Archive

Archive for the ‘Simple algebra’ Category

Nominal and real interest rates

One of the most simple but crucial aspects of learning economics is understanding the relationship between nominal and real interest rates. When you see interest rates quoted for savings accounts or loans, they are usually nominal interest rates, expressed in terms of currency. Eg if you borrow £1000 for one year at 4%, then you pay back £1040, you are paying in interest 4% of the nominal value of the amount you borrowed.

However that does not tell you how the £1040 you repay compares to the £1000 you borrowed in terms of a basket of goods. If inflation was 3.5% then £1040 in a year’s time will be worth $\frac{1040}{1.035}=1004.83$ ie £1004.83 today. So you are actually only really paying $\frac{1004.83}{1000}=1.00483$ ie 0.483% interest in real terms.

This is a concept that most people in the real world won’t think about. They will just think about the nominal rate of interest quoted and not take account of how this will be eroded by inflation.

The relationship between real interest rates, nominal interest rates and inflation is

$(1+r_t) = \frac{(1 + i_t)}{1 + \pi^e_{t+1}}$.

This means that the real interest rate depends on the nominal interest rate and the expected rate of inflation next year. The real interest rate is an ex ante interest rate, because it is based on expectations of inflation. This means at the time you are making a decision (do I save/borrow at this nominal rate of interest?) you have to base your decision on what your expectations of inflation are. A year later, when you know what inflation actually was, you can find out the ex post real interest rate (ie what the real interest rate actually was, regardless of what the ex ante real interest rate suggested it would be).

You can make an approximation to this.

$(1+r_t) = \frac{(1 + i_t)}{1 + \pi^e_{t+1}} \Rightarrow (1 + r_t) (1 + \pi^e_{t+1}) = (1+i_t)$, when you expand the brackets you get:
$1+r_t + \pi^e_{t+1} + r_t \pi^e_{t+1} = 1 + i_t \Rightarrow r_t + \pi^e_{t+1} + r_t \pi^e_{t+1} = i_t$.

As the multiple of real interest rate and expected inflation $r_t \pi^e_{t+1}$ is likely to be small, we can approximate to:

$r_t + \pi^e_{t+1} \approx i_t$ or $r_t \approx i_t - \pi^e_{t+1}$.

Real interest rates are relevant when we are thinking about what something is worth in terms of a basket of goods, so for instance firms that are making decisions about whether or not to make an investment, will think in terms of real interest rates. The real interest rate is the relative price of current consumption compared to future consumption. It tells you how much consumption you gain in the future by sacrificing consumption now.

The IS relation uses the real interest rate: $Y=C(Y-T) + I(Y,r) + G + NX$. As $r \approx i - \pi^e$ we can roughly say $Y=C(Y-T) + I(Y,i - \pi^e) + G + NX$. This is how expected inflation influences the IS curve. A rise in expected inflation will mean that a higher nominal interest rate is needed for every level of output. So a rise in expected inflation will shift the IS curve upwards.

Nominal interest rates are relevant when we are thinking about money markets. When investors are deciding whether to hold money (that pays 0% interest) or some illiquid interest bearing asset like bonds (that pays i% interest) then they will think in terms of nominal interest rates.

The LM relation uses the nominal interest rate: $\frac{M}{P}=YL(i)$.

Categories: Simple algebra

The most basic stuff – supply & demand analysis

Chances are the first type of thing you will do in microeconomics will be along the lines of supply and demand functions. Basically these are functions of quantity in terms of price, ie they tell you how much of a good consumers are willing to buy depending on the price, and they tell you how much producers are willing to supply, depending on the price.

For example you might get a demand function of $Q_D = 320 - 5P$ and a supply function of $Q_S = 50 + 4P$. You can rearrange those equations to put them in terms of P, to get the inverse demand and supply functions. $P = 64 - 0.2Q_D$ and $P = 0.25Q_S - 12.5$

This is telling you that if the price was, say 20, then the quantity demanded will be 220 and the quantity supplied will be 130. There will be an excess of demand over supply, so this will drive up the price – consumers are willing to pay more to get hold of this scarce good, so that higher price will encourage producers to produce more. When we reach a price that brings the quantity supplied in line with the quantity demanded, then we have the equilibrium price.

We can find this just by using simultaneous equations: set $Q_D = Q_S = Q$ so $Q = 320 - 5P = 50 + 4P \Rightarrow 270 = 9P \Rightarrow P= 30$. So now we have our equilibrium price, of 30. We can see then how much quantity is produced at this price by subbing 30 into our demand and supply functions, and in both cases the quantity will come out as 170. So our equilbrium price is 30 and our equilibrium quantity is 170.

Now we can look at the effects of a tax. The government can place a specific tax on the good, which means you pay a tax per unit of output (this is like petrol tax, you pay by unit consumed not as a proportion of the tax). Typically it is the producer that has to pay the tax to the government, ie they have to declare how many units of the good they have sold and pay the tax to the government based on that. So this will affect the producer’s supply function.

Lets say the government puts a specific tax, t, of 5 per unit, so every unit sold means the producer has to pay a tax of 5 to the government. This means that the price the consumer pays and the price the producer gets are no longer the same – the consumer is paying more than the producer ends up with, because he has to pay a portion of it in tax. $P_D -t = P_S \Rightarrow P_D = P_S + t$

So go back to those inverse demand functions: $P_D = 64 - 0.2Q_D$ and $P_S = 0.25Q_S - 12.5$. When we are in equilibrium, so the quantity demanded equals quantity supplied, we can say that $P_D = 64 - 0.2Q$ and $P_S = 0.25Q - 12.5$ and we know that $P_D = P_S + t$ so with t=5, those equations imply that $64 - 0.2Q = 0.25Q - 7.5$.

Solving this for the equilibrium quantity: $64 - 0.2Q = 0.25Q - 7.5 \Rightarrow 71.5 = 0.45Q \Rightarrow Q = 158.889$

Now putting this back into the inverse demand and supply equations gives us:
$P_D = 64 - 0.2(158.889)= 32.222$ and $P_S = 0.25(158.889) - 12.5 = 27.222$. The price the consumer pays, is effectively the ‘market price’, which is 32.222, but the price which the producer actually receives, is 27.222. The gap between them is 5. This is because on each unit that the consumer pays 32.222 for, 5 goes to the government in tax, and 27.222 goes to the supplier.

So the effect of this specific tax, is to increase the market price from 30 to 32.222, and decrease the quantity from 170 to 158.889.

The producing firm’s revenues are equal to price multiplied by quantity, so in the equilibrium without the tax, they made total revenue of $TR = PQ = 30(170) = 5100$ and after the tax they made total revenue of $TR = PQ = 27.222(158.889) = 4325.308$. Meanwhile in the equilibrium which has the tax on it, the government is taking a tax of 5 on all of those 158.889 units sold, so it gets revenue of $GR = tQ = 5(158.889) = 794.444$

Categories: Maths, Simple algebra