Fanny Chevalier

Pierre Dragicevic

Christophe Hurter
email: christophe.hurter@enac.fr

Histomages: Fully Synchronized Views for Image Editing

Keywords: Lightness Hue Saturation Saturation ACM Classification: H5.2 [Information interfaces and presentation]: User Interfaces. -Graphical user interfaces. General terms: Design, Human Factors Image editing, synchronized views, histograms

Lightness

Figure 1: Sky enhancement with Histomages: (a) the image is duplicated and its pixels rearranged into a lightness histogram; (b) bright pixels are selected with the rubber-band selection tool; (c) all pixels are rearranged into a hue histogram and yellow pixels are filtered out with the "subtract" selection brush (bottom). Missing pixels are added with the "add" selection brush on the image (top); (d) the sky is enhanced by resizing the selection on the saturation histogram.

INTRODUCTION

Bitmap image editing software like Photoshop or Gimp is extremely powerful in the hands of experts but is hard to master, because of the wealth of functionality and tools that must be learned. Taken individually, most of these tools are relatively easy to learn and to use (e.g., levels, hue shift, color range selection or selection transforms). However, they each follow their own logics and have a different user interface. In order to make image manipulations more flexible and fluent, image editing software needs more unified, coherent interaction models.

Interaction Models

An interaction model is "a set of principles, rules and properties that guide the design of an interface" [START_REF] Beaudouin-Lafon | Instrumental interaction: an interaction model for designing post-wimp user interfaces[END_REF]. Interaction models can range from general principles (e.g., direct manipulation) to more specific guidelines and mechanisms. They help achieve logical soundness and consistency in user interfaces, with benefits to both developers and end users of complex software. Like many authoring and productivity applications, image editors are primarily based on the swiss army knife principle (i.e., a large set of specialized, independent and complementary tools), along with transversal mechanisms such as selection, undo and redo, copy and paste, scripts and layers.

Software companies mainly focus on improving existing tools and adding more powerful ones (often through plug-in mechanisms), while the underlying interaction models are considered immutable. Since these models only weakly enforce consistency between tools -each new tool such as an image filter typically comes with specific controls -image editing applications are increasingly difficult to master and more and more intimidating to beginners as new versions appear. In this article we explore an alternative model for two types of image editing tasks: selection and color manipulation.

Image Selection

Selecting image regions is a key task in image editing and several tools are generally provided. They include imagespace tools (e.g., lasso or rubber-band selection) and colorspace tools (e.g., color range selection). Recently, a method based on color naming has been proposed [START_REF] Heer | Color naming models for color selection, image editing and palette design[END_REF]. Mixed-space tools combine both approaches. They include the magic wand and other "intelligent" cutout tools whose algorithms are still an active topic of research [START_REF] Li | Lazy snapping[END_REF][START_REF] Wang | Soft scissors: an interactive tool for realtime high quality matting[END_REF]. Some of these tools dynamically select image regions while the image is being painted [START_REF] Olsen | Edge-respecting brushes[END_REF]. Advanced selection tools can dramatically facilitate common image editing tasks, but the more elaborate they are, the more difficult it is for users to build a clear mental model of how they work and to predict their results.

Since no selection tool alone can address all selection tasks no matter how sophisticated it is, it is also essential to be able to combine different types of selection techniques. This is achieved in image editing software through the support of boolean operations on selections, a powerful feature that acts as a bridge between selection methods and allows users to, e.g., select a region using a color picker and tolerance slider, then refine it by lassooing on the image. However, since different types of selections involve different interaction techniques, switching between them can still be tedious.

Color Manipulation

Color manipulation is another key task in image editing, particularly in photo editing. The state-of-the-art interaction model consists in filters whose parameters are set in control panels, with immediate preview on the image. The tools are numerous and include luminosity and contrast correction, color balance, and color replacement. Adobe Photoshop CS5 provides 21 color manipulation tools, 3 of which are automatic and 18 have parameters that are exposed in specialized dialog boxes [START_REF]Photoshop CS5[END_REF]. Techniques have also been proposed for controlling parameters directly on the image [START_REF][END_REF][START_REF] Elmqvist | Color lens: Adaptive color scale optimization for visual exploration[END_REF].

Popular tools among photographers are image histograms, which give a rapid overview of how color components (e.g., brightness) are distributed in an image and what corrections need to be applied. Most image editors include an interactive histogram that can be stretched by moving control points, which in turn adjusts colors on the image. Although useful, current histogram tools are often difficult to relate to the image and require trial and error, or thorough practice.

A few professional applications provide more advanced interactive histograms, such as Adobe Lightroom 4 where histograms can be locally stretched by direct manipulation, although only the vertical boundaries of a few predefined histogram regions can be moved, which effectively amounts to moving control points [2]. Capture NX 2 provides histograms where value ranges can be selected and the corresponding pixels are shown on the image with a blinking animation [START_REF][END_REF]. However, this technique cannot be used to select regions of the image for subsequent manipulation.

Synchronized Views

Early uses of the principle of synchronized views can be found in the visual exploration system by Becker and Cleveland [START_REF] Becker | Brushing scatterplots[END_REF] and in the MVC design pattern from the Smalltalk programming language [START_REF] Krasner | A cookbook for using the model-view-controller user interface paradigm in Smalltalk-80[END_REF], later implemented in a number of GUI toolkits. Today, examples of applications with elaborate support for synchronized views include document editing environments that combine a markup editor with a WYSIWYG window [START_REF] Dragicevic | Gliimpse: Animating from markup code to rendered documents and vice versa[END_REF], and a wealth of visual exploration tools [START_REF] Roberts | State of the art: Coordinated & multiple views in exploratory visualization[END_REF][START_REF] Riche | Understanding interactive legends: a comparative evaluation with standard widgets[END_REF]. These tools support synchronized highlighting, i.e., hovering over an element in one view highlights the same element in the other, and sometimes synchronized editing, i.e., edits in either view are reflected in the other.

Synchronized views have been extensively employed and studied in the field of information visualization under the name of coordinated views (see [START_REF] Roberts | State of the art: Coordinated & multiple views in exploratory visualization[END_REF] for a survey). The use of multiple data representations is thought to promote effective visual exploration, especially if tools are provided to help analysts relate different views. A popular approach in the domain is synchronized highlighting, termed brushing and linking [START_REF] Becker | Brushing scatterplots[END_REF][START_REF] Buja | Interactive highdimensional data visualization[END_REF]. Other solutions involve drawing links between views [START_REF] Collins | Vislink: Revealing relationships amongst visualizations[END_REF] or using animated transitions [START_REF] Heer | Animated transitions in statistical data graphics[END_REF]. Synchronized view configuration mechanisms have also been proposed [START_REF] Riche | Understanding interactive legends: a comparative evaluation with standard widgets[END_REF], but synchronized editing has comparatively received little attention in this field, with a few exceptions [START_REF] Baudel | From information visualization to direct manipulation: extending a generic visualization framework for the interactive editing of large datasets[END_REF].

Image editing applications also involve alternative image representations (mostly histograms) but with very primitive support for synchronized views: interaction with histograms is poor and requires tools that are completely distinct from other tools. Yet histograms are just another way of laying out image pixels and could be therefore treated (at least partly) as images. The similarity between histograms and images has been evoked in art [START_REF] Friedman | Idea: The histogram as the image[END_REF] but to the best of our knowledge, it has never inspired any actual tool. In the following we use simple scenarios to illustrate some of the possibilities offered by such an interaction model for image editing.

HISTOMAGES WALKTHROUGH

Histomages employ a classic multi-window image editor interface and a toolbar that contains a small but fully-functional set of traditional image editing tools (Figure 2). The only visible difference is an extra toolbar at the top of each window. One button is for cloning the window (similar to the Clone Window menu command in Photoshop) while other buttons switch between the image view and several types of histogram views (e.g., red, green and blue channels, LAB lightness or hue) and a 2-D lightness/hue scatterplot.

