# Interpretation of the growth rate of consumption in a simple one-good model

In Perman et al. "The efficient and optimal use of natural resources" it is presented a simple single-good model of welfare dynamic optimisation with non-renewable natural resource where:

Obj: $$max \int_{t=0}^{t=inf}{U(C) e^{-\rho t} dt}$$ s.t. $$\dot S_t = -R_t$$ $$\dot K_t = Q(K_t,R_t)-C_t$$

That is, the economy is comprised of a single output Q that can be either consumed or added to the capital stock K (that doesn't depreciate).

Solving the model and rearranging the first order conditions, the book found that among the optimal path the growth rate of consumption is:

$$\frac{\dot C} {C} = \frac {Q_k - \rho}{\eta}$$

where $Q_k$ is the marginal product of the capital, $\rho$ is the social discount rate and $\eta$ being the elasticity of the marginal utility with respect to consumption is, under bland assumptions, guaranteed to be positive.

Now, my problem is the interpretation of the sign implications of this equation.

If $Q_K$ is greater than $\rho$, the consumption rate increases (and the opposite). What does it mean?

The books explains it considering that $\rho$ (the social discount rate) reflects impatience for future consumption, and $Q_K$ (the marginal product of capital) is the pay-off to delayed consumption.

Under this interpretation the relation implies that along an optimal path when ‘pay-off’ is greater than ‘impatience’ "consumption is increasing", or that "the economy will be accumulating K and hence growing".

But in my experience if $Q_K$ is higher than the discount rate, I have an higher incentive to stock K rather than consume, so how the consumption could increase ? Also, it is said that "when ‘pay-off’ is less than ‘impatience’, the economy will be running down K.", but how can the K reduce in this simple model where there is no capital depreciation?

• How can capital reduce with no depreciation? Presumably because, in this single good model, capital in the form of that single good can at any time be consumed. – Adam Bailey Sep 21 '16 at 9:28

I think you could have posted the question in a simpler manner. The exhaustible resource stock has nothing to do with the question. It is just the basic Euler equation in canonical Solow growth model.

Basically, if your pure rate of time preference $\rho$ (equivalently, your patience level) is higher than the interest rate of capital ($\rho>Q_{k}$), it means that you are impatient and consume most of what you have at earlier dates and you have less to consume in future. In which case, your consumption starts with higher values at date $0$ and dynamically decreases over time.

Another way to see this ; if interest rate of capital is higher, you have more incentive accumulate capital (by having savings) at earlier dates and postpone your consumptions to later dates. In this case, you will consume less in earlier dates and consume more at later dates. Therefore, your consumption will increase dynamically over time.

Hope that it helps.

• Yep, you are right, thank you. The fact is that if you analyse this problem statically you reach the opposite conclusions (higher $Q_k$, lesser consumption) than if you analyse it dynamically. – Antonello Sep 22 '16 at 7:51
• You are welcome ! You can not analyze this problem statically because basically, it is a dynamic framework. You can just look at the steady-state of the problem and make some comparative statics analysis or by the means of implicit function theorem, you can make a phase diagram. – optimal control Sep 22 '16 at 21:50

I got stock with this problem for days.

The fact is that in a statical analysis, increasing $Q_K$ reduces consumption. But in a dynamic analysis, increasing $Q_K$ means more capital is saved and hence next year an higher production function that allows for more consumption than today, hence, a positive growth rate of consumption.

The higher the marginal product of capital, the more productive an additional unit of capital is. The more productive it is, the less savings (=sacrifice of current consumption) we need in order to increase future consumption. Interacting also with the preference parameter (the rate of pure time preference), the model says that when the marginal product of capital is higher than the latter, we save now at a level where
1) Today's consumption is still greater than yesterday's consumption and
2) Tomorrow's consumption will be even higher.

As for "running down the capital stock", in these models it is assumed that "capital" can be consumed as though it was a consumption good. This should already be clear from the differential equation for Capital, where current consumption is allowed to be greater than current production ($\dot K$ can be negative) -but this can only happen if we consume part of the existing capital stock (since the economy is closed and no goods fly in from abroad).