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Lecture 3
Outline
Optimal commodity tax problem with a single consumer
- The general problem
- Walras Law
- Tax vector normalization (uses CRS)
- Problem I: general CRS technology
- Problem II: linear technology
- Ramsey rule for the linear technology model
- Interpretation of the Ramsey rule
- Inverse elasticity rules
- The general problem The government’s budget constraint is:
(q − p)[x[q, π(p)] + xG] = qxG
The left hand side is the tax revenue the government gathers. The right hand side is the cost of its purchases.
(a) Note that the government cannot satisfy this constraint if people engage in no net trades. If people just consume their endowments, then (q−p)(xG) = qxG^ so −pxG^ = 0. This is impossible since all terms in xG^ are positive: the government makes only purchases, it has no endowment to sell. If people were inclined to consume their endowments, the optimal tax vector would have to induce them not to. (b) The general problem is easier to handle if we rewrite this to eliminate xG from the left hand side. This gives (q − p)x[q, π(p)] = pxG.
The most general way of writing the optimal commodity tax problem is then:
Max V [q, π(p)] q 0 , ..., qn; p 0 , ..., pn subject to: xi(q) + xGi = yi(p), i = 0, ..., n (q − p)x[q, π(p)] = pxG
This, however, involves some redundancy and can be substantially simplified.
- Walras Law
Pre-multiply the set of equilibrium conditions by the producer price vector. This gives:
px[q, π(p)] + pxG^ = py(p)
We have:
py(p) = π(p) = qx[q, π(p)]
The first equality is from the definition, the second is from the individual budget constraint. So:
px[q, π(p)] + pxG^ = qx[q, π(p)]
Therefore:
pxG^ = qx[q, π(p)] − px[q, π(p)] = (q − p)x[q, π(p)]
Thus, the government’s budget constraint is redundant if we have all n + 1 equilibrium conditions. In other words:
(a) We can drop the government’s budget constraint from the problem if we have all n + 1 equilibrium conditions. This is how it is usually used. (b) Alternatively, we can drop one of the equilibrium conditions if we include the government’s budget constraint.
- Tax vector normalization (uses CRS)
Under the assumption of CRS (and the restriction to taxing only net trades), one of the tax rates is redundant.
(a) To see the intuition, recall the labor-income model with profit income. Consumption good is the numeraire. Suppose the government places a tax on net trades of both labor supply and consumption. Let w be the producer price of labor This is the gross wage since the producer buys labor. We model what occurs if the gross wage falls by the full amount of the tax. Then:
x =
[ −L Y
] q = p + t =
[ w + t 0 1 + t 1
]
(e) Note! The untaxed commodity may be a good for which we have an endowment. There is still no loss of generality! The fact that we do not tax a commodity with an endowment does not imply we are foregoing a lump-sum tax. This was already implied by our restriction to taxing just net trades. What has just been shown is that, given the restriction to taxing net trades, there is no further loss of generality (under CRS) to not taxing the good with an endowment at all.
- Problem I: general CRS technology
From our previous results, in the general problem:
(a) We know q 0 = 1 and p 0 = 1, so these do not appear as choice variables. (b) We can eliminate the government’s budget constraint.
If we then substitute the market clearing conditions into the technology the producer prices no longer appear. We therefore have:
Max V (q) q 1 , ..., qn subject to: F [x(q) + xG] = 0
This problem determines the demands that must be met. Given this, the gov- ernment chooses producer prices so that firms will supply the required amount. Those producer prices are determined by the demands and the technological relationships derived earlier:
pi = p 0
∂F/∂yi ∂F/∂y 0
∂F
∂yi
, i = 1, ..., n
where we now use p 0 = 1. We will return to this problem after considering the case of linear technology.
- Problem II: linear technology
(a) With a linear technology each derivative ∂F ∂yi is a constant. These constants then determine the equilibrium producer prices through:
pi =
∂F
∂yi
, i = 1, ..., n
So, technology determines the producer prices and these are fixed.