In Histomages, a histogram is a spatial rearrangement of the image's pixels (Figure 1a): all pixels are displayed with their initial color but instead of being laid out on a grid, they are grouped into bins depending on the value of the color channel that is visualized. Pixels overlap vertically in case the histogram does not fit the window, and their stacking order reflects their y-ordering in the original image. Thus, pixels remain coherently grouped, which produces a visual effect similar to stacked graphs [START_REF] Byron | Stacked graphs & geometry & aesthetics[END_REF] and makes it easier to identify image regions. In addition, view switches are smoothly animated by having each pixel rapidly move to its new location. Since histograms are merely images with alternative pixel layouts, all operations that can be done on images can also be done on histograms. We illustrate this with a simple scenario: Pamela was asked to illustrate the cover of a new novel called "The Promised Land". She finds a stock photography (Figure 1a top) and feels it would fit perfectly with a few adjustements. She considers making the sky more blue and the hill more verdant. She also finds that the color of the carpet of yellow flowers is too aggressive and needs to be softened.

Adding and Switching Views

After loading the image in Histomages, Pamela duplicates the window so that one view will be kept to display the image while the other view will show histograms. She first selects the lightness histogram. As the image transitions to the histogram, she sees the sky region quickly move to the right while the mountain region piles up at the center (Figure 1a). On the final histogram, she sees that the two regions exhibit a clean separation in terms of lightness, as opposed to the yellow flowers that clearly span the whole luminosity range.

Selecting Regions of Histograms

Pamela first focuses on the sky. She picks the rubber band selection tool and selects the rightmost region of the lightness histogram (Figure 1b). This effectively selects the brightest pixels on both the histogram and the image view. Unselected pixels fade out to better show the selection.

Although unwanted pixels have been included in the selection, Pamela is aware that the sky and the flowers differ enough in hue. She switches to the hue histogram and indeed sees the flower region and the sky region split out into two separate clusters. Using the brush selection tool and holding the key modifier for the "subtract" mode, she quickly removes all yellow pixels as well as other isolated pixels outside the blue range (see Figure 1c, bottom). As some pixels from the sky are still missing, she switches to the "add" mode and brushes over the missing regions directly on the image (see Figure 1c, top). Now the sky is properly selected.

Dragging and Stretching Histogram Regions

Since the sky appears washed out, Pamela switches to the saturation histogram and indeed notices that the selected pixels are all squeezed to the left, in the low saturation range (see Figures 1d and2). Therefore, she picks the transform tool and drags the selected region to the right, towards higher saturation values on the histogram. This effectively changes the color of all selected pixels so that their saturation match their Although increasing saturation made the sky more blue overall, Pamela does not find the effect very natural. Since clouds are generally supposed to be white, their saturation should probably be left unchanged. Therefore, Pamela drags the selection back to the left, then grabs the right handle and stretches the selection towards more saturated values (Figure 1d). Her idea is to increase the dynamic range of the selected pixels on the saturation channel, and while she is dragging the handle, she does notice that this type of color transformation works much better visually.

More Selecting and Dragging

The various histograms provide many different ways of selecting and recoloring image regions. As another example, when Pamela switches to the hill, she browses through different histograms and finds that it forms a clear bump in the blue channel (Figure 3 top). She brushes the bump then cleans up the selection on the hue and the lightness channels. She then adjusts the hill to a greener tone by dragging the selection to the left of the blue channel. Again, the image is dynamically updated, which in turn updates the histogram in a way that preserves the y-ordering of pixels and makes them appear to "crawl" over the yellow region (Figure 3

bottom).

Two-Axis Scatterplots

To make yellow colors less aggressive and poppies stand out, Pamela decides to retouch the carpet of yellow flowers. She brings up the hue histogram but realizes that the hue channel does not do a good job at separating yellow and green pixels, which in addition to the stems now also include the hill. Therefore, Pamela switches to a 2-D scatterplot visualization of the color space, where the vertical axis is mapped to hue and the horizontal one to lightness (Figure 4 left). Although the scatterplot does not show pixel counts like histograms, it is appropriate for selecting complex ranges of colors. In that case, Pamela needs to select dark to light yellows and orange tones while avoiding red, green and dark pixels. She picks the brush selection tool, applies it starting from a bright yellow color, then uses the immediate selection feedback on the image to help her adjust the trajectory of the brush (Figure 4 middle). She occasionally uses the subtract mode to clean the selection or zooms in to gain more precision.

Once the yellow flowers are properly selected, their hue and lightness can be simultaneously adjusted by freely dragging the selection on the scatterplot. However, since Pamela only needs to wash out the colors, she switches back to the saturation channel and drags the selection to the left. By dragging the selection too far she accidentally discovers that a carpet of white flowers produce a soothing and enchanting effect and thus decides to stick to white flowers.

Painting Histograms

Histomages are also compatible with other traditional image editing tools, which opens up a range of other possibilities in terms of color manipulation. Right after she finished her novel cover, Pamela is asked to produce urgently a cover for a special issue of the Garlic Magazine on a personality.

Pamela downloads a photograph of the personality and decides to produce a Warhol-style posterized cover. She makes four copies of the photo then loads the produced image in Histomages. She then picks the paintbrush tool, and on the red channel histogram, she paints a stripe in a uniform orange (Figure 5). She uses the image preview as a guide, while the histogram is temporarily frozen to prevent the brush from accidentally affecting neighboring pixels. At mouse release, all pixels jump to the corresponding bin of the histogram.

Pamela then paints other stripes of the histogram. Since she wants to preserve the lightness component of the hair, she uses the transform tool and shrinks the selection to zero width instead. Finally, she selects one of the faces on the image view and drags it on the lightness/hue scatterplot until she is satisfied with the variant produced. Through the same method, she produces variants for the two remaining faces.

Quick Comparison with Photoshop

The principal strength of Histomages does not lie in its expressive power (the image manipulations shown can be done with any professional photo editing application) but in the elegance and the concision of its interactions. For example, as and far as our expertise goes, carrying out the task illustrated in Figure 1 with Photoshop CS3 would require a user to:

(1) Select the magic wand or bring up the color range selection dialog (Selection menu) ; click on a pixel of the sky and adjust the tolerance value (text field or slider) until the sky is properly selected ; possibly change the seed pixel and repeat ; if no combination of seed pixel and tolerance is found to work (which is the case in our example), break down the task into several steps using boolean selections through key modifiers ; Close the dialog, pick a tool like polygon selection and clean up the selection on the image. (2) Bring up the Hue/Saturation dialog (Image → Adjustment menu or Ctrl+U) and push the saturation slider to the maximum. This happens to yield a result similar to the one on Figure 1c, but if a result closer to Figure 2 was desired instead, it is not clear how this could be done. Supposing the Levels tool supports saturation, the Hue/Saturation dialog box would have to be closed (all dialogs are modal), and the Levels dialog box would have to be brought up (Image → Adjustment or Ctrl+L) in order to see the histogram and its control handles.

DISCUSSION

We illustrated some of the possibilities of Histomages, a prototype application that is based on a new interaction model for image editing, and rethinks the way selection and color manipulation tasks are carried out. The goal of Histomages is to demonstrate new concepts, more so than to provide a finished product. Therefore, our prototype has limitations and possible extensions that we discuss. We then generalize these concepts and discuss their possible implications for the design of feature-rich applications.

Use by Novices

Retouching images on histograms requires a good understanding of image histograms, i.e., the meaning of different color axes like hue or saturation, and the principle of counting and vertically accumulating pixels of similar value. Although Histomages is initially not meant to be a learning tool, we believe that features like pixel color preservation and animated transitions can help novices understand histograms. We also think that letting users freely explore and manipulate various types of histograms in a consistent manner may have an educational value too. These, however, are only conjectures that need to be further tested in user studies.

Limitations of Selections in Histograms

Selection in histograms is arguably more difficult when there is little contrast between regions. This problem is however common to all color-space selection tools, which are meant to be used when the region of interest differs enough in color. Histomages supports the same tasks but in a more elegant and flexible manner, since it is based on a clear interaction model rather than on black-box color selection algorithms.

Again, no selection tool can solve all selection problems. Intelligent mixed-space selection tools are also limited, as selection is more than automatically detecting contours or color clusters. Ultimately, only the user can tell what is her region of interest so the solution likely lies in familiar, simple and powerful interactions. Boolean operations on selections brought lots of power by allowing selection tools to be combined, while Histomages pushes simplicity and flexibility further by adding logical consistency to these tools.

