Question

5. Stoe-Geary preferences and Ramsey economy] Consider the standard Ramsey model of a closed economy, except that the representative house- hold's instantaneous utility function (felicity function) takes on the following Stone-Geary form, so that preferences are no n-homothetic u(c) 1-6 where č0 represents the subsistence level of per capita consumption. Suppose that the production function has the Cobb-Douglas form, and assume that there's no technological progress e. Does the modification of the felicity function affect the steady-state values of k and c? Explain.

Answer 1

STONE–GEARY UTILITY FUNCTION IN RAMSEY ECONOMY:

A utility function is a functional representation of consumer preferences. To make utility functions easy to use we often also assume some extra characteristics: non-satisfaction, convexity, and same preference. We use utility functions to derive demand curve, by opting the mix of goods that maximises utility subject to a budget constraint. In economics, the utility function measures the welfare or satisfaction of a consumer as a function of consumption of real goods such as food, clothing and composite goods rather than nominal goods measured in nominal terms. Utility function is widely used in the rational choice theory to analyze human behavior. Utility is measured in units called utils.

To assume the utility function, economists generally make assumptions about preferences for various goods. A simple example, in certain situations, oats and cornflakes considered to be perfect substitutes for each other, and the appropriate utility function must reflect such preferences with a utility form of u(o, c) = o +c, where "u" denotes the utility function and "o" and "c" denote oats and cornflakes. A person who consumes 1 dollar of oats and no cornflakes derives a utility of 1 util. Utility functions are often expressed as U(x1,x2,x3…) which means that U, our utility, is a function of the quantities of x1, x2 and so on. The utility functions grows with quantity. The Stone–Geary utility function was first derived by Roy C. Geary, in a comment on earlier work by Lawrence Klein and Herman Rubin. Richard Stone was the first to estimate the Linear Expenditure System.

The Stone–Geary utility function takes the form

$U = \prod_{i} (q_i-\gamma_i)^{\beta_{i}}$

where is utility, $q_{i}$ is consumption of good , and $\beta$ and $\gamma$ are parameters. For $\gamma_i = 0$ , the Stone–Geary function reduces to the generalised Cobb–Douglas function. The Stone–Geary utility function gives rise to the Linear Expenditure system in which the demand function equals

$q_i = \gamma_i + \frac{\beta_i}{p_i} (y - \sum_j \gamma_j p_j)$

where is total expenditure and $p_{i}$ is the price of good .

Most growth models assume homothetic preferences (and quite often Cobb-Douglas model). Homotheticity means in cross-sections, the rich & the poor consume goods in the same proportions. With Cobb-Douglas, each sector accounts for a fixed share of the total expenditure.

Consider the standard Ramsey model of a closed economy, except that the representative house-holds instantaneous utility function (felicity function) takes on the following Stone-Geary form, so that preferences are non-homothetic.

Using Stone-Geary Function

The Stone-Geary function is often used to model problems involving subsistence levels of consumption. In these cases, a certain minimal level of some good has to be consumed, irrespective of its price or the consumer’s income. The Stone-Geary uses the natural log function to model utility. The sum of all the proportions of the goods consumed must equal 1. In the problem above, the subsistence levels of A and B are α and β. The term I is income, and pk {k=a,b} are the prices of A and B. A sequence of {Kt , Yt , Ct , It , Wt , RK,t} ∞ t=0 for a given sequence of {Lt , At} ∞ t=0 and an initial capital stock K0, such that (i) the representative household maximizes its utility taking the time path of factor prices {Wt , RK,t} ∞ t=0 as given; (ii) firms maximize profits taking the time path of factor prices as given; (iii) factor prices are such that all markets clear.

