Exam set: Climate policy in a Solow growth model Flashcards
Problem 2
Case: Climate policy in a Solow-type growth model (IAM, integrated assesment model).
Model:
Output is produced by three factors of production: labour and two types of capital.
- Labour is assumed constant, so it is not in the production function
- The two types of capital are called “black” and “green” capital, where black capital is the only one producing CO_2 emissions.
By substituting green for black capital one can reduce emissions per unit of output.
Question 1.1
Discuss briefly how one can motivate the presence of the term (S^(-gamma))_t in equation (2). Also discuss briefly why equation (6) includes the term -deltaS_t.
Answer: The term (S^(-gamma))_t in equation (2) captures the negative impact of global warming on total factor productivity. A sthe concentration of CO_2 in the atmosphere increases, the average temperature at he surface of the Earth goes up due to the greenhouse effect, causing various damages from climate change, including damages from a higher frequency of extreme weather events such as periods of extreme heat that are harmful to human helath and/or reduce labour productivity. Extreme heat and rainfall can also lower agricultural crop yields; storms and floods can destroy buildings, machinery and infrastructure, and sea level rise can cause a loss of land and physical captial. All of these factors will tend to reduce total factor productivity. The term -deltaS_t equation (6) reflects that CO_2 does not stay in the atmosphere forever, as some of it gets absorbed by the ocean or trees as part of the natural global carbon cycle.
Now suppose for a moment that the ressource allocation in the economy is fully controlled by a social planner who wants to stabalize the concentration of CO_2 in the atmosphere at a constant level Sbar so as to keep global warming at a tolerabel level.
Question 1.2
Derive the constant level of emissions, Ebar, and the constant stock of black capital, Bbar, which will ensure that the CO_2 concentration i kept constant at the desired level Sbar. Show that when the desired CO_2 concentration has been achieved, total output may be written as
see math note 3
Answer: Setting Sprik_t = 0 in (6), due to we assume no evolution, and rearranging, we obtain the level of CO_2 emissions which will ensure that the CO_2 concentration is kept constatnt at the desired level Sbar.
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Inserting (i) in (4), we get the constant stock of black capital which is consistent with the desired constant CO_2 concentration:
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Equation (7) follows directly by instering S = Sbar in (2) and then inserting the resulting expression in (1), using the defintion Abar = A_0 * Sbar^(-gamma)
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Only green capital can now affect an increase or decrease in output, because both factor productivity and and black capital is assumed constant.
Note: The purpose of the next three quesitons is to guide you towards for the economy´s steady state growth rate which will be useful for the subsequent analysis of climate policy
Question 1.3
Define a new variable y_t defined to be Y_t/G_t and use (7) to show that
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The show that when S_t is kept constant, we must have
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So that
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Answer: From (7) we get, when using the y_t defined to be Y_t/G_t
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Dividing by y_t in (iii), we obtain (8)
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From (3) and (5) it follows that for B_t = Bbar, and hence Bprik_t = 0, we have Gprik_t = sY_t. In other words, when black capital is constant, then the the groeth in total capital is only driven by green capital, resulting in Kprik_t equals Gprik_t. The growth is then equal to savings, sY_t. Insterting this in (8) and using the definition y_t defined to be Y_t/G_t, we end up with (9).
Question 1.4
Explain why (9) implies that y_t will stabilize at the constant level
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Answer:
If yprik_t > 0 (growthrate of y_t), equation (9) implies that y_t is increasing over time up to a given thresshold, whereafter the growth rate would ultimately become negative. For the left side of the equation (9) to be positve we need, hence
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Rearanging the equality gives us
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When y_t increases towards the thresshold, the growth rate of y_t, yprik_t, will starts to fall. Eventually yprik_t will can become zero if y_t equals the threshold and negative if y_t is over the thresshold.
Conversely y_t will start to decrease if its over the threshold, due to the yprik_t < 0. Over time y_t will decrease enough to be lees then the threshold, why yprik-t will start to become positive again.
These dynamic forces imply tht yprik will converge on zero. Setting yprik = 0 in (9) and solving for y_t, we obtain the steady state solution (10)
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Question 1.5
Now define the growth rate of output,
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And use (7) plus your previous result
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And (10) to show that in the long run when S_t has been stabilized at Sbar, the growth rate of output will stabilize at the constant steady-state level.
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Give an intuitive economic explenation for the impact of the parameter beta on the steady state growth rate
Answer: From diffentirating (7) we get
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From (10) we know that y_t is constant in the steady state which means that the only growth is coming from
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In steady state. Inserting this in (iv) and solving for g^Y, we obtain (11) which implies that the economy´s long-run growth rate is higher, the greater the value of beta.
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In the steady state, the growth rate of the economy (g^Y) is determined by the factors included in the model, such as green capital and its elasticity (beta). When beta is high, it means that any increase in green capital has a significant positive effect on the overall output of the economy. In other words, the economy benefits more from investments in green capital when beta is higher. With a higher beta, the contributions of green capital to economic growth are amplified. This means that as the stock of green capital grows, it boosts the economy more effectively. This makes the economy more capable of achieving higher long-run growth rates because the positive effects of green capital investments are stronger.
We now assume that ressource allocation in the economy described by equations (1) through (6) is in fact governed by market mechanimes, but that the government can impose a carbon tax at the rate tau_t per tonne of CO_2 emitted, and that the government can also grant a subsidy to green investors at the rate sigma_t per unit of green capital installed.