Currently, Histomages only supports binary selection. A useful extension would be support for fuzzy selections, with support for antialiasing and feathering operations both in the image and in the histogram spaces. Partially selected pixels would travel slower while dragged on a histogram or on a scatterplot, resulting in less prominent edges after color transformation and more natural-looking images. In addition, selected histogram regions are currently manipulated independently from unselected regions, but it could also useful to have a "rubber sheet" feature that preserves boundaries in a similar way to Capture NX 2's direct manipulation tool [START_REF][END_REF].

Non-linear Color Transfer Functions

Our current prototype allows histogram manipulation with the selection transform tool exclusively (move & resize) and therefore only supports linear color transfer functions. Traditional image histograms usually include a third control point and therefore allow for more flexibility in transfer functions, while the Curve tool from Photoshop CS5 supports up to 14 control points. However, in practice 3 to 4 control points are enough to specify transfer functions as complex as S-curves. This is the number of control points supported by interactive geometrical deformation tools such as Photoshop's warp transform, so supporting these tools on histograms would allow specifying non-linear color transfer functions.

Generalization to Other Image Editing Tools

We demonstrated two ways of extending traditional image editing tools so that they can be used on histograms: (a) having the tool affect histogram pixels the same way as image pixels (selection and paintbrush) and (b) interpreting changes in pixel positions within the histogram's coordinate system (the selection transform tool). We experimented with only a few tools, but other common editing tools could be extended to work on histograms following the same two approaches.

A major class of image editing tools are tools that affect pixel colors locally. Beside the paintbrush we implemented, these include the airbrush, eraser, pencil, and brushes that affect pixel lightness or hue. All could be easily modified to work like our paintbrush. We expect these to be useful for retouching non-contiguous close-to-uniform regions. For example, for red eyes removal a user could roughly select the eyes area on the image then paint the hue histogram with a hue brush. This procedure contrasts with current tools, which first require to explicitely select the exact area to be painted.

Another class of tools are pixel displacement tools, of which our selection transform tool is an example. Other geometrical transform tools like the skew, distort, perspective and warp tools could be extended the same way, as well as local displacement tools like the smudge tool. Applied to histograms, these tools could enrich the repertoire of color manipulations. We already mentioned possibilities warp tools could offer in terms of non-linear color transformations.

Other categories of tools would require more experimentation before one can tell whether or not they could be useful on histograms and how to extend their semantics. These include spatial filters (e.g., blur and sharpen) and pixel duplication tools (e.g., copy-paste and the clone stamp). Finally, some tools such as writing text might not make sense at all.

Alternative Pixel Layouts

We illustrated our interaction model on histograms and scatterplots, but other types of pixel layouts can be explored. Visual exploration and complex selections might be facilitated with multidimensional visualizations techniques such as parallel coordinates [START_REF] Hoffman | Dimensional anchors: a graphic primitive for multidimensional multivariate information visualizations[END_REF] or scatterplot matrices [START_REF] Elmqvist | Rolling the dice: Multidimensional visual exploration using scatterplot matrix navigation[END_REF], where data points would be pixels and the dimensions would be color components and pixel coordinates. Some layouts could also be user-defined, in order to let users create meaningful pixel groups and easily reuse them for subsequent manipulations. Especially promising are techniques that combine both approaches such as star plots [START_REF] Kandogan | Star coordinates: A multi-dimensional visualization technique with uniform treatment of dimensions[END_REF], where users directly manipulate axes to create and explore linear combinations of dimensions, or dynamic filtering lenses [START_REF] Hurter | Moleview: An attribute and structure-based semantic lens for large element-based plots[END_REF], where users can switch between different dimension mappings locally.

An Interaction Model for Rich Multi-View Applications

The interaction model behind Histomages reuses and refines the paradigm of synchronized views and is based on the following four general principles:

1. Consistent View Structures. All views are made of similar graphical primitives (pixels in our case) which try to retain their graphical attributes across views. This gives a sense of "conservation of matter" and can make it easier for users to build a mental model of how views are constructed (e.g., what an image histogram means), and how regions of different views relate to each other once they are constructed.

2. Consistent View Interactions. Any tool used to interact with a view can also be used on other views, with identical or similar effects. This is especially relevant for selection tools, because if both multi-view selections and boolean operators are supported, powerful "selection sculpting" tasks can be carried out [START_REF] Elmqvist | Rolling the dice: Multidimensional visual exploration using scatterplot matrix navigation[END_REF]. Furthermore, tool consistency can be beneficial to facilitate learning, since what has been learned for one view can be transferred to other views with little effort (e.g., our paintbrush). Finally, the semantics of some tools can be adapted to the specificities of different views in order to enrich their expressive power (e.g., our transform tool). Overall, applying this principle requires breaking traditional master/slave relationships between views [START_REF] Roberts | State of the art: Coordinated & multiple views in exploratory visualization[END_REF] in favor of multidirectional dependencies [START_REF] Riche | Understanding interactive legends: a comparative evaluation with standard widgets[END_REF][START_REF] Dragicevic | Video browsing by direct manipulation[END_REF], and removing traditional couplings between tools and views in favor of tool polymorphism [START_REF] Beaudouin-Lafon | Instrumental interaction: an interaction model for designing post-wimp user interfaces[END_REF][START_REF] Beaudouin-Lafon | Reification, polymorphism and reuse: three principles for designing visual interfaces[END_REF].

3. Constantly Synchronized Views. The visual consistency between views and the underlying data is maintained at all times, except during modifications that require a view to be temporarily frozen (e.g., using a paintbrush on a histogram). This principle, together with principles 1 and 2, ensures that the system fully supports the traditional synchronized views paradigm: synchronized highlighting (or brushing and linking) is supported through selection tools, while synchronized editing is supported through regular editing tools.

4. Smooth View Transitions. Transitions between views within the same window are smoothly animated, which can further help explain how views are built and clarify how they are mapped [START_REF] Heer | Animated transitions in statistical data graphics[END_REF][START_REF] Chevalier | Using text animated transitions to support navigation in document histories[END_REF][START_REF] Dragicevic | Gliimpse: Animating from markup code to rendered documents and vice versa[END_REF]. The design and implementation of animated transitions are greatly facilitated by the principle 1. We believe this interaction model can make it easier for both experts and beginners to use multi-view interfaces, and can also make interaction less tedious and more rewarding.

However, this interaction model alone cannot support the large variety of tasks supported by feature-rich commercial applications and is not meant to be used as a replacement. It is rather a complement to existing tools and mechanisms and a way to "polish" their logics. Although integrating this model in existing software might require significant refactoring, it does not necessarily involve many changes from the user's perspective. Histomages is fully integrated with the traditional GUI of image editors: it uses the same multiwindow environment and similar visual representations (with some visual enhancements to histograms), and the selection and editing tools it supports are essentially the same.

CONCLUSION

There has been much focus on the problem of "software bloat", often with the assumption that not all features of large applications are useful, or at least not for everyone and not all the time [START_REF] Mcgrenere | Are we all in the same bloat ?[END_REF]. However, complex activities such as image editing cannot be properly carried out without mastering a wealth of tools that are all complementary and all relevant to many tasks. We believe that the major issue with most feature-rich applications is not an excess of functionalities but a lack of logics and consistency in the way they are exposed at the user interface. Often, most development efforts are spent designing new tools, without much concern for the interaction models that allow these tools to work together.

With Histomages, we show that with a proper interaction model, a small number of tools can provide a large set of functionalities (with sometimes unexpected features), while ensuring some logical coherence. Histomages also suggests that new interaction models do not necessarily have to disorient expert users and can be sometimes integrated into existing GUIs without major changes.

Figure 2 :

 2 Figure 2: Increasing the saturation of the sky.

Figure 3 :

 3 Figure 3: Reducing the blue component of the hill.new x-position on the histogram (see Figure2). While she drags the selection, Pamela sees the sky change on the image, and she stops once she is satisfied with the color.

Figure 4 :Figure 5 :

 45 Figure 4: Selection in the Lightness/Hue space.Red

ACKNOWLEDGMENTS

We would like to thank Dustin Freeman and Jean-Daniel Fekete for their comments on this paper.


Gregory Ponthiere
email: gregory.ponthiere@ens.fr.

