta Q^{d}/c_{d}}
$$

These equations define supply and demand curves. We'll assume the markets for apples and bananas are independent in the sense that the supply and demand curves for one market don't depend explicitly on the other. This means that if we plot the supply and demand curves for e.g. bananas in the 3D space $(\Delta A^{(s,d)}, \Delta B^{(s,d)}, P)$, they are constant surfaces in the $\Delta A$ (apples) direction, and every slice at constant $\Delta A$ is the same supply and demand curves for bananas. For visual people, pictures are best (bananas on left, apples on right):


The same goes for the apples, but it's constant along the $\Delta B$ direction (on right, above). If I add the supply surfaces prices (times the quantity supplied) for both goods, I get the Production Possibilities (PPS) surface (in relative terms)

$$
PPS = a_{0} \Delta A^{s}  e^{\Delta A^{s}/a_{s}} + b_{0} \Delta B^{s} e^{\Delta B^{s}/b_{s}}
$$

If I do the same for the demands, I get the Indifference Surface (IS):

$$
IS = a_{0} \Delta A^{d}  e^{-\Delta A^{d}/a_{d}} + b_{0} \Delta B^{d}  e^{-\Delta B^{d}/b_{d}}
$$

I plot the PPS in orange and the IS in blue here:


These two surfaces are tangent at the black dot. The locations where $PPS = 0$ represent the production possibilities frontier (PPF), dotted orange curve. This is the line where the total cost to supply the given amount of apples and bananas at particular location on the curve is the same as the cost to supply the amount in equilibrium (i.e. at $\Delta A = \Delta B = 0$ recall we are looking at $\Delta Q^{s}$, not the absolute level $Q^{s}$). Likewise, $IS = 0$ represents the indifference curve (IC). One way to think about these lines is that the represent lines of constant aggregate supply (PPF) and constant aggregate demand (IC) with a changing mix of apples and bananas produced/consumed. In fact, the vertical axis is basically aggregate demand (supply) relative to equilibrium.  By "aggregate" here, we mean just apples and bananas in this two-good economy.

The last piece required to reproduce Rowe's graph is the "budget constraint". This is basically the equation:

$$
a_{0} \Delta A + b_{0} \Delta B = 0
$$

It's the amount you can change quantity of apples or bananas without spending more or less money (note that $a_{0}$ is the equilibrium price of apples). Here are the two curves and the budget constraint, plotted on the same (2D) graph (essentially the slice through zero on the z-axis = relative aggregate demand = relative aggregate supply):


That's all for now. I'll get to the rest of Rowe's post later.

PS Here's another view of those tangent surfaces:

2 comments:

  1. The apple market could depend on the banana market if hypothetically bananas were needed to make apples. However we assume independence.

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