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Consider a closed economy with perfect competition, no government and no fiat money. Consumer goods are aggregated as a single type of goods and prices are measured in units of goods. For such an economy, there are basically two intertemporal equilibrium models of capital accumulation.

In the renting model:

  1. It is the households that accumulate capitals.
  2. In each period $t$, the firms rent capitals, say $K_t^f$ units, from the households in the factor market at the market rental rate $R_t$. There is no transfer of ownership of capitals.
  3. The firms do not borrow. In each period $t$, the rental cost of capitals, $R_t K_t^f$, is directly deducted from the profit in the current period.
  4. Every unit of consumption that the households forgo becomes one unit of new capital in the next period, which is added to the existing stock of capitals, say $K_t^h$ units, owned by the households.
  5. At equilibrium, the supply of capitals by the households equals the demand for capitals by the firms, i.e. $ K_t^s = K_t^d = K_t $.

In the investing model:

  1. It is the firms that accumulate capitals.
  2. In each period $t$, the firms invest in new capitals, which are added to the existing stock of capitals owned by the firms. The investment $I_t$ is financed by borrowing from the households at the market interest rate $r_t$.
  3. In each period $t$, the firms need to repay the principal plus interest, $(1 + r_t)I_{t-1}$, of the loan from the last period. This repayment is deducted from the profit in the current period.
  4. Every unit of consumption that the households forgo becomes one unit of household saving. This saving, $S_t$, can then be loaned to the firms.
  5. At equilibrium, the supply of loans by the households equals the demand for loans by the firms, i.e. $ S_t = I_t $.

If we assume that the capitals depreciate at a rate $\delta$ in both models, then the two models are actually mathematically equivalent via the following relations:

$$ R_t = r_t + \delta \quad \text{and} \quad K_{t+1} = I_t + (1 - \delta) K_t $$

Why are there two equivalent models? Which model of capital accumulation do macroeconomists prefer?

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  1. Why are there two equivalent models?

First, the models are not necessarily equivalent, and they are not even full models only descriptions of capital accumulation by firms that would be typically nested in larger model. The mechanism there is different, in the first one firms rent capital directly from household, in second one there is an financial market that equilibrates savings and investment.

Sure if you do not add anything else to the model, a market where firms rent capital directly from household will look exactly like financial market. However, if you would want to expand these models in model 1 there would be very little role for banking sector since capital is acquired directly from households, whereas in the model 2 you could quite naturally add additional banking sector that would serve as intermediary between households and firms. This could then lead to some interesting results that would be different than just in model 1 (even though $R_t=r_t+δ$ and $K_t+1=I_t+(1−δ)K_t$ might stil hold the equilibrium values of $K$, $I$ and $r$ could end up being different.

Second, just because two models share some equations that does not mean they are equivalent either. Let me give you an example, for any firm profit can be described as:

$$\pi = (p(q)-c(q))q \tag{*}$$

and every model of competition (monopoly, perfect competition, monopolistic competition etc) will have some version of the equation *. However, even though the profit equation is same you will get different outcomes once you specify what the number of firms are, what entry conditions are etc. For example, assuming 1 firm and no entry the you would end up with:

$$\pi_q' = p'(q)q + p(q) -c'(q)= 0$$

However, assuming number of firms $n \to \infty$ and free entry $p(q)=p$, and so the equilibrium would be given:

$$\pi_q' = p -c'(q)= 0$$

Now I am not trying to say that there are not a cases or models where both model 1 and 2 will become completely equivalent, but that is not a general result that will hold every single time once you integrate these models capital accumulationinto a larger model.

Third, sometimes there is more than 1 way to express equivalent ideas. Math is nothing else than complex symbolic language. Like in English, in math you can say the same idea in two different ways. For example, saying $x$ belongs to either $A$ or $B$ can be written as:

$𝑥∈𝐴∪𝐵$ or $(𝑥∈𝐴)∨(𝑥∈𝐵)$, there is no substantial difference there (aside from lenght of the statement) $𝑥∈𝐴∪𝐵 ⟺ (𝑥∈𝐴)∨(𝑥∈𝐵)$.

  1. Which model of capital accumulation do macroeconomists prefer?

This is impossible to answer, there are no surveys that would be so detailed as to be able to put some concrete number on this.

However, as described under section 1 there are two cases:

A) cases where difference between these models actually matters once they are integrated into larger macro models that have further assumptions (e.g. adding banking sector, aggregate supply and demand and integrating them together)

B) cases were the difference does not matter at all.

In A) modeller should decide on what they believe is the most appropriate. E.g. you can ask yourself are there some rigidities in banking sector that would make qualitative/quantitative difference? Or are markets without intermediation less perfect? etc. Which of these situations is more close to what we can observe in real life?

In B) it is purely aesthetic choice. You can go with either one.

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