David Hippolyte D'albis

Marc De La Croix

Paola Fleurbaey

Paolo Gobbi

Alice Piacquadio

Thomas Schoonbroodt

Bertrand Seegmuller

Stephane Wigniolle

Zuber

Fair Accumulation under Risky Lifetime

Keywords: egalitarianism, di¤erentiated mortality, optimal capital accumulation, Golden Rule, fertility. JEL classi…cation: D63, E22, I12, J13

Individuals save for their old days, but not all of them enjoy the old age. This paper characterizes the optimal capital accumulation in a twoperiod OLG model where lifetime is risky and varies across individuals. We compare two long-run social optima: (1) the average utilitarian optimum, where steady-state average welfare is maximized; (2) the egalitarian optimum, where the welfare of the worst-o¤ at the steady-state is maximized. It is shown that, under plausible conditions, the egalitarian optimum involves a higher capital and a lower fertility than the utilitarian optimum. Those inequalities hold also in a second-best framework where survival conditions are exogenously linked to the capital level.

Introduction

Following early contributions by [START_REF] Phelps | The golden rule of accumulation: a fable for growthmen[END_REF][START_REF] Phelps | Second essay on the golden rule of accumulation[END_REF] and [START_REF] Diamond | National debt in a neoclassical growth model[END_REF], the study of the optimal capital accumulation consists in deriving formal conditions under which the capital stock maximizes the consumption per head at the stationary equilibrium. In a two-period overlapping generations (OLG) model with no technological progress, the Golden Rule states that steady-state consumption per head is maximized when the marginal productivity of capital equals the sum of the cohort growth rate and the depreciation rate of capital.

The standard Golden Rule concerns the long-run dynamics of an economy. Given that the population is likely to vary strongly over long periods of time, it is natural to explore whether the Golden Rule remains valid in a more general framework with a varying demography. In a pioneer contribution, [START_REF] Samuelson | The optimum growth rate for population[END_REF] showed that, when the social planner chooses the fertility rate in addition to capital, the Golden Rule still prevails, with the di¤erence that the cohort growth rate in the Golden Rule formula takes now its optimal value. More recently, the Golden Rule was also shown to be robust to the inclusion of the survival rate in the social planning program (see [START_REF] De La Croix | How powerful is demography? The Serendipity Theorem revisited[END_REF].

But the introduction of a variable population size raises serious di¢ culties regarding the selection of an adequate social objective. Those di¢ culties are general, and occur each time we face what [START_REF] Par…t | Reasons and Persons[END_REF] called a "di¤erent number choice", including in the particular context of optimal capital accumulation. 1 For instance, the "optimal" fertility derived by [START_REF] Samuelson | The optimum growth rate for population[END_REF] depends on the particular social objective -average utilitarianism -that he assumed. Other normative criteria could have been used instead, as there is no decisive argument supporting that particular social welfare criterion. 2 The existence of risky and unequal lifetimes tends also to question the social objective to be pursued when considering optimal capital accumulation. 3 To see this, consider a 2-period OLG economy, where the survival from the …rst period (working period) to the second period (retirement) occurs with an exogenous probability . Under standard assumptions on individual preferences, all young agents save, at the laissez-faire, some resources for their old days. The problem is that, at the next period, only a fraction of savers will enjoy their savings, whereas a fraction 1 of the savers will not. 4 It follows from this that, at the laissez-faire, there exist substantial welfare inequalities across members of a given cohort, between agents who turn out to be long-lived and agents who turn out to be short-lived. The latter ones su¤er not only from a shorter life, but, also, from the lost savings due to unanticipated death.

From a normative perspective, welfare losses due to longevity inequalities are problematic. Indeed, even if it is true that all agents did, at the young age, what they planned to do, and saved as much as they wanted, it remains nonetheless that those savings decisions were made ex ante, that is, before one knows one's own longevity, and, as such, had to rely on survival probabilities. Those probabilities are useful information when making decisions in front of risk, but the problem is that, at the individual level, those probabilities all turn out to be false ex post. In the second period, each agent is either alive or dead, and if he had known, in advance, his actual longevity -rather than his expected longevity -he would probably have saved di¤erently. One can thus, following [START_REF] Fleurbaey | Assessing risky social situations[END_REF], interpret situations of risk as situations of incomplete information: young agents who save resources for their old age do not know their actual length of life, and this lack of knowledge casts some doubt on the capacity of those agents to do what is the best for them.

Interpreting situations of risk as situations of incomplete information supports, as argued by [START_REF] Fleurbaey | Assessing risky social situations[END_REF], the replacement of (standard) ex ante social welfare criteria, which rely on individual expected outcomes, by ex post social welfare criteria, which are based on individual realized outcomes. Note that those two families of social criteria can, in some cases, lead to similar optimal allocations. For instance, as emphasized by [START_REF] Hammond | Ex ante and ex post welfare optimality under uncertainty[END_REF], the average utilitarian social objective can be interpreted either as the expected utility of a representative agent, or as the average realized welfare in the society. However, once there is some ethical concern for equality, the distinction between ex ante and ex post social welfare criteria becomes crucial. 5In the example of prematurely dead savers, where longevity inequalities are exogenous, there is a strong support for adopting an egalitarian social objective. Indeed, as argued by [START_REF] Fleurbaey | Fairness, Responsibility and Welfare[END_REF], welfare inequalities are acceptable only if these result from di¤erences in responsibility characteristics, but not if these result from circumstance characteristics, for which individuals cannot be held responsible. Given that prematurely dead savers cannot be regarded as responsible for their realized longevity, there exists a strong ethical support for adopting egalitarian social objectives in that context. Therefore, if we interpret the situation of risky longevity as a situation of incomplete information, there is a strong support for adopting an ex post egalitarian social objective, which favors the reduction of inequalities in ex post lifetime welfare.

That rationale has led [START_REF] Fleurbaey | Compensating the dead? Yes we can[END_REF] to study, in a two-period model with risky lifetime, the optimal allocation of resources, under a social objective focusing on the realized welfare of the individuals who turn out to be the worst-o¤ ex post, that is, after individual longevities have been revealed. 6 In that paper, it is shown that the ex post egalitarian optimal allocation achieves a full compensation for premature death, that is, it makes, ex post, short-lived and long-lived agents equally well-o¤. That result is somewhat surprising: compensation for a premature death may seem a priori unfeasible, because short-lived individuals cannot be identi…ed ex ante, and cannot be a¤ected by any transfer ex post. However, it is possible to provide a full compensation to the short-lived by concentrating the consumption of resources early on the lifecycle.

Whereas that ex post egalitarian social objective may seem quite plausible in a static framework, it is not obvious to see what its consequences are for the optimal capital accumulation problem. How would a fair capital accumulation rule look like? To what extent would this di¤er from the standard Golden Rule? More precisely, does egalitarianism lead to a lower or, alternatively, to a higher optimal capital in comparison to standard average utilitarianism?

The goal of this paper is to reexamine the optimal capital accumulation in an economy where individual lifetime is risky, and where individuals turn out to have unequal longevities. For that purpose, we consider a two-period OLG model where the probability of survival to the old age (second period) depends on the level of preventive health investment per worker made during the young age (…rst period). Within that framework, we characterize the optimal capital accumulation rule under two distinct long-run social optima: on the one hand, the average utilitarian criterion, which focuses on the steady-state average welfare; on the other hand, the ex post egalitarian criterion, which focuses on the welfare of the worst-o¤ individuals born at the steady-state.

The reason why we introduce here an endogenous survival probability has to do with the recent emphasis on the determinants of mortality variations in the OLG literature. 7 Note, however, that the above discussion on the choice of an adequate social objective under unequal longevities is also relevant for that literature. For instance, models by [START_REF] Blackburn | A model of longevity, fertility and growth[END_REF], [START_REF] Chakraborty | Endogenous lifetime and economic growth[END_REF] and [START_REF] Galor | Natural selection and the evolution of life expectancy[END_REF] are frameworks where agents do not choose their life expectancy, which depends on the level of human capital (in Blackburn and Cipriani), on the level of physical capital (in Ckakraborty), or on the genetic type of the agent (in Galor and Moav). Other models, such as [START_REF] Bhattacharya | Public and private expenditures on health in a growth model[END_REF] and Boucekkine and La¤argue (2010) include agents who can a¤ect their life expectancy through some investment in their health, but still cannot choose a certain lifetime. In those models, agents with the same health investment can actually turn out to have unequal longevities.8 Therefore, those economies, where agents can hardly be regarded as responsible for inequalities in realized longevity, invite an ex post egalitarian social objective. The present paper proposes to characterize optimal capital accumulation under such a social objective, and contrasts it with the one under standard average utilitarianism.

