Why do prices and interest rate still exist in the example with a single household? Does the household trade and borrow with itself?
In the model you present above it is ambiguous how goods are produced, so there is ambiguity there. You could have a model of an economy where single household owns the firm and trades with itself. However, that would normally be mentioned, unless your description is missing context I think the problem just assumes there are some 'person-less' markets, that work like a wending machine, where this single household can get its goods and services.
When it comes to interest rate, interest rate is just an exchange rate between present and future value $PV= FV/(1+r) \implies PV(1+r)=FV $. There does not need to be any financial market for interest to exist. There is always some real interest rate anytime you have more than 1 time period. Interest rate is just an exchange rate between present and future. In addition, the market clearing conditions in your case, will make any period 1 saving grow, so you can also imagine here economy has some 'drop box-wending-machine' like financial system. You deliver anything you do not consume in period 1 to the machine and in period 2 the machine multiplies all your savings of first product with factor 2 and second product with factor 4.
These are all simplifying assumptions, because once you explicitly assume there are both workers, firm owners and banks, you have to model not just household utility and decisions, but also that of firm and bank separately. Unless this added complexity would qualitatively change the results of interest from the model it would be waste of time to model it all in. Models are supposed to be simplified versions of reality.
However, note prices and interest rate do not require trading. For example, in a simple Robinson Crusoe economy, where Crusoe has option to either:
- a) go fishing, and he can catch 1 fish per hour
- b) collect wood, and he can collect 10 pieces of wood per hour
Given the parameters of the Robinson Crusoe economy above, the price of fish would be 10 sticks and price of a stick would be 1/10 of a fish.
To measure inflation you would have to declare one good a numeraire good, otherwise you will have a problem that you will get different CPI when different goods are selected. Let us say that we declare sticks to be numeraire. In that case we can say that price of fish is 1/10=0.1 sticks, and the price of stick is 1 stick.
Interest rate can exist as well if you allow for multiple time periods and if you allow RC to save and invest some of his sticks (for example he can use them to make capital investments into better fishing rods and fishing boat etc).
Although, in your model there are no explicit assumptions how the goods are produced so again there is ambiguity there, in a simple model like yours they might as well just appear on the market which functions like an wending machine.
Why does the absolute price inflation depend on an exogenuous variable, the interest rate r, while the relative price inflation can be endogenuously determined, regardless of the value of r? I assume it has something to do the fact that there is no money in the example?
First, inflation in the economy with money also depends on $r$. For example, in standard textbook IS-LM the money market equilibrium will be given by:
where $M$ is money supply, $P$ aggregate price level, $Y$ output and $i$ interest rate, and $L$ is the demand for loanable funds/money. If we solve for $P$: $P=M/L(Y,i)$ (at least to the degree you accept IS-LM model, but most macroeconomists would). Next, by Fisher equation $i \approx \pi +r$, so price level should depend on real interest rate generally in economy with money as well.
Second, now to give you specific answer for your particular model the reason is as follows:
- You are dealing with endowment economy there is fixed endowment of both commodities (aka imagine Hermes by his grace filled the 'wending machines' for your household).
- Next, the market clearing conditions specify the rate at which one can save and invest parts of the endowment for the second time period.
- The interest rate again just tells you the exchange rate between present and future value of your consumption. The higher the interest rate the lower present value of your period 2 consumption (the household is more impatient).
As a consequence, ratio of prices for good 1 across different time period will depend on interest rate $p_2^x/p^x_1 = (1+r)/2$, because if you value future consumption less (higher $r$), you will consume more of the $x$ good today and there will be less $x$ good tomorrow which will affect the prices.
Moreover, the whole point of this is captured by the equilibrium condition:
This condition just says that future value of consumption $x_1$ needs to be equal to future value of good $x_2$ which needs to be equal to the value of $x_2$ and $y_2$ (which are already in the future). Or alternatively you can rewrite the same equation as:
which would just state that today's value of consumption $x$ has to equal to today's value of consumption $y$ which has to be equal to present value of future consumption of $x$ and $y$. Economically this is quite intuitive result.