In other words, the owners of black capital must pay a total carbon tax bill equal to tau_t * E_t = tau_t * B_t per period, while the owners of green capital recieve a total subsidy amount equal to sigma_t * G_t per period.
The Government´s net revenue is returned to the private sector as a lump sum transfer that does not effect investment decisions.
Quesiton 1.6
Capital owners are free to invest in the type of capital that yields the highest marginal return net of taxes and subsidies. Explain (by using (1)) that a capital market equilibrium therefore requires that
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What is the magnitude of the MAC, that is, the marginal social cost of reducing CO_2 emissions by one tonne?
Answer: The marginal return on investment in a unit of black capital (net of the carbon tax paid on the 1 tonne of CO_2 emissions from the use of this capital) is
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Where
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Is the marginal product of black vapita, derived from (1). The marginal return on investment in a unit of green capital, accounting for the subsidy to the use of this type of capital is
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Where
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Is the marginal product of green capital, again derived from (1). In a capital market equlibrium investment in the two types of capital must yield the same marginal return, since otherwise investment would be reallocated from the type if coutal with the lower marginal return to investment in the type of capital the higher marginal rate (the perfect reallocation market). Due to the declining marginal prodcutivity of both types of capital, this reallocation of investment would drive the marginal rates of return into equality. In equlibrium, the right hand sides of (v) and (vi) must therefore be equal to each other, implying that (12) must hold.
By reallocating a unit of investment from black capital to green capital, CO_2 emissions can be reduced by 1 tonne. The social cost of this abatement of CO_2 green capital, CO_2 emissions can be reduced by 1 tonne. The social cost of this abatement of CO_2 emissions is the loss of output caused by the reallocation of investment to a place where it is less efficient used (lower marginal product of capital). This output is equal to the difference between the marginal product of black capital and the marginal product of green capital, so the MAC is
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From (v) and (vi) plus the capital market equlibrium condition that the marginal returns on investment in the two types of cpaital (net of taxes and subsidies) must equal each other, we have
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Thus the MAC can be measured by the sum of the carbon tax rate and the rate of subsidy to green capital.
Question 1.7
Now suppose the government does not levy a carbon tax (i.e., τ_t=0) but only uses the subsidy σ_t to achieve its climate policy target. Suppose further that the subsidy rate σtσt is continuously adjusted so that CO2 emissions are kept at the constant level derived in Question 1.2 ensuring that the CO2 concentration is kept constant at the level Sbar (implying that Bt=Bbar). From our previous analysis we know that the economy will then converge on a steady state where (10) and (11) hold. Now use (12) with τ_t=0 to derive an expression for the ratio of the government’s total subsidy bill to output, σ_t * G_t / Y_t. How will this ratio evolve over time? Will the government be able to maintain the subsidy policy in the long run?
Answer: Setting tau_t = 0 and B_t = Bbar in (12), we finde
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In the steady state, it follows from (10) that G_t / Y_t = y_t is constant and equal to s(1-beta)/g. Inserting this in (ix), we see that in the steady state,
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Since output Y_t is steadily growing in the steady state and the other variables on the right hand side of (x) are constant, it follows that the government´s subsidy will be growing systematically relative to GDP in the long run and ultimately absorb all of GDP. Mathemathically is Y_t increasing on both sides, why the subsidy needs to grow as weel. This is obviously not sustainable, so the subsidy policy will not be viable in the long run.
Question 1.8:
Suppose instead that the government does not offer a subsidy to green capital (σ_t=0) but only imposes a carbon tax (τ_t>0) which is continuously adjusted to ensure that Et=Ebar, implying that the stock of black capital is kept at the level Bbar which stabilizes the CO2 concentration at Sbar so that, once again, the economy converges on the steady state (10) and (11). Now use (12) to derive an expression for the ratio of total carbon tax revenue to GDP, τ_tBbar/Y_t. How will this ratio evolve over time? Will the government be able to maintain its carbon tax policy in the long run? Is the tax policy preferable to the subsidy policy, or vice versa? Motivate your answer.
Answer: Setting sigma_t = 0 and B_t = Bbar in (12), we get
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Again, we can go back to (10) and note that Y_t/G_t = y_t is constant and equal to g/s(1-beta) in the steady state.
Inserting this in (xi), we find that in the steady state,
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Because of the steady growth in output Yt, the last term on the right-hand side of (xii) will converge on zero in the long run, so the government’s total carbon tax revenue relative to GDP will converge on the constant ratio α which is strictly less than one (important for stability). The term Bbar/Y_t on the right-hand side of (xii) will approach zero as Y_t grows and this means that the left-hand side, τ_tBbar/Y_t, will also approach zero in the long run convergin to the constant ratio α.
In contrast to the subsidy policy, the carbon policy is therefore economically sustainable in the long run and for that reason it is clearly preferable to the unsustainable subsidy policy. In addition, the carbon tax policy is consistent with the polluter-pays principle and therefore seems more fair from an environmental policy perspective. Finally, in the real world a subsidy to green investment would have to be financed by distortionary taxes rather than a non-distortionary lump sum tax, so from an economic efficiency perspective the subsidy policy would likewise be less attractive than the carbon tax policy, since the revenue from the carbon tax could be used to reduce other distortionary taxes.