Anticipating on our results, we …rst show that, when the fertility rate is exogenously given, the egalitarian social optimum involves the same optimal capital as under average utilitarianism. However, once fertility is chosen by the social planner, the egalitarian optimum involves, under plausible conditions on production, a higher optimal capital and a lower optimal fertility than the average utilitarian optimum. Thus being egalitarian reinforces the necessity to accumulate capital. Finally, we show that our results remain globally valid in a second-best framework where longevity investments are not directly chosen by the social planner, but are exogenously related to the capital level.

The rest of this paper is organized as follows. Section 2 presents the model. Section 3 derives optimal capital and fertility under average utilitarianism. Section 4 characterizes the social optimum under an ex post egalitarian objective. Section 5 considers the second-best framework where health investments are exogenously linked to the capital level. Section 6 concludes.

The model

We consider an OLG model with the same population structure and production technology as in the model developed by [START_REF] Chakraborty | Endogenous lifetime and economic growth[END_REF]. Time is discrete and goes from 0 to in…nity. Longevity is risky: all agents live a …rst period (young age), but do not necessarily survive to the second period (old age).

Demography. The size of the cohort born at time t is denoted by L t . Each young adult has n children (n 0):

L t+1 = nL t (1)
All members of cohort t live the …rst period for sure, but only a proportion t+1 of that cohort will survive to the second period. The proportion of survivors t+1 depends on the amount of health expenditures per young adult h t :

t+1 = (h t ) (2)
We assume, as usual, that 0 < (h t ) < 1 for all h t 0, and that 0 (h t ) > 0. It is also assumed that (0) = ~ > 0 and that lim ht!1 (h t ) = < 1.

Preferences. Agents have time-additive lifetime welfare, which depends on consumption per period. If the utility of being dead is normalized to zero, the lifetime welfare of the short-lived agents of cohort t, denoted by U sl t , is:

U sl t = u(c t ) (3)
where we assume, as usual, that u 0 (c t ) > 0 and u 00 (c t ) < 0, that is, that temporal welfare is increasing and concave in consumption c t .9 For the long-lived members of cohort t, lifetime welfare, denoted by U ll t , is: 10

U ll t = u(c t ) + u(d t+1 ) (4)
where d t+1 is the consumption at the old age.

When considering the optimal health investment, a central aspect of preferences concerns the intercept of the temporal utility function. Following the literature, we assume that there exists a consumption c 0 such that: 11 u( c) = 0

(5)

Put it di¤erently, there exists a welfare-neutral consumption level c that makes an individual indi¤erent between further life with that consumption and death.

Production. The production of an output Y t involves labour L t and capital K t , according to the production function:

Y t = F (K t ; L t ) (6)
where K t denotes the total capital stock, while L t is the labour force, which coincides here with the whole cohort at birth. The function F ( ) is a positivelyvalued production function, increasing and strictly concave with respect to capital. Capital depreciates at a constant rate , with 0 < 1. Under constant returns to scale, the production function can be written as:

y t = f (k t ) (7)
where y t is the output per worker, and k t is the capital stock per worker, while f ( ) = F (k t ; 1) denotes the production function in its intensive form. Under the above assumptions on F ( ), we have, for

k t > 0, that f (k t ) > 0, f 0 (k t ) > 0 and f 00 (k t ) < 0.
12 The marginal productivity of labour is:

!(k t ) = f (k t ) k t f 0 (k t ) (8)
From the concavity of f (k t ), we have:

!(k t ) 0 and ! 0 (k t ) = f 00 (k t ) > 0.
Resource constraint The resource constraint of the economy, stating that what is produced is either consumed or invested, is:

F (K t ; L t ) = c t L t + h t L t + d t (h t 1 )L t 1 + K t+1 (1 )K t (9)
At the steady-state and in intensive terms, we have:

f (k) k(n + 1) = c + h + (h) n d (10)
Note the roles played by demographic variables in the resource constraint. The fertility rate n has two e¤ects: on the one hand, n puts some pressure on the available resources (the LHS), by reducing, ceteris paribus, the sustainable capital available for each worker (i.e. the "Solow e¤ect"); on the other hand, n also a¤ects the spending part of the constraint (the RHS), by weakening the weight of the elderly's consumption (i.e. the "Samuelson e¤ect"). The survival rate only a¤ects the economy's spending (RHS), by making the old's (aggregate) consumption larger ceteris paribus.

The utilitarian …rst-best problem

To characterize the optimal capital accumulation, we will focus here on a stationary equilibrium, where all per worker variables k t , h t and t are constant over time, whereas all aggregate variables, such as production Y t , total capital stock K t , and total consumption C t grow at a rate n. 13 We consider the problem of a utilitarian social planner, who selects capital, consumptions, health investment and fertility in such a way as to maximize the average lifetime welfare prevailing at the stationary equilibrium, subject to the resource constraint of the economy. That planning problem is: [START_REF] De La Croix | A Theory of Economic Growth. Dynamics and Policy in Overlapping Generations[END_REF] on this. 1 3 Therefore we will, in the rest of this section, delete time indices.

max c;d;k;h;n u(c) + (h)u(d) s.t. c + (h) n d = f (k) k (n + 1) h s.t. h 0 1 2 See de la

6

The planner's problem can be rewritten by means of the following Lagrangian:

max c;d;k;n;h u(c) + (h)u(d) + f (k) k (n + 1) h c (h) n d + h
where is the Lagrange multiplier associated to the resource constraint, while is the Lagrange multiplier associated with the non-negativity constraint for h. The …rst-order conditions (FOCs) can be rewritten as:

u 0 (c) u 0 (d) = n (11) f 0 (k) = + n 1 (12) k = (h) (n) 2 d (13) u 0 (c) 0 (h) [u(d) du 0 (d)] = (14)
as well as the conditions

0; f (k) k (n + 1) h c + (h) n d (15) 0; h 0 (16)
with complementary slackness.

From condition (11), we see that the marginal rate of substitution between consumptions at the two ages of life should be equal to the optimal fertility rate n. Expression (12) is the standard Golden Rule for optimal capital accumulation, stating that the marginal productivity of capital should be equal to the sum of the depreciation rate of capital and the cohort growth rate n 1.

Condition (13) states that the optimal fertility n should be such that the (negative) capital dilution e¤ect (the LHS) equals the (positive) intergenerational e¤ect (the RHS). Note that, on the basis of equation ( 13), nothing guarantees that the optimal fertility n is higher (or equal to) the replacement fertility level n = 1. Thus it is possible to have an optimal fertility n strictly below the replacement fertility rate (n = 1), with the corollary that the population becomes extinct asymptotically. That case is especially likely to prevail when the intergenerational redistribution e¤ect is small in comparison with the capital dilution e¤ect.

Condition ( 14) states that an interior optimal health investment h exists only if there exists some level of h > 0 such that the marginal welfare loss from foregone consumption (1st term of the LHS) equals the marginal welfare gain from health investment (2nd term of the LHS), yielding = 0. That condition holds if and only if we have, under h > 0:

u 0 (c) = 0 (h) [u(d) du 0 (d)]
That equality requires that the second period is, at the optimum, worth being lived, that is, that u(d) > 0 or d > c. This condition is satis…ed either when individuals assign a high value to life (i.e. c is low), or when the economy is very productive, insuring that the optimal second-period consumption exceeds c. 14 That latter assumption is quite weak, and so we can expect, under general conditions, an interior optimal health investment.

Proposition 1 summarizes our results.

Proposition 1 The average utilitarian …rst-best optimum fc ; d ; k ; n ; h g is such that:

u 0 (c ) u 0 (d ) = n f 0 (k ) = + n 1 k = (h ) (n ) 2 d u 0 (c ) = 0 (h ) [u(d ) d u 0 (d )] c + (h ) n d = f (k ) k (n + 1) h h > 0
Proof. See the above FOCs.

