My Math Forum Relating the Interest Rate to the Supply Curve

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 March 16th, 2012, 10:28 PM #1 Member   Joined: Sep 2010 Posts: 63 Thanks: 0 Relating the Interest Rate to the Supply Curve Today, I was scratching my head as to how one might model supply and demand changes based on price elasticity. This was inspired by the suggestion someone had that speculation played a significant part in oil prices: http://robertreich.org/post/19353120672 But, I figured it was more important to look at basic supply and demand first. Unfortunately, when trying to relate this to price elasticity I got some disappointing results: viewtopic.php?f=46&t=28699 Which suggests a mistake in that effort but I think I was able to come up with some other useful ideas. First lets make the rather boring observation that Revenue (profit would be more suitable but just keeping it simple for now) is equal to price multiplied by quantity. To make it somewhat interesting lets use a linear model: $R=\left (P_o + \Delta Q \frac{dP}{dQ} \right )\left( Q + \Delta Q \right )$ $R= P_oQ_o+\Delta Q \left ( P_o + \frac{dP}{dQ} Q_o \right ) + \frac{dP}{dQ} \left ( \Delta Q \right )^2$ and do another curve for cost: $C=C_o+\frac{dC}{dP} \Delta Q$ Now if you take the profit (I'm using revenue here for simplicity) and divide it by the cost this gives you the average rate of return. In a competitive market the average rates of return should be equal to the risk free rate of return plus a risk premium. In a non competitive market the marginal rate of return will instead be the quantity which is equal to the risk free rate of return plus the risk premium. We can estimate the marginal rate of returns with difference quantities. The return people will require will depend on the interest rate and the perceived risk of the investment. So the average rate of return is: $i_{(avg)}=\frac{P_oQ_o+\Delta Q \left ( P_o + \frac{dP}{dQ} Q_o \right ) + \frac{dP}{dQ}\left ( \Delta Q \right )^2}{C_o+\frac{dC}{dP} \Delta Q$ And the difference approximation to the marginal rate of return is: $i_{(dif)}=\frac{\Delta Q \left ( P_o + \frac{dP}{dQ} Q_o \right ) + \frac{dP}{dQ} \left ( \Delta Q \right )^2}{\frac{dC}{dP} \Delta Q$ In both equations we set the rate of return $i_{(avg)}$ or $i_{(dif)}$ equal to a constant then solve for $\Delta Q$. Now one might wonder why not just include the borrowing cost in a typical supply curve. The answer I'll give is that in an efficient market the borrowing cost should be a function of the demand as investers should consider the impact on total market returns from additional investment. Trying to account for this in a supply curve is an unnecessary and cumbersome step when we can directly apply this principle. This is as clear as I can make it tonight as my mind is tiered today so let me finish the post with some URLs to numbers which I think will be helpful in extending this model to the oil market: The cost curve of oil can be found here http://stochastictrend.blogspot.com/201 ... e-oil.html: The price elasticity of oil is given here: http://15961.pbworks.com/f/Cooper.2003. ... udeOil.pdf Knowing price and demand we could use it to get the slope of the demand curve. Total oil inventories are given here: http://www.bloomberg.com/quote/DOESCRUD:IND I haven't included inventories in my model yet but the amount of space we have to store excess oil will determine how much outside of equilibrium we can operate in. This could be relevant when considering speculation. Well, that's all for tonight. I hope someone else finds this interesting.

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