The Lagrangean and the First-Order conditions are:

Lagrange function (normalizing L0): L = X∞ t=0 β˜t C˜ 1−θ t 1 − θ + λt (1 − δ)K˜ t + Wt + RK,tK˜ t − C˜ t− −(1 + n)K˜ t+1

First order conditions (FOCs): ∂LL ∂C˜ t = 0 =⇒ C˜ −θ t = λt

∂LL ∂K˜ t+1 = 0 =⇒ βλ˜ t+1 (1 − δ + RK,t+1)) = (1 + n)λt

implying C˜ t+1 C˜ t !θ = β˜ RK,t+1 + 1 − δ (1 + n)

Using the definition of β˜ and rewriting in intensive form: ct+1 ct θ = β RK,t+1 + 1 − δ (1 + g) θ

For consumption per capita, using also the definition of the interest rate rt : C˜ t+1 C˜ t !θ = ct+1At+1 ctAt θ = β (1 + rt+1)

For θ > 0: C˜ t+1 > C˜ t ⇐⇒ 1 + rt+1 > β−1

Interpretation: For (per capita) consumption to grow the (market) interest rate must exceed households’ rate of time preference

Interpretation: It is optimal for households to postpone consumption (i.e. save in the current period and consume more in the next period) if the related utility loss is more than offset by the rate of return on savings.

The higher θ the less responsive consumption to changes in the interest rate. In other words: The higher θ the stronger the consumption smoothing motive (the lower intertemporal substitution).

In additionally, separable utility functions, any deviation from CES would give us non-homothetic preferences.

Suppose that the utility function, U:R^J$\rightarrow$ R, is quasi-concave, increasing, and separable then, it is homothetic. Stone-Geary is highly tractable, because the marginal propensity to consume for all goods is independent of the income, which allows for aggregation across households. However, this same feature of Stone-Geary makes it also highly restrictive: Income distribution across households has no effects on the aggregate demand. The average propensity to consume each good is either monotonically increasing (luxury), monotonically decreasing (a necessity), or constant for all income levels, making it ill-suited for capturing the rich patterns of structural change. Asymptotically homothetic, suggesting that non-homotheticity is merely a transitional problem.

Consider a closed economy without exogenous technology and population growth, where firms produce a generic good Yt with the production function Yt = F (Kt , L) = K α t L 1−α , 0 < α < 1,

where Kt is aggregate capital and L is the number of workers in the economy. The law of motion for aggregate capital is given by Kt+1 = (1 − δ) Kt + It , K0 > 0,

where It denotes aggregate investment, and 0 < δ < 1 the depreciation rate. For simplicity, let aggregate labor supply (population) be equal to one, L = 1, such that consumption per worker, ct , is the same as aggregate consumption, Ct = ct = ctL.

the Ramsey growth model where household utility is maximized ∞ ∑ t=0 β tu (Ct), such that aggregate investment (savings) is endogenous It = Yt − Ct.

Thus in this model markets are competitive, thus input factors Kt and L are paid their marginal products.

The Lagrangian reads L = ∞ ∑ t=0 β tu (Ct) + λt [K α t + (1 − δ) Kt − Ct − Kt+1],

with associated optimality conditions 0 = ∂L ∂Ct = β tu 0 (Ct) − λt 0 = ∂L ∂Kt+1 = −λt + λt+1 h αK α−1 t+1 + (1 − δ) i 0 = ∂L ∂λt = K α t + (1 − δ) Kt − Ct − Kt+1.

Eliminating the Lagrange multiplier the second optimality condition can be written as u 0 (Ct) = βu 0 (Ct+1) h αK α−1 t+1 + (1 − δ) i

Thus the optimality conditions can be summarized as u 0 (Ct) u 0(Ct+1) = β h 1 + α(Kt+1) α−1 − δ i Kt+1 − Kt = K α t − δKt − Ct.

Suppose that the production function has the Cobb-Douglas form, and assume that theres no technological progress e. Does the modification of the felicity function affect the steady-state values of k and c.

The equilibrium dynamics of the model at any time t can be characterized by 3 equations:

ct+1 ct θ = β f 0 (kt+1) + 1 − δ (1 + g) θ

kt+1 kt = 1 − δ (1 + g)(1 + n) + 1 (1 + g)(1 + n) f (kt) − ct kt

and

lim t→∞ kt+1Y t s=1 (1 + n)(1 + g) f 0(kt+1) + 1 − δ ! = 0

At t = 0 capital is fixed. For given initial k0 and c0, equations above describe the future evolution of these variables: kt and ct. In the steady state equilibrium kt and ct must be constant.