The above conditions are necessary, but not su¢ cient for an interior optimum. As it is well-known, su¢ ciency requires to consider second-order conditions, which may not be satis…ed under endogenous fertility and mortality. Given that the literature has already examined those second-order conditions, we will here assume that second-order conditions are satis…ed, so that the above …rst-order conditions characterize the interior social optimum. 15 Finally, it should be stressed here that the conditions mentioned in Proposition 1 may be satis…ed by several distinct vectors fc ; d ; k ; n ; h g, so that those conditions do not guarantee the uniqueness of the social optimum. Actually, in the simple case where demography is …xed, i.e. n = n and = , simple assumptions on the production function f (k) [f (k) > 0, f 0 (k) > 0, f 00 (k) < 0 for k > 0] and on the utility function u(c) [u(c) > 0, u 00 (c) < 0 for c > 0, as well as lim c!0 u 0 (c) = +1], together with the assumption f 0 (+1) < + n 1 < f 0 (0), su¢ ce to guarantee the uniqueness of the utilitarian social optimum (see de la Croix and Michel 2002 p. 78-79). However, once n and are variables, the uniqueness issue becomes much trickier, since standard assumptions on production and utility functions do no longer guarantee the uniqueness of the social optimum. We do not explore the uniqueness issue further here, since our emphasis in this paper lies on the comparison of that utilitarian optimum with the egalitarian optimum, and, for that purpose, the uniqueness issue is not crucial. Actually, the comparison of the two optima, based on …rst-order conditions, keeps its entire signi…cance even when the uniqueness of the social optima under comparison does not prevail, because we know that the optimum, whatever it is, must necessarily satisfy the associate set of …rst-order conditions. 1 4 Note that assigning a high value to life coïncides with a low level of c, and not with a high level of c. The reason is that, if agents value strongly the mere fact of being alive, even extremely low consumption levels will make them regard life as better than death. Therefore the threshold c must be very low in that case.

1 5 See [START_REF] Deardor¤ | The optimum growth rate for population: comment[END_REF] and [START_REF] Michel | Population growth and optimality: when does the serendipity hold[END_REF] on the interiority of the optimal fertility rate. See de la Croix et al (2012) on the interiority of the optimum survival rate.

The egalitarian …rst-best problem

Let us now contrast the average utilitarian social optimum with the ex post egalitarian social optimum. Whereas average utilitarianism can be interpreted as the maximization of the expected welfare of a representative agent from an ex ante perspective (see [START_REF] Hammond | Ex ante and ex post welfare optimality under uncertainty[END_REF], the ex post egalitarian optimum considers, on the contrary, what maximizes the realized -rather than expected -welfare of the worst-o¤ individuals at the stationary equilibrium.

The motivations for focusing on the worst-o¤ individuals are the following. Clearly, the social objective behind the standard Golden Rule -i.e. the maximization of the steady-state average welfare -makes a lot of sense in the context where the Golden Rule was initially studied: a context where cohorts are all made of identical agents. However, once risky mortality is introduced, taking the maximization of average welfare seems less adequate from an ethical perspective. In our model, individuals are all identical ex ante, but some persons will, ex post, turn out to have a shorter life than others. Given that being long-lived or short-lived is here a matter of pure luck, it makes sense to take, as a social objective, not the maximization of average welfare, but, rather, the maximization of the welfare of the unlucky, worst-o¤ individuals. 16 Since the worst-o¤ agents at the steady-state can be either the short-lived, who enjoy a lifetime welfare equal to u(c), or the long-lived, who enjoy a lifetime welfare u(c) + u(d), the social planner's problem is here:

max min c;d;k;h;n fu(c); u(c) + u(d)g s.t. c + (h) n d = f (k) k (n + 1) h s.t. h 0
That social planning problem is less straightforward than the standard utilitarian problem. The reason is that the objective function to be maximized is continuous but not di¤erentiable. However, the identi…cation of the worst-o¤ agents in the population can help us to rewrite that planning problem in a more convenient manner.

Actually, whereas common sense identi…es the worst-o¤ individuals with the short-lived ones, this is not necessarily the case. Whether the short-lived or the long-lived is the worst-o¤ depends on whether second-period consumption d exceeds the welfare neutral level c or not:

U sl 7 U ll () u(d) ? 0 () d ? c
In the light of that condition, we can rewrite the above social planner's problem under a much simpler form. Actually, the egalitarian planning problem can be rewritten as the maximization of the welfare of the short-lived individuals (i.e. u(c)) subject to the constraint that the lifetime welfare of the long-lived individuals is not inferior to the lifetime welfare of the short-lived ones (i.e. u(c) + u(d) u(c)). Indeed, if we had u(d) < 0, long-lived agents would be worse o¤ than short-lived agents, so that the objective of maximizing the welfare of the short-lived would not make sense. Hence the egalitarian social planner's problem consists in maximizing the welfare of the short-lived subject to the constraint u(d) 0, and can thus be rewritten as:

max c;d;k;n;h u(c) + f (k) k (n + 1) h c (h) n d + [u(d)] + h
where denotes the Lagrange multiplier associated to the resource constraint, is the Lagrange multiplier associated with the non-negativity constraint for welfare at the old age, whereas is the Lagrange multiplier associated with the non-negativity constraint for health investment. The …rst-order conditions are:

u 0 (c) = (17) (h) n = u 0 (d) (18) f 0 (k) = n + 1 (19) k = (h) (n) 2 d (20) 1 + 0 (h) n d = (21)
as well as the conditions

0; f (k) k (n + 1) h c + (h) n d (22) 0; u(d) 0 (23) 0; h 0 (24)
with complementary slackness.

From FOC ( 19), it appears that the optimal capital accumulation remains determined by the standard Golden Rule. FOC (21) suggests that the optimal level of health investment h is a corner solution, and equal to 0. 17 That corner solution for health investment may seem surprising, but is actually quite intuitive: for the unlucky agents who do not reach the old age, preventive spending is a waste of resources, so that it is optimal to set those expenditures to zero. Regarding the last two FOCs, it is clear that it is optimal to exhaust the economy's resources, so that the budget constraint is, at the optimum, a strict equality. Moreover, regarding consumption at the old age, we must have, at the optimum, that second-period consumption d equals c, in such a way as to obtain u(d) = 0.

Substituting for h = 0 and for d = c in (20), we obtain:

k = ~ (n) 2 c
We see that, ceteris paribus, the capital dilution e¤ect (i.e. the LHS) is the same as under average utilitarianism, but the intergenerational redistribution e¤ect 1 7 Indeed, an interior health investment would require the LHS of ( 21) to be equal to 0, that is:

u 0 (c) + u 0 (d)d 0 (h) (h) = 0
The …rst term of the LHS is necessarily positive, and the second term as well, so that this equality is always violated.

(i.e. the RHS) is here weakened in comparison to what holds under utilitarianism. The reason is that, under an interior utilitarian optimum, we necessarily have u(d) > 0, implying d > c. 18 As a consequence, we can expect, ceteris paribus, fewer children under egalitarianism than under average utilitarianism.

Thus the possibility of an optimal fertility rate strictly smaller than the replacement rate n = 1 is, ceteris paribus, more plausible here than under the average utilitarian optimum. The possibility of an optimal fertility lower than the replacement rate is, ceteris paribus, larger when the minimum survival chance ~ is low, and when the welfare neutral consumption level c is also low. 19 Note, however, that this result is only true under a given k, so that further calculations will be necessary to have an unambiguous comparison of the two social optima. Prior to this, Proposition 2 summarizes our results.

Proposition 2 The egalitarian …rst-best optimum fc ; d ; k ; n ; h g is such that:

c = f (k ) k (n + 1) ~ n c d = c f 0 (k ) = + n 1 k = ~ (n ) 2 c h = 0
Proof. See the above FOCs.