It must be the case that all factor demand curves and all produced goods supply curves are horizontal. This is what Karl Marx was after – prices are determined independently of demand! Denote the equilibrium producer price vector by:
p∗ (b) Regarding linear technology: i. In general equilibrium, CRS alone does not imply a linear technology. ii. If labor is the only factor in production and all technologies are CRS, then the production possibilities frontier (as a relationship among the produced goods) is linear in every direction. iii. For more general results on linear technology, see Kemp et al. (1978). (c) If all producer prices are fixed, then consumer prices are the producer prices plus taxes: q = p∗^ + t
With p∗^ given and q determined by t, the market clearing conditions can be dropped from the original problem. All that remains is the government’s budget constraint. Since both p∗^ and xG^ are exogenous, we can eliminate p∗xG^ in the con- straint and replace it with R, “revenue.” i. Note! If producer prices change then we can not do this. A given quantity of numeraire would purchase different quantities of goods depending on equilibrium prices. ii. Of course, if the government were not providing a vector of goods, but were merely making a lump-sum return of numeraire, then we would have a revenue constraint with just CRS. The problem in which the government levies commodity taxes on a single individual to make a lump-sum return of numeraire is less in- teresting than the problem we consider. (d) Our problem now becomes:
Max V (p∗^ + t) t 1 , ..., tn subject to: tx(p∗^ + t) = R
- Ramsey rule for the linear technology model
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i = 0, ..., n
σi : [0, ti] → n+
σi(τ ) =
q 0 (t) + t 0 .. . qi− 1 (t) + ti− 1 qi(t) + τ qi+1(t) .. . qn(t)
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i = 0, ..., n
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p∗ 0 + t 0 .. . p∗ i− 1 + ti− 1 p∗ i + τ p∗ i+ .. . p∗ n
∑^ n
i=
ti
∂xk ∂qi
The two expressions are not the same. Regular demand need not have the kind of symmetry that would allow us to convert the latter expression into the former. So, we cannot use the latter to interpret the former. The potential lack of symmetry reflects the fact that the vector of regular demands, (x 0 , ..., xn), is not itself the gradient of something. In particular it is not the gradient of indirect utility. This is (−αx 0 , ..., −αxn), where α is the marginal utility of income.
(d) Using all of the earlier results:
xck(p∗^ + t, U∗) − xck(p∗, U∗) xck(p∗^ + t, U∗)
∑n i=1 Skiti xck (p∗^ + t, U∗)
= −θ < 0 , k = 1, ..., n
It follows that, for any pair of goods k and l: xck(p∗^ + t, U∗) − xck(p∗, U∗) xck(p∗^ + t, U∗)
xcl (p∗^ + t, U∗) − xcl (p∗, U∗) xcl (p∗^ + t, U∗) This gives the Ramsey rule: If all Sij are constant in the relevant range, then if t is an optimal commodity tax vector, then eliminating all taxes should cause an equal percentage change in the compensated demand for all goods. Notice that the post-tax situation is the reference point since this is used in the denominator. Technically the change is from the post-tax vector to the pre-tax vector, not the other way around. (e) In general, optimal taxation implies intervening in every market and at a different tax rate. This leads to a presumption against the efficiency of broad based taxes. On the other hand, there are models of government behavior (positive political economy) that suggest that real governments should perhaps be restricted to broad based taxes since they choose rates to solve a different problem from the one stated above. (f) The formula should not be interpreted as saying that, in the real world, necessities “should” be heavily taxed. The real world has many additional properties including the fact that there is more than one person. While the presumption is that there would be an efficiency loss from not taxing necessities, there may be an equity gain that produces a net gain in social welfare.
(g) Since compensated demand for good k depends on all prices, it is not true that goods that are more own-price inelastic necessarily get higher taxes.
However, if we assume that these cross effects are zero, then we do obtain this result.
- Inverse elasticity rules
(a) Version for Compensated Demand i. Recall from the theory of the consumer Hicks’ three laws: Sik = Ski Skk < 0 ∑^ n
i=
Sikqi = 0
The last follows immediately from the fact that compensated demand curves are homogeneous of degree zero in prices. ii. The last two imply that we cannot assume that all cross effects are zero. For example, if there are just two goods “0” and “1”, they must be Hicksian substitutes (S 01 > 0). However, there is no contradiction in assuming that the only cross effects that are nonzero are with the numeraire. Thus: Sik = 0, i = 1, ..., n, k = 1, ..., n, i = k ∑^ n
i=
Sikqi = Skk qk + S 0 k = 0, k = 1, ..., n
iii. This is a substantive behavioral assumption about cross effects with a particular good. It may be more empirically reasonable for some goods than others. The choice of numeraire “matters” if we are going to express a behavioral restriction in terms of it. iv. The derivation of the Ramsey rule gave us for good k: ∑^ n
i=
tiSik = −θxck, k = 1, ..., n
If the only cross effects are with the numeraire, then: tkSkk = −θxck, k = 1, ..., n Rewriting Skk as the derivative of compensated demand and multiply- ing both sides by qk gives: ∂xck ∂qk
qk xck
≡ ckk = −θ
qk tk The elasticity is independent of all prices other than qk, and it is evaluated at post-tax prices and post-tax utility U∗. Taking reciprocals and using qk = p∗ k + tk: 1 ckk
θ
tk p∗ k + tk