Using above equations, the long-run solution to the Ramsey model:

f 0 (k ∗ ) = (1 + g) θ β − 1 + δ

c ∗ = f (k ∗ ) − (n + g + δ + ng)k ∗

Transversally, the condition written in the steady-state: lim t→∞ k ∗ (1 + n)(1 + g) f 0(k ∗) + 1 − δ t = 0

This implies: f 0 (k ∗ ) > n + g + δ + ng

Since fⁿ(k) < 0 for any k > 0

fⁿ(k ∗ ) > f 0 (kG ) =⇒ k ∗ < kG

Equivalently, we can show that the steady-state savings rate s ∗ falls short of the savings rate consistent with the golden rule:

the steady-state savings rate is: s ∗ = 1 − c ∗ f (k ∗) = (n + g + δ + ng) k ∗ f (k ∗)

Using above: s ∗ < f 0 (k ∗ ) k ∗ f (k ∗) = α(k ∗ )

Since households are impatient (β < 1) and smooth consumption (θ > 0, relevant if g > 0). Higher β implies more patient consumers. If β goes up, f 0 (k ∗ ) goes down, which means that k ∗ goes up The ct+1 = ct locus on the (k, c) chart shifts right Steady-state consumption goes up Intuition: if households are more patient, they save more, which brings them closer to the standard golden rule.

For simplicity consider a model without labor or technology growth max:

ct,kt+1,yt U0 = X∞ t=0 β t u(ct) subject to yt = f (kt) and ct + kt+1 = yt + (1 − δ)kt

L = X∞ t=0 β t u(ct) − X∞ t=0 λt [ct + kt+1 − f (kt) − (1 − δ)kt ]

FOC: ct : β t uc,t = λt kt+1 : −λt + λt+1 f 0 (kt+1) + 1 − δ = 0 TVC (imposed) : lim t→∞ λtkt+1 = 0

The optimal alocation is given by uc,t βuc,t+1 = f 0 (kt+1) + 1 − δ TVC : lim t→∞ β t uc,tkt+1 = 0 yt = f (kt) ct + kt+1 = yt − (1 − δ)kt

Use zero profit condition: kt+1 = (1 − δ) kt + yt − ct

Note that uc,t+1 uc,t = β(1 + rt+1) so that Y t s=1 1 1 + rs = βuc,1 uc,0 βuc,2 uc,1 · · · βuc,t uc,t−1 = β tuc,t uc,0.

Hence TVC condition is equivalent to lim t→∞ β t uc,tkt+1 = 0

Thus Ramsey economy reflects two fundamental welfare theorems. First welfare theorem: every competitive equilibrium is efficient. 2 Second welfare theorem: any efficient allocation can be supported as a competitive equilibrium (possibly with lump-sum transfers). In modern economies governments are big players G Y is 40-50% (once all government expenditure is accounted for) Some problems worth analysing: difference/ similarity between tax and debt financing of gov. expenditure reaction of economy to government shocks (spending, taxes).

In the Ramsey model Ricardian equivalence holds: Government plans sustainable - initial debt plus net present value of expenditures equals net present value of revenues T0 + X∞ t=1 Tt Rt = (1 + r0)B0 + G0 + X∞ t=1 Gt Rt where Rt ≡ Y t i=1 (1 + ri).

Household lifetime budget (with government) C0 + X∞ t=1 Ct Rt = (1 + r0)K0 + (1 + r0)B0 + W0L0 + X∞ t=1 WtLt Rt − T0 − X∞ t=1 Tt Rt.

Thus Steady state: f 0 (k ∗ ) = (1 + g) θ − β β(1 − τk ) + δ c ∗ = f (k ∗ ) − (n + g + δ + ng)k ∗ − g ∗

Thus permanent increase in government expenditure (financed with lump-sum taxes), temporary increase in government expenditure (financed with lump-sum taxes) and permanent increase in capital tax.