As under the utilitarian optimum, it may be the case that several vectors fc ; d ; k ; n ; h g satisfy the above conditions. Hence the uniqueness of the egalitarian optimum is not guaranteed. However, in order to compare the utilitarian and the egalitarian optima, it is not necessary to be certain that each of those social optima is unique. Indeed, given that each social optimum necessarily satis…es a set of …rst-order conditions (mentioned in Proposition 1 for the utilitarian optimum, and in Proposition 2 for the egalitarian one), the comparison of the two optima based on the study of those …rst-order conditions leads to conclusions that hold whatever those optima are unique or not.

When comparing the egalitarian and utilitarian …rst-best optima, some obvious di¤erences appear. First, the optimal level of prevention under the egalitarian optimum, which is equal to zero, is lower than under utilitarianism (when the utilitarian optimum is an interior solution). The reason is that, from the point of view of average welfare, it makes sense to spend some resources to raise the survival probability. On the contrary, from the point of view of a short-lived person, preventive health spending are just a waste of resources.

The comparison of optimal consumption levels is more complex, since it depends on the optimal level of capital and fertility under the two social objectives. Let us take, as a starting point, the case where fertility is not chosen by the social planner, but is equal to an exogenous level n = n. In that case, the optimal capital satis…es, under utilitarianism and egalitarianism, the same condition, which is the standard Golden Rule:

f 0 (k ) = + n 1
Given that both and n are constant, the optimal capital is the same whatever the social objective is. That result is quite intuitive. Our egalitarian social objective cares about the worst-o¤ individuals living at the steady-state, and, as such, focuses on intragenerational justice. On the contrary, the Golden Rule, which aims at maximizing consumption possibilities at the stationary equilibrium, concerns the intergenerational allocation of resources, and remains thus relevant under our (intragenerational) egalitarian objective. Indeed, the Golden Rule guarantees the largest aggregate consumption possibilities at the steadystate, and those consumption possibilities can be used di¤erently depending on the adopted conception of intragenerational justice.

But that kind of intra-versus intergenerational separation result collapses once fertility is chosen by the social planner. Indeed, in that general case, the optimal capital and fertility levels satisfy respectively: 20

f 0 (k u ) = + n u 1 k u = (h u ) (n u ) 2 d u
for the utilitarian optimum, and

f 0 (k e ) = + n e 1 k e = ~ (n e ) 2 c
for the egalitarian optimum. From those conditions, it can be shown that the two optimal capital and fertility levels di¤er under the two social objectives. While calculations are left to the Appendix, Proposition 3 summarizes our results.

Proposition 3 Under the utilitarian and the egalitarian optima, we have: Proof. See the Appendix.

if f 0 (k) + 1 > 2k! 0 (k),
Hence, the egalitarian optimum may, in general, involve a lower or a higher capital than the utilitarian optimum once the fertility is also chosen by the social planner. The prevailing inequality depends on the level of the depreciation rate of capital , as well as on the properties of the production process.

As stated in the second part of Proposition 3, we can see that, under a full depreciation of capital = 1 and a Cobb-Douglas production function with < 1=2 (as usually calibrated), the optimal capital under egalitarianism is superior to its level under average utilitarianism. Inversely, the optimal fertility is larger under utilitarianism. Note that this comparison does not tell us whether a particular social optimum involves a fertility rate strictly lower than the replacement fertility n = 1. However, we know for sure that, in the case of < 1=2, if the utilitarian optimum involves a fertility rate lower than the replacement level, this will be also the case for the egalitarian optimum, since this optimum involves a lower fertility than the utilitarian one.

Hence, even if the optimal capital remains, under egalitarianism, still determined by an accumulation rule that looks pretty much like the standard Golden Rule, i.e. f 0 (k) = + n 1, the mere fact that egalitarianism involves a lower optimal fertility implies also a higher optimal capital than under average utilitarianism. The reason is the following. Egalitarianism, by focusing on the worst-o¤ individuals in the cohort, recommends a welfare-neutral consumption at the old age, which restricts the intergenerational gains from fertility. The "Samuelson e¤ect" being reduced, the optimal fertility under egalitarianism is lower, implying a higher optimal capital. Thus, contrary to what one may expect, egalitarianism reinforces the necessity to accumulate capital.

In sum, the egalitarian social objective recommends, in comparison to average utilitarianism, a lower old-age consumption, and a lower preventive health spending. Those di¤erences leave the optimal capital equal to its utilitarian optimal level when the fertility rate is exogenously given. However, once fertility is also a part of the social planner's optimization problem, the egalitarian optimum involves, in general, a higher capital and a lower fertility than under utilitarianism. Thus, it is through the optimal fertility decision that the shift to egalitarianism a¤ects the optimal capital accumulation. Without that channel, egalitarianism would imply the same capital level as under utilitarianism.

The egalitarian second-best problem

The egalitarian …rst-best solution may face strong resistance among policy circles. Imposing a zero investment in health in the name of the -possibly fewunlucky individuals who turn out to be short-lived is questionable. This is the reason why we now turn to a second-best framework, where health expenditures take an interior value. In this section, we will consider the social planner's problem in a second-best context, where health spending are not directly chosen by the social planner. For that purpose, we consider three distinct cases: 21 A1 Health spending per worker are constant: h t = h > 0. A2 Health spending per worker are a constant fraction of labour productivity: h t = !(k t ).

Concluding remarks

The goal of the present paper was to revisit the optimal capital accumulation in the context where lifetime is risky and where individuals turn out to have unequal longevities. For that purpose, we compared, in a two-period OLG model, two long-run social optima. On the one hand, the average utilitarian criterion, which consists of maximizing the average welfare at the stationary equilibrium; on the other hand, the ex post egalitarian criterion, which consists of maximizing the welfare of the worst-o¤ born at the steady-state.

It has been shown that whether egalitarianism leads to a higher optimal capital or not depends on whether fertility is chosen by the social planner or not. When fertility is not chosen, the optimal capital is, under egalitarianism, exactly the same as under utilitarianism. Indeed, in that case, egalitarian concerns, which remain intragenerational, are not relevant for discussing the optimal capital accumulation, which is an intergenerational issue. However, that intragenerational versus intergenerational divide vanishes once fertility is chosen by the social planner. Indeed, in that case, the egalitarian optimum involves, under general conditions, a higher optimal capital than under utilitarianism, and, also, a lower optimal fertility. The reason is that egalitarianism, by implying lower old-age consumption, weakens intergenerational gains from fertility, i.e. the Samuelson e¤ect, and, hence, the optimal fertility. Thus egalitarian concerns do not reduce, but do actually raise optimal capital accumulation. That result is somewhat surprising, since one would expect egalitarian concerns to lower the optimal capital. Nonetheless, the present study shows that, contrary to what one may expect, capital accumulation becomes even more necessary when one is only concerned with the worst-o¤s. That result was also shown to be globally robust to the shift to a second-best set up where health investment is exogenously related to the capital level, even if the introduction of some constrained level of health investment reduces, in general, the optimal capital in comparison to the …rst-best.

Finally, let us notice some limitations of the present study. A …rst issue that remains to be explored consists of the implementability of the egalitarian optimum characterized here. That question can be formulated as follows: under which …scal instruments could a government decentralize the egalitarian social optimum, and induce the fair capital accumulation? That question is not a trivial one, especially when fertility is also endogenous, since, in that case, the study of the decentralization is likely to be sensitive to the structure of parental preferences in terms of fertility. As such, the study of decentralization would require another paper on its own. Besides the implementation issue, one should also remind that the capital accumulation rule studied here is "fair" only in a particular sense. That capital accumulation rule, based on the maximization of the worst-o¤'s welfare, is concerned with correcting for the arbitrariness of the time of death. However, there exists, from the point of view of history, another major source of welfare inequalities, which is also arbitrary: the time of birth. Given that the present study focused on the stationary equilibrium only, this could not deal with that issue, which would require to explore the Generalized Golden Rule and its egalitarian variants in the context of an endogenous demography. This task is also left on our research agenda.

  we have: k u > k e and n u < n e ; if f 0 (k) + 1 = 2k! 0 (k), we have:k u = k e and n u = n e ; if f 0 (k) + 1 < 2k! 0 (k), we have: k u < k e and n u > n e .Assuming a Cobb-Douglas technology y t = Ak t , we have, under a full depreciation of capital, that:if > 1=2, we have: k u > k e and n u < n e ; if = 1=2, we have: k u = k e and n u = n e ; if < 1=2, we have: k u < k e and n u > n e .

On the distinction between equality of opportunity ex ante and equality of opportunity ex post, see[START_REF] Fleurbaey | Ex ante versus ex post equality of opportunity[END_REF].

Note that, in general, the worst-o¤ individuals are those who turn out to have a shorter life than others. But, as we discuss below, this is not necessarily the case.

See the survey by[START_REF] Boucekkine | Growth economics of epidemics: a review of the theory[END_REF].

This is also the case in the real world, where longevity inequalities are largely due to factors on which individuals have no in ‡uence, such as genetic background (see[START_REF] Christensen | The quest for genetic determinants of human longevity: challenges and insights[END_REF]. Note also that, even though empirical studies, such as[START_REF] Kaplan | Mortality among the elderly in the Alameda county study: behavioral and demographic risk factors[END_REF],[START_REF] Contoyannis | Socio-economic status, health and lifestyle[END_REF] and[START_REF] Balia | Mortality, lifestyles and socio-economic status[END_REF], highlight the impact of some behaviors on the average (or expected ) longevity, those studies identify only risk factors, but not factors a¤ecting, for sure, the longevity realized at the individual level.

We also assume lim c!0 u 0 (c) = +1 (see de la Croix and Michel 2002 p. 5). 1 0 For the simplicity of presentation, we abstract here from pure time preferences. 1 1 See[START_REF] Becker | The quantity and the quality of life and the evolution of world inequality[END_REF].

6 See[START_REF] Fleurbaey | Compensating the dead? Yes we can[END_REF] on the ethical foundations behind the maximization of the ex post welfare of the worst-o¤ individuals under risky longevity.

8 See supra. 1 9 Indeed, if c is very low, there is little welfare gain from fertility, and so the social optimum involves a fertility rate less than the replacement level.

0 Here we use the superscripts u and e to refer to, respectively, the utilitarian and egalitarian optima.

1 Those cases are also studied in de la Croix and Ponthiere (2010), who focus on the utilitarian optimum.

3 That negative e¤ect of capital accumulation on consumption possibilities depends on the shape of the survival function, through the derivative 0 (h(k)).

4 For simplicity of presentation, we focus here on case A1, but similar conditions can be derived for cases A2 and A3.

A3 Health spending per worker are a constant fraction of the output per worker: h t = f (k t ).

The case A1 coincides with standard OLG models with …xed life expectancy 1 + ( h). 22 The case A2 includes the economy considered by [START_REF] Chakraborty | Endogenous lifetime and economic growth[END_REF]. The case A3 makes health spending depend on the whole social income.

Under case A1, the egalitarian social planner's problem can be rewritten as:

where is the Lagrange multiplier associated to the resource constraint, whereas is the Lagrange multiplier associated with the non-negativity constraint for old-age welfare u(d). The …rst-order conditions are:

as well as the conditions

with complementary slackness. From ( 27), it appears that, as in the …rst-best problem, the optimal capital accumulation remains determined by the standard Golden Rule. Old-age consumption remains equal to c. Note also that, on the basis of (28), we obtain that, because of the higher preventive spending, there is now a higher intergenerational redistribution e¤ect, which leads to a higher fertility ceteris paribus.

Turning now to cases A2 and A3, the social planner's problem becomes:

where denotes the Lagrange multiplier associated to the resource constraint, whereas is the Lagrange multiplier associated with the non-negativity constraint u(d) 0. The …rst-order conditions are:

The standard Diamond model relies on = 1.

as well as the conditions

with complementary slackness. As above, second-period consumption equals the welfare-neutral level c. Substituting for d = c, one can rewrite (33) as:

That equality characterizes the optimal second-best capital level. Note that, under h 0 (k) = 0, we are back to the standard Golden Rule. However, once capital accumulation raises health investment, the optimal capital becomes, ceteris paribus, lower than under the standard Golden Rule. That comparison holds only ceteris paribus, that is, for an equal fertility level n. One can, nonetheless, expect the optimal fertility to di¤er in the …rst-best and in the second-best, simply on the grounds that a higher health investment h raises the intergenerational redistribution e¤ect, inviting a higher optimal fertility for an equal level of capital k. The following proposition summarizes our results.

Proposition 4 The egalitarian second-best optimum fc ; d ; k ; n g is such that:

Regarding optimal capital, we have:

Proof. See the above FOCs.

If fertility was …xed to some exogenous level n = n, the optimal capital at the second-best would be, under case A1, the same as it is at the …rstbest. Moreover, under cases A2 and A3, the optimal second-best capital would be strictly inferior to its …rst-best level. The reason is that a higher capital tends here to foster health investment, and, hence, raises the inactive to active population ratio n , which reduces consumption possibilities ceteris paribus. 23 Hence, it follows that the second-best egalitarian optimal capital is, under case A1, equal to the one under utilitarianism, and, under cases A2 and A3, it is lower than the utilitarian optimal capital. When fertility is a part of the planner's choice, it is more complicated to compare the …rst-best and second-best egalitarian optima. To see this, let us consider the case A1. Comparing the condition for …rst-best (F B) fertility and second-best (SB) fertility, we have:

it appears that, for a given k, the larger survival probability at the second-best raises the Samuelson e¤ect (RHS), and, as such, the optimal fertility is, at the second-best, larger than at the …rst-best for a given k. That result is important, but, here again, the optimal second-best capital is a function of the optimal fertility, so that a ceteris paribus rationale is not satisfactory. Proposition 5 summarizes our results. 24 Proposition 5 Consider case A1. We have:

Assuming Cobb-Douglas production y t = Ak t and a full depreciation of capital, we obtain:

if > 1=2, we have:

Proof. See the Appendix.

Hence, if we take the realistic case where < 1=2, the optimal second-best capital under egalitarianism is lower than at the …rst-best, and the optimal second-best fertility is larger than at the …rst-best. The intuition is that the Samuelson e¤ect is reinforced by the larger proportion of survivors associated with h > 0, which pushes, at the end of the day, towards a higher optimal fertility and a lower optimal capital than in the …rst-best. Hence, the extent to which egalitarianism leads to a higher optimal capital than utilitarianism is lower at the second-best than at the …rst-best. Note, however, that, if < 1=2 and h < h u , it is still the case, even at the second-best, that egalitarianism recommends a lower fertility and a higher capital than utilitarianism. The addition of the constraint h > 0 reduces the gap, but it still exists.

In sum, the way in which the introduction of exogenous health investment a¤ects the optimal accumulation depends on whether fertility is part of the planner's choice or not. This dependency on fertility is well illustrated in the case A1 (…xed health investment). When fertility is exogenous, the second-best capital is equal to its …rst-best level. Once fertility is a choice variable, the second-best capital is generally inferior to its …rst-best level, because of a larger Samuelson e¤ect, which pushes towards a larger fertility and a lower capital.

Appendix

Proof of Proposition 3

Under utilitarianism, the optimal capital and fertility levels satisfy respectively:

On the contrary, at the egalitarian optimum, these satisfy respectively:

Substituting for the Golden Rule in the optimal fertility condition, we have:

2 is increasing (resp. decreasing) in k, we have k u > k e (resp. k u < k e ). We have:

It is positive if and only if:

Taking the example of the Cobb-Douglas production y t = Ak t , we obtain:

Hence, under = 1, k u > k e holds if and only if > 1=2. The rest of Proposition 3 follows from similar rationales

Proof of Proposition 5

Under A1, the optimal capital under egalitarianism is still given by the standard Golden Rule, but the di¤erence is that the optimal fertility rate n may di¤er signi…cantly in the …rst-best and the second-best. To see this, note that, if one substitutes for the Golden Rule in the FOC for optimal fertility, one obtains:

Hence, if the LHS is increasing in k sb , a higher RHS in the second-best in comparison to the …rst-best (i.e. c ( h) > c~ ) must imply a higher optimal capital level. Therefore, if the condition:

is true, then the second-best optimal capital is larger than the …rst-best optimal capital, and the second-best fertility is lower than the …rst-best fertility. Under a Cobb-Douglas technology and a full depreciation of capital, k sb exceeds k f b if and only if:

The rest of Proposition 5 follows from grouping that condition with the second part of Proposition 3.